Mathematics For Junior High School Volume 2 Part I

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MATHEMATICS* FOR JULI3R HIGH SCL IGVL

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PART I

SCHOOL ICMTliEMATbCS STUDY T-dlm

YALE UNlVERSTTY PR€SS

School Mathematics Study Group

Mathematics for Junior High School, Volume 2

Unit 5

Mathematics for Junior High School, Volume 2

Student's Text, Part I

Przp~red under the supervision of the

Patlcl on Scventh and Eighrh Grades

ot'the School Mathcniatics Study Group:

K. D. Anderson Louisiana Srate University

j. A. Rrown Universiry of Delaware

Lenorc John University ol' Chicago

B. W. Jones Univcrsit): af Colorado

P. S. Jones University at Michigan

J. R. Mayor Anlerican Association for the

Advancemcn t of Scietlce

P. C. Roserlbloom Univerdry of Minnesota

Vervl Schult Supervisor of Mathematics,

Washington, D.C.

Ncw Haven and London, Yalc U~liversity Press

Copyright O 1960, 1961 by Yde University. Printed in the United States of America.

AD rights reserved. This book may not be reproduced, in whole or in pan, in any form, without written permission from the publishers.

Financial support for the School Mathematics Study Group has been provided by the Nadonal Science Foundation.

I FOREWORD

I The increasing con t r ibu t ion of mathematfcs to the culture of the modern world, as well as its importance as a vital p a r t of ac ient i f ic and hurnanietlc education, has made it easent ial t h a t the mathematics i n our schools be both well selected and w e l l taught.

With t h l s in mind, the va r ious mathematical organizations in the United States cooperated in the formation of the School

I Mathematics Study Group (sMsG). SMSG includes college and university rnathematiciana, teachers of mathematics at all l eve l s , experts in education, and representatives of science and technology. The general objective of SMSG is the Improvement of the teaching of mathematics In the school8 of thls country. The National Science Foundation has provided substantial funds f o r the support of th is endeavor.

One o f the prerequisite8 for the improvement o f the teaching of mathematics in our schools is an Improved curriculum--one which takes account of the Increasing use of mathematics in science and technology and In other areas of knowledge and at the same t i m e one which reflects recent advances in mathematics i t se l f . One of the first projects undertaken by SMSG was to enlist a group of outstanding mathematicians and mathematics teachers t o prepare a series of textbooka which would illustrate such an improved curriculum.

I The professional mathematlclans in SMSG believe that the I mathematics presented i n t h i s t e x t is valuable f o r all well-educated ; c i t l z e n s In our society t o know and that it I s Important f o r the i precollege student t o learn i n preparation for advanced work In the field. A t the aame time, teachers in SMSG b e l i e v e that it is preaented in such a form that it can be readily grasped by students.

In mast instances the material w l l l have a familiar note, but the presentation and the point of view will be different. Some material wlll be entirely new to the traditional c u ~ r i c u l u m . This is as it should be, for mathematics is a living and an ever-growing subject, and not a dead and frozen product of antiquity. This healthy fusion of the old and the new ahould lead students to a better understanding of the basic concepts and structure of mathematics and provide a firmer foundation for understanding and use of mathematics in a acientiflc-society.

It is not intended that this book be regarded as the only d e f i n f t l v e way of presenting good mathematics t o students at this level . Instead, it should be thought of as a sample of the kind of improved curriculum that we need and as a source of suggestions f o r t h e authors of commercial textbooka. It is sincerely hoped that these t e x t s w i l l lead the way toward i n s p i r i n g a more meaningful teaching of Mathematics, the Queen and Servant of the Sciences.

The preliminary ed i t i on of this volume wac prepared at a writing session neld at the

University of Michigan during the Bummer of 1959. Reviaions were prepared at Stanford University ln the summer of 1950, taklng into account the c l a s s m ~ m experience with the preliminary edition during the academlc year 1959-60. T h i s edition wax prepared at Yale

University In the summer of 1961, a ~ a l n taking into account t h e classroom experience ulth

the StanFord edi t ion during the academic year 1960-61.

The following is a 1-Lst of a l l those who have participate6 in the preparation of thls volume.

R.D. Anderson, Lauislana State Univeralty

B.H. Arnoid, Oregon State College

J.A. Browrl, University nf Delaware

Kenneth E. Brown, U.S. Office o f Education

Mildred B . Colt, K.D. Waldo Junior High. School, Aurora, Illinois

B . H . CoLvin, Boeing Scientific Research Laboratories

J.A. Cooley, University of Tennessee

Richard Dean, California Institute of Technnlog~

H.M. Gehman, Univeralty of Buffalo

L. Roland Genise, Brentwood Jur.10~ H i g h School, bentwood, New York

E. Glenadlne Cibb, Iowa State Teaehepa College

Richard Good, University of Maryland

Alice Hach, A n n ArWr Public Schools, Ann Arbor, Michigan

S.B. Jackson, University of Maryland

Lenore John, Univeralty High School, University of Chicago

B.U. Jones, University of Colorado

P.S. Jonea, Univereity o f Michigan

Houaton Karnes. Louisiana State University

Mildred KeifreP, Cincinnati Public Schools, Cincinnati, Ohio

Nlck Lovdjleff, Anthony Junior High School, Minneapolis, Minnesota

J .R. Mayor, American Aaaociation for the Advancement of Science

Sheldon Meyers, Educational Testing Service

Muriel Milla, Hill Junior High School, Dsnver, Colorado

P .C . Rosenbloom, University of Minnesota ~ l i z a b e t h Roudebueh M i t c h e l l , Seattle Public Schools, Seattle, Wasi?lngton

V e r y 1 Schult, Washington Public Schools, Washington, D . C .

George Schaefer, Alexis I. DuPont High School, Wilmington, Delaware

Allen Shields, U n i v e ~ e i t y of Michigan

Rothwell Stephena, Knox College

John Wagner, School Mathematics Study Group, New Haven, Connecticut

Ray Welch, Westport Public h h o o l s , Weatport, Connecticut

O.C. Uebbep, University of Delaware

A .B. Uillcox, Amherst College

FOREWORD . . . . . . . . . . . . . . . . . . . . . . . . . . v

. . . . . . . . . . . . . . . . . . . . . . . . . . PREFACE ix

Chapter

NUMBERS AND COORDINATES . . . . . . . . . . . 1 . . . . . . . . . . . . . . . TheNumberLine 1 Negative Rational Numbers . . . . . . . . . . 7 . . . . . . . . Addition of Rational Numbers 12 . . . . . . . . . . . . . . . . . Coordinates 21 . . . . . . . . . . . . . . . . . . . Graphs. 30 . . . . . Multiplicat~on of Rational Numbers 34 . . . . . . . . Division of Rational Numbers 42 Subtraction of Ratlona 1 Numbera . . . . . . . 46

. . . . . . . . . . . . . . . . . . . . . . 2 . EQUATIONS 51 . . . . . . . . . . . 2- 1 . Writing Number Phrases 51 . . . . . . . . . . 2- 2 . Writing Number Senfences 58 . . . . . . . . . . . . 2- 3 . Findlng Solution Sets 75 . . . . . . . . . . . . 2- 4 . Solving Inequalities 91 2- 5 Number Sentences with Two Unhowns . . . . . 93

3 . SCLENTIFIC NOTATION. DECIMALS. AND THE METRIC SYSTEM 3- 1 . Large Numbers and S c l e n t i f l c Notation . . . . . . . . . 3- 2 . Calculating wlth Large Numbers 3 - 3 . Calculating with Small Numbers . . . . . . 3- 4 . Multiplication: m r g e and Small Numbers . . . . . 3- 5 . Division: Large and Small Numbera 3- 6 . Use of mponents in Multiplying and . . . . . . . . . . . . . Div id ing Decimala 3- 7 . The Metric System; Metric Units of Length . . . . . . . . . . . . . 3- 8 Metric Units of Area 3 - 9 . Metric Units of Volume . . . . . . . . . . . . . . . 3.10 . Metric Units of Ma~s and Capacity

4 CONSTRUCTIONS. CONaRUeNT TRIANGLES. . . . . . . . . . . . . . AND THE PYTHAGOREAN PROPERTY 157 4- 1 . Introductfon t o Mathematical Drawings . . . . . . . . . . . . . . and Constructions 157 . . . . . . . . . . . . . 4- 2 . =sic Constructions 162 . . . . . . . . . . . . . . . . . . 4- 3 . Symmetry 172 . . . . . . . . . . . . . . 4- 4 Congment Triangles 176 . . . . 4- 5 . Showing Two Triangles To Be Congruent 184 . . . . . . . . . . . . . 4- 6 The Right Triangle 192 . . . . 4- 7 . One Proof of the Pythagorean Property 202 . . . . . . . . . . . . . . . 4.8. Quadrilaterals 206 4.9 . s o l i d s . . . . . . . . . . . . . . . . . . . 208

Chapter . . . . . . . . . . . . . . . . . . . . . 5 . ~ L A T I V E ERROR 215 . . . . . . . . . . . 5- 1 . Greatest Possible Error 215 . . . . . . 5- 2 . Precision and Significant Digits 219

5- 3 . Relative Error, Accuracy, and . . . . . . . . . . . . . . Percent of E r r o r 223 . . . . . . . 5- 4 . Adding and Subt rac t ing Measures 227 . . . . . . 5- 5 . Multiplying and Dividing Measures 229

. . . . . . . . . . . . . . . . . . . . . 6 . REAL NUMBERS 235 . . . . . . . . . . 6 - 1 Revlew of Rational Numbers 235 . . . . . . . . . 6- 2 . Density o f Rational Numbers 240 6- 3 . Decimal Representations f o r the Rational

Numbers . . . . . . . . . . . . . . . . . . . 245 6- 4 The Ratlonal Number Corresponding to a . . . . . . . . . . . . . . Periodic Decimal 250 . . . . . 6 - 5 . Rational Points on the Number Line 255 . . . . . . . . . . . . . 6 - 6 Irrational Numbers 257 . . . . . . . 6 - 7 A Decimal Representation for 8 262 6 - 8 . f r r a t f ona l Numbers and the Real Number . . . . . . . . . . . . . . . . . . . system 267 6- 9 . Geometric Properties of the Real Number Lfne . 276

fol lowing

PREFACE

Key ldeas of junior high school mathematics emphasized in this text are: s t ruc tu re of arithmetic from an algebraic view- point ; t he real number system as a progressing development; metric and non-metric re la t ions In geometry. Throughout t h e materials these ideas are associated w l t h their appl ica t ions , Important a t t h i s level are experience with and appreciation of abstract concepts, the role of definition, development of precise vocabulary and thought, experimentation, and proof. Substantial progress can be made on these concepts in the junior h igh school.

Fourteen experimental u n i t s fo r use in the seventh and eighth grades were written in t h e s m e r of 1958 and tried out by approximately 100 teachers in 12 centers in va r ious parts of the country in the school year 1958-59. On the basis of teacher evaluations these unlts were revlsed during the summer of 1959 and, with a number of new units, were made a part of sample textbooks for grade 7 and a book of experimental units f o r grade 8. In the school year 1959-60, these seventh and eighth grade books were used by about 175 teachers in many parts of the country, and then further revised in t h e s m e r of 1960. Again during the year 1960-61, this text f o r grade 8 w a s used by nearly 200 classes in a l l parts of the country, and then this edition was prepared I n t h e summer of 1961.

Mathematics is fascinating to many persons because of its oppor tuni t ies f o r c r e a t i o n and discovery as well as for i t s utility. It i s continuously and r ap ld ly growing under the proddlng of both Intellectual curiosity and practical applica- t ions . Even junior high achool students may formulate mathematical ques t ions and conjectures which they can test and perhapa settle; they can develop systematic attacks on mathematical problems whether or not the problems have routine or immediately determinable solu t ions . Recognition of these important factors has played a considerable part in selection of content and method In this text.

We firmly be l ieve mathematics can and should be studied with success and enJoyment. It is o u r hope that this text may greatly assist a l l teachers who use it to achieve this highly desirable goal .

Chapter 1

RATIONAL NUMBERS AND COORDINATES

1-1. The Number Line 7

The idea of number is an abstract one. The development o f a good number system required centur ies in-the civilization of man,

To help understand numbers and their uses, many schemes have been

used. One of the most auccessful of these ways to pic tu re numbers is t h e use of the number l i n e . Suppose we recall the cona tmc t ion of a number l i n e as a starting point f o r further discussion of numbers .

We think of the llne in t he drawing below as extending end-

lessly i n each d l r e c t l o n . We choose any point of the line and label it 0. Next, choose another po ln t t o the right of 0 and l a b e l it 1. This determines a unit of length from 0 to 1.

S t a r t i n g at 0, we lay of f this unit length repeatedly toward the

r i g h t on the nwnber l l ne . This determines t h e loca t ion of the

points corresponding to the counting numbers 2, 3 , 4, 5 , . . . . 1 The number we associate w i t h the point midway between 0

and 1. By laying o f f this segment of length one-half un i t over and over again, we determine the additional points corresponding

1 3 5 to , , -2, . . . Next, by using a length which is one-third of

t he unit segment and measuring t h i s length successively to the 1 2 4

r i g h t of zero, we locate the po ln t s 5, 7, 7, 3, . . .* Similarly, we l oca te the points to the right of 0 on the number l i ne cor responding to f r a c t i o n s having denominators 4, 5 , 6 , 7, . * - . Some of these are shown in the following f igure .

By this natural process we associate with each rational nun- ber a point on the line. Just one p o i n t of the line i a associated

with each rational number. We thus have a one-to-one correspond-

ence between these ra t ional numbers and some of the points of the l i n e . We speak of the point on the number line corresponding t o

the number 2 as the p o i n t 2. Because of t h i s one-to-one correspondence between number and poin t , we: can name each po in t by the number which labels it, This is one of the great advantages of the number l i n e , It al lows us to i d e n t l f y po ints and numbers and helps us use geometric points to picture r e l a t i o n s armrng numbers. We a h a l l illustrate some of these uses in the next f e w paragraphs.

Remark. You might think that t h i s one-to-one correspondence assigns a number to every point on t h e line, to the right of 0. This is far f rom true. In fact, there are many, many more poi'nts unlabeled than labeled by t h i s process. These unlabeled points correspond t o numbers like a, @, a, f i , which are not ra t iona l numbers. In Chapter 6 . we shal l study more about such numbers.

Properties of the -- Number Line - The number line locate6 numbers by means of points in a very

na tu ra l way, The construction of the number line locates t h e

r a t i o n a l numbers in order of increasing size. Hence we can always tell where a number belongs on the l i n e . The larger of two nunbers always lies to t he right. Thus: 5 > 3 (5 is greater than 3), and on the number line 5 l ies to the right of 3. A number greater than 3 corresponds to a point located to the right of 3.

Since 2 < 4, the point 2 lies t o the l e f t of the point 4. We

can easily check the relative positions of numbers such as 3, 0, 5 4 3 , 1, 3, 2. Once we have located the corresponding points on the line, it is easy to t e l l at a glance whether one number is greater than another or less than another.

The point corresponding t o 0 1s chosen as a po in t of reference and ca l l ed t h e origin. (some people ca l l it the f i d u c i a l polnt, and you may call it t h i s t o impress your f r i ends , If you wish!) The h a l f - l i n e extending to the right from the o r i g i n along the number l i n e Is called t h e p o s i t i v e h a l f - l i n e , Any number which is greater than zero l i e s on t h i s p o s i t i v e h a l f - l i n e and is c a l l e d a p o s i t i v e number. In particular, we speak of the counting nun- - bers 1, 2, 3, 4 as t h e p o s i t i v e Integers. Note that to say a number is p o s i t i v e simply means that it l i e s to the right of zero on t he number Ilne,

Addition on the Number Line -- - Addition of two numbers can eas i l y be pictured on the number

line. To add 2 to 3, we start at 3 and move 2 units to the

right. In t h l e way the opera t ion 3 + 2 = 5 is represented by a m o t l o n along the number line, The motion ends at the po in t corresponding to the sum,

We may also think of the number 3 as determining an arrow [or directed line s e p e n t ) starting at 0 and ending at 3. To represent the addition of 2 to 3, we simply draw an arrow of length 2 to begin at 3 instead of at 0, The arrow (directed l i n e sewent) representing 3 + 2 thus begins at 0 and enda at 5. To avoid confusion, we frequently Indica te these arrows s l igh t ly above the number line, as in t h e following figure f u r the sum 4 + 2.

[sec . 1-11

The arrows suggest a move of 4 units followed by a move of 2

u n i t s in reaching the point (4 + 2). We a l s o interpret the pic-

ture as suggesting the addition of two directed line sepents of lengths 4 and 2 to form t h e sum segment of l eng th 6.

Applications -- of the Number Line - The number l i n e is used in

many f o m s in our everyday life.

A ruler is a fine example, of course. The house numbers along

some city streets define a rcugb

form of n m b e r line, (1f you llve

on a curved street, you w i l l have t o take more mathematics before you study distance and numbering along a curve !)

One of the most common applica-

tions of the number line appears in a thermometer, Here we use two

number scales, one in each d i r e c t i o n along the l i n e , Temperatures above zero appear above zero on the line; temperatures below zero are measured below zero on the line.

+ 100 Degrees Above

10 Degrees Below

20 Zero

Temper at ure

To compare profits and losses of various divisions of a large company for a given month, we

mZght use scales as shown at the right. D i v i s i o n A, which lost 4000 dollars would appear at the point labeled A. D i v i s i o n B, which showed a p r o f i t of 5000

d o l l a r s , appears at B.

4 000 P r o f it8

in Dollars

1000

Losses 2000 in

Dollars

3000

What other examples of the use of number l l n e can you think of?

Exercises 1-1 - 1, For each o f the following numbers draw a number llne. Use one

Inch as the uni t of length. Locate a po in t of o r i g i n on the l l n e , and then locate the point corresponding to the number. Just above the number l i n e , draw the corresponding arrow.

(a) 4 (c , , % ( e ) 3.75

1 (b) 7 (4 4 (f) 1.125

2. Represent each of the following additions by means of arrows on a number l i n e , Use a separate number llne f o r each addi t Ion.

(a] 2 + 3

(b ) 1 + 6 3 ( a ) r + 1.25

3. Locate the fo l lawlng numbers on a number line and determine which is the largest in each set .

4, On some recent automobiles, speedometers use a form of the hmber l i n e to indicate speed in miles per hour. On these speedometers, a l i n e , similar to an arrow, is used to indfcate the speed. Choose a convenient unlt of l ength and construct

such a number l i n e showing speeds up to 70 m.p.h . L a b e l on It

the p o i n t s correspondlng to the following po in t s . Draw a corresponding arrow fo r each number Juat above the number line.

(c) 35 m.p.h. (d) 60 m.p.h.

5. Locate on a number l i n e the midpoints of the fol lowing segments,

(a) From 0 to 2, I 7 ( c ) From 2 to 5 .

1 5 (b) From 8 t o 8. (d) ~ ~ l m 2 to 6.5,

6. Use a diagram representing addit lon by means of m w s on the

[sec. 1-11

number line to show t h a t 2 + 3 = 3 + 2. What proper ty of

addif ion does t h i s illustrate?

7. Using arrows to represent add i t i on on the number l i n e , show

t h a t

What property of addition does t h i s illustrate?

8, Think of a way to repyesent the product 3 - 2 by means of 1 arrows on t h e number l i n e . Try it also for 5 - 2 and 6 .

9 . How would you show t h a t 2 3 = 3 2 by means of arrows on the

number line? What property of multiplication does t h l s illustrate?

1-2. Me~at ive Rat iona l Numbers

In the preceding discussion of number 1Lne there is a very seri- ous omission. We did not label the p o i n t s to the l e f t of zero. We used only the hal f - l lne from the o r i g l n In the p o s i t i v e direc- t i o n . To suggest how to label these p o i n t s (and why we want to!), let us look at the familiar example of temperature,

A number line representing temperature, such as we find on a thermometer, of t en looks l i k e t h l s .

Temperature i n Degrees Fahrenheit

Here temperatures less than zero are represented by numbers to t h e

left of the origin and designated by the symbol " ' " . Tempera- tures greater t h a n zero are Identified with the sign '' + ". Thus, - 10 refers t o a temperature of 10 deg~ees below zero (to t he l e f t

of zero), and '10 refers to a temperature of 10 degrees above

zero (to the r i g h t of zero), Actually, above zero and b e l o w zero seem more natural terms to use when the scale is ve r t i c a l .

T h i s idea of distance (or of p o i n t s ) along a line on opposite a i d e s of a f ixed po in t occurs frequently In our ordinary tasks. Thlnk how o f t e n we speak of distance8 t o the l e f t or to the r i g h t ,

l o c a t i o n s n o r t h or south of a given p o i n t , altitudes above or be low sea l eve l , longitudes east or west, or the time before or after a certain event . In each of these situations, there is a

suggestion of points loca ted on opposite s i d e s of a glven p o i n t

(or number), or distances measured In opposite directions f r o m a given po in t (or number). A l l of them suggest the need for a num- b e r l i n e which uses p o i n t s to the l e f t of the or ig in as w e l l as pointa to the right of the origin,

The n a t u r a l way to describe such a number line I s easy to see.

We start with the number line for p o s i t i v e rationals which we have already used. Using the same u n i t l engths , we measure o f f dls-

tances to the left of zero as shown below:

We locate -1 as opposite to '1 in the sense that It is 1 unit to the l e f t af zero. Similarly -2 is opposite to '2, I s located opposite to +I ( ) is opposite t a +$, e t c . These '9' tiopposite' numbers, corresponding to p o i n t s t o the left of zero, we call negative numbers. Each negative number lies to the l e f t of zero and corresponds to the opposite pos i t ive number. Thls

direction "to the left" is c a l l e d the negative direction. - - - 1 We denote negative numbers as 1, 2, (4, -(;), ~ ( 8 ) e t o . ,

by use of the raised hyphen. We read ( - 2 ) as "negative two ." Thls negative symbol " - " tells us t h a t the number i a less t h an

zero (lles to the l e f t of zero). We sometimee emphasize tha t a number is positive (greater than zero) by writing the symbol ' "

4" i n a raised position as in 2 , '6. e t c . Usually we do not this unless we want t o emphasize the p o s i t i v e character of a num-

be^. The new numbers we have introduced by t h i s process are the

negative rational numbers. The s e t cons i s t ing of positive rational numbers, negative r a t i o n a l numbers, and zero, we c a l l the rational numbers.

The special set of r a t i o n a l numbers whlch c o n s i s t a of the p o s i t i v e Integers, the negative Integers and zero is c a l l e d the s e t of Integers, We frequently denote t h i s set as: --

I = [ a * . - - -

- 4 , 3, 2, 1, 0,1,2, 3 , 4 , , - 9 ] .

Note t h a t the set of integers consists of only the count ing num-

bers and t h e i r o p ~ o s i t e s together with zero.

Examples of the Use of Neaative Numbers ---- The negative numbers are as real and aa useful as the posl-

tLve numbers we have used before. In fact we have used them many times without callfng them negative numbers. The i r special use- fulness is in d e n o t i n g the idea of "oppositet1 or ''oppositely

directed" which we mentioned. L e t us use pos i t ive numbers to denote distance east of

Chicago. The negative numbers wlll denote distances west o f Chicago. A number l i n e like the one below

Dlstance f r o m Chicago in mlles

can therefore be used to plot the position of an airliner flying an east -west course pasaing over Chicago. F o r an a i r l i n e r f l ~ i n g

a north-south course over Chfcagc, how could you i n t e rp re t this number l i n e ?

[sec . 1-21

The t i m e before and af te r the launching of a satellite can be indicated on a number line l i k e the following:

(seconds before launching) ( ~ e c o n d e after launching)

- +wo - ' 400

- '300

- "200

- +TOO - 0 Distance

Note tha t the number line we use need not be placed horizon-

t a l l y . If we speak of altitude above sea level as positive and a l t i t u d e below sea level &a nega-

Kansas C i t y

tive, It may seem more natura l to use a number l i n e in the v e r t i c a l

position.

Also, for distances nor th or sou th from Kansas City you may wish to use a vertical l ine .

t 0 Al t 1 tude

- 5000

- 4000 - 3000

- 2000

in feet -1000 w l th

1 -moo reference

To represent buslnesa prof i ts and loases, a vert ical line is more convenient. A Rlgher pos i t ion I n the l i n e seems naturally to correspond to greater prof l t . Of course, in other instances some other orlentatf on (not necessarily hor i zon ta l or vertical) may seem natural. for the number line.

1. Lacate on the number l i n e points corresponding to the followfng numbers.

Are there any pairs of "opposites" in t h i s l i s t?

2 , Sketch the arrows determined by the fa l lowing rat ional numbers.

3. Arrange the following numbers i n the order in which they - - appear on the number line: 4, i, -[$), $? -6 , ($1 , Which is the largest? Which 1s the smallest?

4 . How could you represent the folLowing quantities by means of positive and negative numbers?

(a) A p r o f l t of $X)OO; a loss of $600.

(b) An alt i tude of 100 ft. above sea level; an altitude of 50 ft, below sea level.

(c) A loss o f 15 yards, a gain of 10 yards.

(d) A distance of 2 miles east, a distance of 4 miles West.

5 . The elevator control board of a department s to re l i s t s the f l o o r s as B 2, B 1, G, 1, 2, 3 . Here G refers t o ground l e v e l and B 1, B 2 denote basement l e v e l s . How could you

use p o s i t i v e and negative numbers to label these f l o o r s ?

6 . Draw a number l i n e indicating altitudes from -1,000 f t . to '10,000 ft. U s e intervals of 1,000 ft. Locate alt i tudes of -800 ft., '100 ft,, +2500 ft., -500 ft.

1-3. Addition of Rational Numbers

We saw that the addition of t w o p o s i t i v e numbers Is easily represented on the number l i n e . To refresh your memory, try find- i ng the sums 2 -+ 4, 3 + 2, 1 + 7 on t h e number l i n e . On the

number l i n e , the sum 4 + 2 ia represented by the p o l n t 2 unita

beyond 4 (or 4 units beyond 2 s ince it makes no difference which number is chosen first). Note that in adding a pos i t ive number to another p o s i t i v e number, we always move t o the right (in 'the p o s i t i v e direction) a long the number l i n e . So we describe t h i s process of addition by saying t h a t in adding 2 to 4, we

start at 4 and move 2 units to the right, or 2 unlts I n the positive direct ion . We saw tha t a convenient way to represent t h i s process is by means of armws (directed l i n e segments) of appropriate length, Thus, toe sum 4 + 2 corresponds to thla p ic tu re .

To thFnk of moving f r o m 0 to 4, we draw the Cirec ted l l n e

segment corresponding t o 4. Then beginning a t 4, we draw the directed line segment of length 2 whlch corresponds t o 2. In this way we find the arrow o r length 6 corresponding t o ( 4 + 2).

Think now of our construction f o r t h e negative numbers i n t h e number line. Remember tha t -2 l a the opposite of 2. To say

- *

that 2 fs the opposite of 2 means that 2 is the same dis-

tance from 0 aa 2 but In the negative d i r e c t i o n as shown here:

- The arrow associated with 2 is 2 u n i t s in length and specifies the negative direct ion as indicated in t h e sketch. How would you - - 1 sketch 4; -3; '(5)? Remark. Many times it is valuable to Indicate the approximate

positi~n of numbers on the number line in order to compare

their loca t ions re la t ive t o one another. In such cases only a rough p ic tu re l a necessary, and careful measurements of length are not justified, We refer t o such a

rough picture as a aketch.

What would we mean by the sum 5 + ( 1 3 ) ? Uslng directed

arrows, we can find the polnt corresponding t o 5 + ( - 3 ) by start- irg a t 0, moving 5 unlts in the positive direction snd then 3 unlta in the negative (opposite) d i rec t ion . Thus, 5 + (-3) = 2,

as shown in the following sketch:

- Here, 3 is associated with an arrow of length 3 units directed

in the negative d i r ec t ion . In adding ( -3) to 5, we slmply draw - the arrow for 3 as o r i g i n a t i n g at 5 (that is, beginning at the end of the arrow corresponding to 5 ) . -

To add 3 and 4, draw a sketch like the following:

Thus, 3 + (-4) = -1, Find the sum 2 + (-5) in the same way, Consider the sum 2 ) + ( 4 . Here the arrows are both in the

negative direct ion . We see f r o m a sketch that (-2) + (-4) = ( - 6 ) .

In the same way, find the sums: (-3) + (-2); (-1) + (-6); ('6) + ('2).

One property of special interest is illustrated by the sum ( - 2 ) + (+2).

Here -2 corresponds to an arrow of length 2 units i n the nega- t i v e direction. Adding (-2) and (+2) corresponds t o moving 2 u n i t s to the l e f t f r o m 0 and then 2 units back to 0. Thus

(-2) + (2) = 0. Check tha t (-1) + 1 = 0, 3 + (-3) = 0,

('8) + 8 = 0,

We see by the use of the number l i n e t h a t the addition o f numbers, whether pos i t ive or negative, is real ly very simple. We

need only keep in mind the l o c a t i o n of the numbers on the number

l i n e to carry out t he o p e r a t i o n . We see that:

When both numbera are p o s i t i v e the sum ia positive,

as i n t 5 + '3 = "8;

and, when both numbers are negatlve the sum I s negative,

as in ( - 5 ) + ( - 3 ) = -8.

When one number is p o s i t l v e and one number is nega t l ve ,

it is the number farther from the o r i g l n which determines whether the sum is positive or negative.

F o r example:

In ( - 5 ) + +3 = -2,

the sum is negative because the -5, whlch is + farther from zero than 3, i s negatlve.

+ the sum is p o s i t i v e , because the 5 , which is - farther from zero than 3, is poaitlve.

Another way of saylng th i s l a , the arrow of greater l e n g t h

determines whether the sum is p o s i t i v e or negatlve. In fact , t h i a

one rule also works Fn the case when both numbers are positive or when both numbers are negative. Notice that In cases like

( - 2 ) + 2 and 3 + (-31, the arrows are of equal lengths but opposite in d i rec t ion . In these cases the sum wlll be zero.

Exercises 1-3a - 1. Find the following suns and sketch, using arrows on the number

line.

( a ) 9 + (-5) (4 5, + (-10)

(c ) (-8) + 11 (P) 3 + (-11)

2. Supply the m i s s i n g numbers in each of the following statements so that each statement w i l l be true.

(d) ( - 7 5 ) + 74 = ( 1 (h) (-0.45) + 0.45 = ( )

3. Obtain the sum in each of t h e following problems.

(a> 25 + (-6) (dl (-20) + (-10)

4. Supply a number in each blank space so t h a t each sum w f l l be

correct.

5. A company r e p o r t s income for the first s i x months of a year

a3 follows:

January $900 pmfi t A p r i l Q ~ O O O profit February $2000 p m f i t May $4000 l o s s

March $6000 loss June $3000 l o s s

(a) How could you represent these Income f l p r e s in terms of

positive and negative nwnbe~s?

(b) What is the t o t a l income for the slx-month per iod?

(c ) What is the t o t a l income for the first three months of t h e

year? (d) What i s the t o t a l income for the four-month period,

March, Apri l , May and June?

6. A boy rows upstream at a speed of 4 miles per hour against a current of 2 m i l e s p e r hour.

(a) How could you use positive and negative numbers in repre- aen t ing these s p e e d s ?

( b ) What would represent his actual speed upstream?

7. In four successive plays from scrimmage, startlng at its own 20 yard l i n e , Frankl in High makes

a gain of 17 yards, then

a loss o f 6 yards, nexb

a gain of 11 yards, and f i n a l l y a loss o f 3 yards,

(a) Represent the gains and losses in terms of positive and negative numbers.

(b) Where is the ball after the four th play?

( c ) What is the net gain after the four plays?

8. An airplane t ravel ing at 13,000 ft. makes a cl lmb of 5000 ft.

followed by a descent of 3000 ft.

(a) Represent the plane's final a l t i t u d e as a sum of positive and negative numbers.

(b) What Is the plane's altltude after the descent?

9 . ( a ) Think of a way to represent the product 3(-2) by means of arrows on t h e number l i n e . Try it also for:

Ib) 5 (-1).

(4 2 . -(&

Inverse Elements under - Addit ion

Recall that '2 + ( -2 ) = 0. This sentence says that 2 is

the number which when added to 2 yields 0. We saw in Volume I t h a t 0 is t h e ident i ty element under the operation of addition. Any two numbers with sum 0 are s a i d to be inverse elements under - addition. Hence 2 is the inverse element corresponding t o '2

under the operation of addit ion. We call -2 the additive

inverse of +2. Likewise '2 is the addi t ive inverse of - 2 . - Taken together, the elernenta "2 and 2 are cal led additlve inverses.

Class Exercises 1-3a - - 1. Find the additive inverse of each of the following numbers. - - - - 2

7, 9, 11, 12, -6, 15, 20, 0,

[aec. 1-31

2 Which of the following paira are additive Inverses? +

( a ) 20s +20 [ c ) -5, -($I (b) 5 -5 - I 2 (a) 7 s

On the number l l n e we see t h a t any number and i ts addlt lve

inverse w i l l be represented by arrows of the same length and

opposlte direction, as indicated by t he sketch of '2 and -2.

When added, these "oppositett arrows of equal length always + - give 0. To add 2 to 2, we t h i n k o f m o v i n g 2 units inthe

negative dlz'ection from 0 and then 2 unfts back in the p o s i t i v e

d i rec t ion . The two movements are precisely the opposite of each

o t h e r and bring us back to the starting point. The motions

descrlbed are true "inverses" of each other , f o r when one is added to the other, t h e final reault is 0 . The number l i n e thus pro-

vides a geometric picture for the meaning of addit ive inverses; the

a m of two oppositely directed arrows (motions) of equal length is

zero. In t h e addition of a positive and a negative number, we noted

that the longer arrow determines whether the sum is p o s i t i v e o r negative. The length o f the arrow for the sum can be obtained by the picture of additive Inverses. For example, in the aum

5 + ( -2 ) = 3 wemaywrite

by introducing the addi t ive inverse of the shorter arrow. Then, s ince 2 + (-2) = O we have

5 + (-2) = 3 + 2 + (-2) = 3 4-0 = 3 .

Note that t he other arrow of the two i n t o which 5 is separated

represents the sum 3 .

731s procedure Is a general one, as t he following examples illustrate.

In each case, the a d d i t i v e inverses add up t o zero, and the remain-

ing number is the sum.

Claas Exercises 1-3b - 1. Sketch the arrows corresponding to the numbers in t h e above

three examples. In each case, determine the arrow correspondlng to t h e sum.

2 Perform the following additions by introducing the additive

Inverse f o r the s h o r t e r arrow, and sketch the operat ion on the

number l i n e .

Exercises 1-3b - 1. Complete the following:

(a) 10 + -7 = ? + 7 + -7 (a) 23 + -28 = ? + ? + -18 - (b) 1 4 + 5 4 = - 1 4 + 1 4 + ? ( e ) -36 + 20 = ? + ? + 20 ( c ) 12 + -14 = 12 + ? + -2

2. Perform the fol>owing additions using the additive inverse.

-ample: 28 + -20 = 8 + 20 + -20 = 8 t 0 = 8.

(a) 42 + -12 (b) -12 + 8

- I c ) 7 + 35

Id) 6 + -17 ( e ) 344 + -140 (f) -172 + 96

3 . S t a t e whether the sum will be posltive or negative. Example : -17'2 + 37 4 negative . (a) 26 + -24

(b) 7 2 + -92

( c ) -376 + 374 (j) -0.132 + 0.0132 (a) -4,312 + 4,324 (k) -3.17'2 + 3.1724

[ e ) 1,436,312 + -1,436,310 (1) 0.0012 + -9

4. we know chat +3 > -17. 15 the following statement true or false? "The aum of a poeitive number and a negative number always has the sign of the greater number." &plain your answer.

5 . The sum of a positive number and a negative number is zero when

1-4. Coordinates

Coordinates on a Line - L - -

IRt us consider the number line from a di f fe ren t point of

view. As we have seen, a rational number can always be associated

with a point on the number line. The number associated In this way w i t h a point of the l ine is cal led a coordinate of the poin t . In

the following drawing, the number zero is associated w i t h the

reference point ca l led the or ig in .

Point A is denoted by the number (+3). Point B I s denoted by (-2). We write ~ ( ' 3 ) t o mean t h a t A is the point w i t h coordinate '3. Likewise B(-2) means that 8 is t h e point with coordinate (-2) on

t h e line.

Recall that every positive rat ional number is associated with

a poin t on the positive half-line. Every negative r a t i o n a l number

corresponds to a point on the negative h a l f - l i n e . The coordinate

we have assigned to a po in t i n t h i s way tells us two th ings . It t e l l s us t h e distance from the origin t o t h e p o i n t . It a l s o te l l s us t h e direction f rom the or ig in to the po in t .

Exercises 1-4a

1. Draw a segment of a number line 6 inches in length. Mark of f segments of length one i n ch and place the origln at its mid-

po in t . On t h e llne loca te t h e following points :

A(-1 ) . ~(31, ~(11, T(o). L(-(;)). P ( - 2 ) .

2. ( a ) In Problem 1, how far I s it in inches between the point

labeled T and the point labeled L?

(b) between P and B? ( c ) between L and B?

(d ) f r o m t he o r ig in to A?

3 . Using a number line with 1 inch a s the unit of length, mark the following points:

4 . If the l ine segment in Problem 3 were a highway and if it were drawn to a scale of 1 inch representing 1 mile, how far in miles would it be between these points on the highway:

( a ) F and R? ( b ) D and E?

[sec. 1 -41

5 . D r a w a number line in a v e r t i c a l instead of ho r i zon t a l pos i t i on .

Mark your number scale w i t h p o s i t i v e numbers above t he o r i g i n and negative numbers below the origin. Label points t o corre-

- - spend w i t h the r a t i o n a l numbers 0 1 2 3, 1 , 2, 3, -4.

Coordinates in the Plane --- Recall from your prev ious work in t h l s chapter tha t number

l ines can be drawn v e r t i c a l l y as well as horizontally.

You have learned that a single coordinate locates a poin t on t h e number line. A point l i k e S in the next figure is not on the number line and cannot be located by a single coordinate. How-

ever, we see that S is d i r e c t l y above the point ~ ( ' 3 ) . To locate 'S

point S, draw a v e r t i c a l number line perpendicular t o t he horizon- t a l number line and intersecting it at t h e or ig in . Your drawing

should look like t h i s :

The hor i zon ta l number line is called the X-axis and the vertical number line is c a l l e d the Y-axis. When we refer to both number llnes we call them the axes.

To determine the coordinates of point S , draw a line segment from po in t S perpendicular to the X-axis. It intersects the X-axls at (+ 3). Now draw a perpendicular f rorn polnt S to

the Y-axis . It Intersects the Y-axis at (+2) . Point S is said

t o have an x-coordinate of (+3) and a y-coordinate of ('2),

which we write as ('3, '2). We use parentheses and always write the x-coordinate before t h e y -coordinate.

In t he diagram below, observe how the coordinates of points .

A, B, C, and D were located.

Thus ~ ( x , y) represents the point P in terms of its coordinates. This may be done f o r any point P in t h e plane.

T h i s system of coordinates is called a rectangular system because

the axes are at r i g h t angles to each o t h e r and distances of points

from the axes are measured along perpendiculars from the points to

t h e axes. Each ordered p a i r of rational numbers 1s assigned to a

point in the coordinate plane. Locating and marking the point with

respec t t o t h e X-axis and t h e Y-axis i s called p l o t t i n g t h e point. - The Idea of a coordinate system is not new to you. When you

locate a point on t h e earth's surface, you do so by iden t i fy ing the

longi tude and latitude of t h e po in t . Note t h a t the order in which you write these numbers is important. For example, suppose you

looked up t h e longitude and l a t i t ude of your home town and acciden- tally switched t he numbers around. It is possible t h a t your de-

s c r i p t i o n would place the l oca t ion of your home town in the middle

of the ocean.

Suppose you were giv lng directions to help a friend locate a

cer ta in place in a c i t y laid out in rectangular blocks ( s t r e e t s a t

right angles t o each o t h e r ) . You tell him to start a t t h e c e n t e r of the city, go 3 blocks east and 2 blocks nor th (see diagram

b e l o w ) . Would t h i s be the same as t e l l i n g him t o go 2 blocks

east and 3 blocks nor th? Of course not! Do you see why it i s important t o be careful with the order when writing a p a i r of coor- d i n a t e s ?

Exercises 1-4b - 1. Given the following set of ordered pairs of rational numbers,

locate the points in t h e plane associated with these pairs.

[ ( Q # l ) # O , ~ ) , m,~), (2,41# (4,4)* - 1 - ( -3 ,3) , (49-31, E-5,3), ( 0 , - 5 ) r (-6301 1

2. On squared paper draw a pair of axes and label them. Plot the

points in the following sets . Label each point with its coordinates. Use a different pair of axes f o r each set .

s e t A = 1(6,-3), 7 (-9r17)r ( 5 r - 1 ) s (-8,lo)~ (o,o), - 1 - I4,3)1*

set B = [ ( ~ r l ) , ( 6 s - 5 1 , ( - 3 , - 3 ) # (4,-1013 (-9,-6), ( '8*0), (0, '5) # ( -'D - 5 ) 3

3. ( a ) Plot the points in the following set:

c = [(o,o), ( - 1 , (+LO), (-2.o), ( -3,0), 1+3,o) 1.

(b) Do a l l of the points named in Set C seem to lie on the same line?

( c ) What do you notice about the y-coordinate f o r each of t h e PO int s?

(d ) Are t h e r e any points on this line for whfch the y-coordinate is different f r o m zero?

4. (a) Plot the paints in the following set:

D = ((0*0), 0 - (0.+1), (0.--2), (0.+2).

(0, -31, (0.+3) 1 (b) Do a l l of the points named in Set D seem to l i e on the

same line?

( c ) What do you notice about the x-coordinate for each of the pointa?

( d ) A r e there any points on t h i s line for which the x-coordinate l a different f r o m zero?

Did you notice that the half planes above and below the X-axis

intersect the half planes to the right and to the l e f t of the

Y - a x i s ? These Intersections are ca l l ed quadrants and are numbered In a counter-clockwise d i r ec t i on wi th Quadrant I b e i n g the lnter- sect ion of the h a l f plane above the X-axis and the ha l f p lane to

the r i g h t 0.f the Y-axis, This quadrant does not include the p o i n t s on the pos i t i ve X-axis or pos i t ive Y-axis, nor does it Include t he

o r ig in .

P o i n t s in the Intersection s e t of theae two half planes are in the first quadrant or Quadrant I. The intersection of the h a l f plane above the X-axia and half plane to the le f t of the Y-axis is

Quadrant 11. Quadrant I11 I s the intersection of the h a l f plane below the X-axis and the half plane to the l e f t of the Y-axis. Quadranl IV 18 the intersection of the h d f plane below the X-axls

and the half plane to the right of the Y-axis. Note tha t t h e coordinate axes are not a part of any quadrant.

The numbers in ordered paira may be p o s i t i v e , negative, or

zero, as you have noticed in the exercises. Both numbers of t h e

p a i r may be positive. Both numbers may be negative. One may be pos i t ive and the o the r negative. One may be zero, or bath may be

zero.

Class - Exercises 1, Given the following ordered pairs of numbers, write the number

of the quadrant in which you f i n d the point represented by each of these ordered pairs.

Ordered Pair

( a ) (3, 5) ( b ) 0 3 - 4 1 ( 4 (-4, 4)

(a) (-39-11

(4 (8, 6 ) (f) <7,-11 (g) ( - 3 , -51

Quadrant

2. (a) Both numbers of t h e ordered pair of coordlnates are positive. The poin t is In Quadrant

(b) Both numbers of the ordered pair of coordinates are negative. The poin t is in Quadrant

( c ) The x-coordinate of an ordered p a i r Is negative and the

y-coordinatk is p o s i t i v e . The point is In Quadrant - ( d ) The x-coordinate of an ordered pair is positive and the

y -coordinate i s negative. The point is in Quadrant

3. (a) If the x-coordinate of an ordered pair I s zero and the - y-coordinate is not zero, where does the point l i e ? --

(b) If t h e x-coordlnate of an ordered pai r 1s not zero and -- the y-coordinate is zero, where does t h e point l i e?

( c ) If - both coordinates of an ordered pair are zero, where - is the point lacated?

4. Points on either the X-axis or the Y-axis do not l i e in any of the f o u r quadrants, Why not?

Exercises 1-4c - + + + + Plot the points of s e t L = [ A ( 2, I), B( 2, 3)].

Use a straightedge to Join A to B. Extend line seg-

ment AB. Line AB seems to be parallel to which axis?

+ + + f P l o t t h e polnts of set M = [ A ( 2, 3 ) , B[ 5, 3 ) ) . U s e a straightedge t o join A to B. Extend line seg-

ment AB.

Llne AB seema to be parallel to which axis? + + Plot the poin ts of set N = [A(o,o), B( 2, 3) 1 .

Join A to B, Extend l i n e segment AB.

Is llne A 8 paral le l t o e i ther axis? -t + Plot the points of s e t P = [ A ( 4, 4), ~ ( + 2 , 0) 1.

Join A t o B. Extend line segment AB. 4- f Plot the po in t s of s e t Q = [ C ( 6, 3 ) , D ( o , + ~ ) 1.

J o i n C to D. Extend l ine segment CD.

What I s the intersection s e t of lines AB and CD?

Plot the points of set R = [A (o ,o ) , ~ ( + 6 , O), ~(+3,+4)1

on the coordinate plane.

Use a straightedge to join A to B, B t o C, C to A .

Is the triangle (1) scalene, (2 ) isosceles, or (3) equi-

la teral?

Plot t he po in t s of s e t S = [~(+2,'1), B(-2,+1),

( 2 , 3 ) ~('2, -3) I . Use a straightedge to join A to B, B to C, C t o D, and D to A .

Is the figure a square?

D r a w the diagonals of the f igure . The coordinates of the point of i n t e r s e c t i o n af t h e diag-

onals seem to be ?

7. (a) Plot the points of s e t T = [~('2,'1), 3(+3,+3),

C( -2,'3), D( -3,'l)l. (b) Use a straightedge to join A to B, B to C,

C to D, and D to A.

( c ) What is t h e name of the quadrilateral formed? (d) Draw the diagonals of quadrilateral ABCD.

(e) The coordinates of the poin t of Intersect ion of the diagonals seem to be ?

Consider the s e t of po in t s whose coordinates (x,y) sat isfy t h e condition:

y = x.

Draw a pair of coordinate axes and label them. Locate these

points: ( 0 0 ( 1 , 5 5 , ( 2 2 , ( 4 . The condition y = x is sat isf ied f o r each of these po in t s because in each case the

y-coordinate is equal to the x-coordinate. Can you f i n d another point in t h e plane whose y-coordinate is equal to its x-coordlnate?

If you have plotted correctly the points listed above, you can draw a llne containing them and also containing other points fo r

which y = x .

Graph of

y = x

Is there any point on t h l s line whose y-coordinate is d i f f e r -

en t from its x-coordinate? The graph of the s e t of points described

by the condition y = x I s the line through the o r i g i n extending

into quadrants I and I11 and making angles with the coordinate axes that are equal In measure.

Class Exercises 1-5

In these exercises, use t h e f i g -

ure to the r i g h t .

We have just seen tha t a l ine contains a l l the points satisfy- ing the condition y = x . Let us now consider the condition

y > x . The ordered pair (3,5) fita t h i s condition. It is

named by what l e t t e r in the diagram? Another ordered p a i r that satisfies y > x is ( 1 , 4 ) . What letter names t h l s

point in the diagram? Some o the r ordered pairs t h a t f i t this

condition are: ( 2 , (4,6), ('4, O), (0,6), ( '6 , 6). What lettera are used t o name these points?

2. Thus far a l l t h e points whose ordered pairs satisfy the condi- t i o n y > x seem t o l i e above the line. We should examine the coordinates of points C, H, and R also since they are above the line. Is the ordered pair named by , ( 2 ) or

2 Naturally, you said ( 4 2 Don't forget that this - means x = 4 and y = -2. Do these values satisfy the condition y > x? Check to see whether points C and R "belong" to the condition y > x . Is there any difference between the condition y > x and the condition x < y?

3. You should recall t h a t a line in a plane determines two half- planes. How can this idea be used t o deacribe the set of points whose coordinates satisfy the condition y > x?

4. Whene do the points whose coordinates satisfy y < x l i e ? Check by mentally p l o t t i n g the points associated with the ordered pairs ( 1 3 ) ( 4 , f ) , (2,0), 5 (4,-7). Do you f ind that all of these points l i e in the half-plane below the lfne?

5. Where are the points whose coordinates satisfy x > y? Do the

pointa whose coordinates aatisfy x = y l i e in either half- plane determined by the line? Explain.

Consider the condition y = 4. It is sat isf ied by these ordered pairs: (2,4), (5,4), ( 2 4 ) (7,4), (-4, 4 ) . The condition is satisfied by any ordered pair whose y-coordinate is 4, regardless of the value of its x-coordinate.

The graph of the set of points described by the condition y = 4 is the llne parallel to the X-axis and 4 units above it.

Now let u s consider these condi t ions:

(a 1 Y > 4 (b) Y < 4

In diagram (a) line -.k is t he graph of t he set of po in ts described by the cond i t i on y = 4. Choose a point 6 , Does

the condition y = 4 describe t h i s ordered pair? Since the

y-coordinate in t h i s ordered p a i r is greater than 4, the condition y > 4 describes it. Are there other poin ts in the plane with y-coordinates greater than 4? Locate two o the r points, K and M, w i t h y-coordinates greater than 4, Are these poin ts above the l ine R ? Yes, they are in the shaded region which I s

one of the half-planes determined by t h e llne y = 4. The graph of t h e s e t of' points described by t h e condition

y > 4 lies in the half-plane above the l i n e 4 units above and parallel to the X-axis. The l i n e is not part of the graph.

In the shaded part 00 diagram (b) are located points f o r which the coordinates satisfy the condition y < 4. Locate point ( 2 , l )

i n t h i s region. Since t h e y-coordinate I s less than 4, the condi-

t i o n y < 4 describes it. Locate o t h e r po ln ts in the plane w i t h

y-coordinates less t h an 4. Are these points in the half plane

[sec. 1-51

below line/ ? Try other points In t he half-plane below line/

to see if they satisfy the condition y < 4. The graph of the s e t of points deacrlbed by the condition

y < 4 lles in the half-plane below the l ine 4 units above and

parallel t o t h e X-axls.

Exercises 1-5 I, Sketch the graph of the a e t of po in t s selected by each condl-

tion below. U s e di f ferent coordinate axea for each graph,

(a) y = + 2 ( g ) x S - 3

(d) x = 3 ( J ) Y = -2

(m) Compare the graphs obtained in (a). (b) , (c ) . (n) Compare the graphs obtained i n (d ) , (e) , (f).

2. Some ordered pairs that satisfy the condition y = 3 + x are

(0931, 5 (-3,o)~ (-1,2)# ( - 7 p - 4 )

(a) Plot the points associated with these ordered pairs.

(b) Draw the line through these points.

( c ) Shade in with a colored pencil the half-plane containing

the points whose coordinates satisQ the condition y > 3 + x .

1-6. Multiplication - of Rational Numbers

In the preceding sec t ion we have sketched graphs of a f e w

simple equations. If we wish t o graph an equation like y = -2x,

using posit ive and negative numbers f o r x, we will need to f i n d

products like ( 2 3 and 2 4 ) . The work we have

already done in this chapte r and In ear l ier grades has prepared us f o r multiplication in whlch one or more of the f a c t o r s are negative numbers. In Exercises 1-3, you used the number line t o find prod-

u c t s involving negative numbers in several cases. In Problem 12

of t h i s sec t ion , you will be asked to find products like (-2) 3

using the number line. Flrst, however, l e t us consider t h i s ques-

tion from another polnt of v i ew , In earlier grades you may have considered the set of multiples

of 2 If the elements of t h i s s e t S are written i n order, each number may be obtained from the one which precedes it by adding 2:

Similarly, any number (except the first) in the s e t of multi-

p l e s of 7 , when they are ordered, may be obtained by a d d i n g 7 to

t h e number which precedes it,

For the set T we see t ha t we can also say t h a t each number may be

obtained from the one which follows it by sub t rac t ing 7. For

example, 28 follows 21, and 21 = 28 - 7 , Some of you may have made multiplication tables l i k e this:

In this table , 30 is the product of 6 ' 5 . This multiplleation

table suggests a way of thinking about products of two ra t iona l

numbers when one or both of the numbers are negative.

Some of the c e l l s in t h e above table have been f i l l e d from

ou r howledge of arithmetic. Also we have used the property that

the p~aoduct of a negative number and 0 is 0. Mow to complete

the table, let us observe, f o r example, as we go up in the right- hand column that each number is 3 less than the number below it.

We shall refer to this column as the "3 column." Thus, the

'3 column'' would become

Similarly the "3 row" would become

Applyfng this notion to the remainder of the cells, the table

would be completed as shown on the followtng page.

In particular, we see t h a t the t o p row, which i a the I I - tt 2 row, is completed as shown here:

In o the r words, if we are to keep the property of multiplication of counting numbers, which we recalled earlier far multiples of 2 and 7 , we must accept the following products:

You abould notice similar results in other parts of the t a b l e . For each column and each row, the difference between two consecutive numbers is a fixed amount. As far as t h i s table is concerned, t h e

product of two negative numbers is a p o s i t i v e number, and the prod- u c t of a negative and a positive number (in either order) is a negative number. These conclusions are actually correct for all positive and negative r a t l o n a l numbers. It should be clear,

[ sec . 1-61

however, that we have not proved this resul t , nor have we given it

as part of a d e f i n i t i o n . We have only shown one reason why t h e

conclusions are plausible. Another reason for adopting the above rules for multiplication

is t h a t these rules enable us to re ta in the usual commutative, distributive, and associa t ive properties. For example:

From t h i s we see t h a t 2 (-3) is the addi t ive inverse of 5. Also, we know that -6 is t h e add i t ive inverse of 6 ,

Therefore, we must have 2 ' ( - 3 ) = -6. In genera l the product

of a negative and a positive number must be a negative number,

Also 4 + - 4 = 0 -

-5 ( 4 + 4) = -5 ' O = O

-5 ( 4 + - 4 ) = ( - 5 ) ' 4 + ( - 5 ) ' ( -4)

We have shown above t h a t ( - 5 ) 4 = 4 ' ( - 5 ) = -20, -

and therefore 20 + ( - 5 ) ( -4 ) = 0.

This means that ( 5 ' ( 4 ) is the add i t ive Inverse of -20. - We b o w t h a t '20 I s the additive inverse of 20, and therefore

these two numbers m u s t be equal and ( - 5 ) ( -4 ) = '20; t h a t is, the product of two negative numbers must be a positive number.

Exercises 1-6 - 1. Look at the large mul t lp l i ca t f on table which we completed in

this sec t ion . In which rows do the products increase a s we

move to t h e right?

2. Using the same table, in which columns do the products decrease

as we move down?

3, Give, in correct order, the products of 7 and the integers -

from 4 to 6 .

[sec. 1-61

- 4. Give, in correct order, the products of 4 and t he integers -

from 5 to 5 . 5. Complete the following table. If it he lps you see the pattern,

add appropr ia te columns on t h e rlght, or add appropriate rows

at the bottom of the t a b l e . It may also help you to first fill in t h e rows and columns for 0 and 1.

Using the above table, v e r i f y the commutative property of multiplication fo r : - (a ) 2and 1 (b) -3 and 0 ( c ) -2 and -3 ( d ) -1 and -3 .

Illustrate the a s s o c i a t i v e property of multiplication f o r - numbers: 2, -1, and 5 . Illustrate the distributive property of multiplZcation over

add i t i on Sor the sets of numbers, us ing the last two numbers i n each s e t as a sum.

* - - (a) -4, 3, 8 [S) 2, 3, 6 (c) 10, -8, -1

9, Find the products:

(a) - 4 . 0 (f) 49 - -5 (k) 4 . 3 * - 5

(b) -4 2 (&I -6 - '9 (1) '6 8 -12

(c ) - 4 - 5 (h) 'lo '60 {m) -3 -2 -11

(d) 8 0 ' 3 ( 1) -21 -43 ( 1 - l o m -8 -($)

(e ) 17 '2 ( J ) (-0.6). (-1.4) ( 0 ) -(16)

10. Find the producta:

a 1 ) 4 (b) -1 0 5 c 1 (d ) 8 - (-1) (e ) 77 (-1)

1 . Show the use of the number line in finding the products by repeated addition:

(a) 3 ' -2 (b) 5 ' -2 ( c ) 4 -3 .

2 State in your own worda how one could use the number line to find the product -4 ' 3. (Hint: Use the commutative prop-

erty of multiplication.)

13. A foo tba l l team has the ball on i t s own 45-yard line and then loses two yards on each of the next three successive plays.

(a) What will its new pos i t ion be?

(b) Write an expression involving negative numbers to obtain the answer to (a).

14. What must n be, if 2n - -18? 15. What is n in each of the fo l lowing equations?

- (a) 3n = -36 (d) 3n = 30

- ( b ) 5n = -75 (e) 2n E -8

( c ) -2n = 10 (f) "6n='12

16. In the following problems 5n multiplication put a number in

t h e parentheses so tha t the statements will be correct .

(a) ( ) 6 = -12 (i) 1 ) = -1

(b) 5 - 0 = -15 ( J ) 6 ) = '36

(c) ( 1 0 ) = 100 (k) (-9). = 81.

(4 ( - 5 ) * ( = 20 (1) 5 ' ( ) = -30

(e) - 5 1 = -20 (m) ( ) (7.0) = -90

( f 11 . ( ) = - n o (n) ( ) * ( - s o ) = loo

(g) - 1 ) 1 = 1 t o ) (-6). ( ) = -60

(h) (-7) * ( ) = o (P) * ( 1 = -1

17. Find the products:

(a) . ( - 6 ) (-10) (n) (-16) (-12)

(b) 3 t - 4 ) ( 0 ) (-45) (-3)

(4 3 6 (P) 25 (-3)

- 21 -6 (4 (9) (-27) 0

1-7. Division - of Rational Numbers

We h o w that if 3 n = 39, then n = 13, since 3 13 = 39. Also, in the d e f i n i t f o n of rational numbers (Chapter 6 , Volume I), . .

we c a l l (or 39 + 3) the ra t ional number n for which

Let us apply the methods we have used i n d i v i s i o n of r a t i o n a l numbers i n the seventh grade as we think of div i s ion of r a t i o n a l numbers i n v o l v i n g pos i t i ve and negative numbers.

F i n d n if = '18.

We know 2 ('-9) = '18

- Hence, n = 9 or (-9) A ~ B O -18 ; 2 = '9.

In t h i a section, we wall discuas d i v i s i o n only as the opera- t i o n which is the inverse of multiplication. To f 1nd -8 + -2, we

think - -8; 2 = n or - 2 - n = -8

n = 4, since - 2 . 4 = -8

'8 ; -2 = 4.

The question, "What I s 16 divided by -41'' ia the same as - the question, "BY what number can 4 be multiplied to obtaln 1 6 ? I t We know, -4 -4 = 16. Hence,

Which of the following are true statements?

(a) -63 ;. -9 = 7 (d) '2 -($) = 3

( e ) -2 + 3 = -(;I

I c ) '8 -13 = 104 (f) 3 + - ( $ ) = - 2 [sec. 1-71

You should be able to ahow that a l l of these are true statements except (b) and (e).

Before starting t o do the exercises, study the fol lowing and be sure that you know why they are true- statements.

4 4 What I s the reciprocal of -($? We h o w that and n

are reciprocals If

- 4 ($ n = 1.

and -[$I *-(+I = 1,

we have n =

therefore, ) I s the reciprocal of

hercises 1-7

1. Find the products:

(b) -4 '3 I e) -8 '9 (h ) -10 -($

( c ) 2 * - 6 (f) -21 -35 (i) -@I -(%I

[sec. 1-71

2. Find the quotients:

(a) - 2 8 i 7

(b) 12 ; -3

( c ) - 1 2

3. Complete t h e table:

( d ) -72 + 3 ( g ) -3 + 4

( e ) 72 + '9 (h) 4 + 3 )

4. Complete the table:

5 . Find r so that these sentences will be true statements.

- (a) 3 r = 17 ( d ) 3 r = - 2 1

- (g) ? r = 8

6. Write the reciprocals of each number In P:

7. Divide i n each of the following:

i 8, Find n if;

(a) - 2 $ 3 = n ( b ) 2$-3 = n

9. Write ( $ 1 as a quo t i en t in two ways.

10. Find n if

(a) 7 n = -6 ( b ) -7 n = 6

7 11. Write two sentences, using n, in which n = (=) would make U

the sentence a true statement.

12. Find n f o r each of these equations,

(a) (-25)n = "92 ( c ) -4 n = -(:)

(b) (-92) + (-25) = n ( dl + ( -4) = n

92 13. Write - as a quotient In another way, using t w o of the 25

numbers: 92, -92, 25, or -25.

14. Write two number aentences, uslng n, in whlch n = - 92 would make the sentence a true statement,

25

15. Complete the statements:

(a) If a and b are pos i t ive or negative integers, then

is the r a t i o n a l number x f o r which b

(b) If is a r a t i o n a l number than is p o s i t i v e if a a n d

b are

a ( c ) If is a ra t ional number then i; i a negative, if

either a or b f a and t h e other

1-8. Subtraction Rational Numbers

The number l ine will be he lp fu l in the understanding of sub-

t r a c t i o n of rational numbers. In arithmetic we learned, f o r example, t ha t if 7 + 4 = 11, then 11 - 4 = 7 . This statement nay be explained by referring to the following figure.

[aec . 1-01

We will use t h i s proper ty for negative numbers as well . Since

8 + (-5) = 3, then 3 - (-5) = 8. The f igure may he lp you to understand this.

The property we are using is that; whenever a + b = c , then

c - b = a. To f ind 7 - 4, we must f i n d the number which, when added to 4, gives 7. Since 4 + 3 = 7, it fol lows t h a t 7 - 4 = 3 .

To find ( -7 ) - 4, we must think of the number which, when

added to 4, gfves -7. Since (-11) + 4 = -7, we know t h a t

( - 7 ) - 4 = -11.

By the commutative property of addition a + b = b + a. Any

statement about add i t ion gives two statements about subtraction. For example :

- 3 + ( - 2 ) = 1 , s o t h a t 1 - 2 ) 3 and 1 - 3 = 2.

N o t i c e a l s o t h a t 3 - 2 = I and 3 + ( -2) = 1.

So we see t h a t : 3 - 2 = 3 + (-2). In other words the result of subt rac t ing 2 is the same as

adding the additive inverse of 2. The examples given above can

be used t o v e r i f y this general property.

Since 3 - ( - 5 ) = 8 and 3 4 - 5 = 8 , then 3 - ( - 5 ) = 3 + 5. Since 7 - 4 = 3 and 7 + (-4) = 3 ,

then 7 - 4 = 7 + ( -4 ) . -

Since ( - 7 ) - 4 = 11 and ( -7 ) + ( -4 ) = -11,

then ( - 7 ) - 4 = ( -7 ) + ( - 4 ) . Sfnce 1 - ( -2 ) = 3 and 1 + 2 = 3 ,

then 1 - (-2) = 1 + 2 . -

Since 1 - 3 = 2 and 1 + ( -3 ) = -2,

then 1 - 3 = 1 + ( - 3 ) .

1 In each case we see t h a t subtracting a number I s the same as add-

Ir lg i t s additive inverse.

Exercises 1-8 - 1. Add the numbers in each s e t ,

" (a) 2, 5 (d) 4 1 8 ( g ) -8, -13, -24

( b ) 7 , -7 (el -2, -3, 15 -

(h) 7 , 50, -110 -

( c ) 5, -2 -

(f) 21, 6, -7 (i) -23, 3 9 , 14

2. Find the aum of ) and ) and write t w o equations

involving subtraction which can be obtained from t h i s sum,

3. Find x in t h e fo l lowing:

( a ) ( - 5 ) + 2 = x - 2 - 3 (4 + ( 1 = x

,(b) ( - 3 ) + x = 8 - 13 (f) ( $ 1 x =

( c ) 8 + x = ' 3 (91 ) + x =

4 3

(a x t ( - 4 ) = 11 - I1

(h) x + 3 =

4. Supply the mlssfng number in each case,

5. Perform t h e subtractions in the following:

6 . What are the addi t ive inverses of

( a ) 10 (b) -100 ( a ) 4 (e) (f) -(g) 7 . Change each part of Problem 5 t o a problem of addlng the

additive inverse. For example, ( -4 ) - 2 = (-4) + (-2).

8. Perform the fo l lowing subtractions,

( a ) ( 1 0 - ( 3 = (h) 9 - ( -3 ) =

(b 4 - 6 = (1) 7 - (5) =

(4 16 - 12 = ( J ) 7 - ( "5 ) =

( dl 8 - 2 = (k) 2 - g =

( e ) ('8) - 2 = (1) 2 - (-9) =

(f) ( 0 - ( 2 ) = (ra) 3 - 10 =

(g) (-91 - 2 - - (n) 3 - (30) =

50

9 . Complete the following tables.

( a 1 (b 1

Chapter 2

EQUATIONS

1. 1 2-1. 'Writing Number Phrases

DO you like mystery s tor ies? Have you ever lmaglned yourself

to be a detec t ive l i k e Sherlock Holmes or &ncy Drew? Sometimes a mathematician must flnd one or more unknown numbers from certain

c lues . Then the mathematician works l i k e a detect ive trying to solve a mystery.

For example, suppose you are t r y i n g to flnd a certein number.

Let us call the number x. Mathematicians o f t e n use l e t t e r s l i k e 11 11 "x," y, "v," and so on t o represent unknown numbers. You are

given the following clue:

x + 5 = 7 .

Ln w o r d s we may say t h a t 7 is 5 more than the unlaown number.

In t h i s example can you f i n d the unlmown number? You probably

can.

I Sometimes the unknown number is not so easy t o flnd. For example, suppose you were gfven this problem:

Tom bought a ticket for a football game. Altogether he

paid $1.10 (or 110 cents), including the t ax . If the c o s t of the t icket is $1.00 more than the amount of the

tax, what is the amount of tax on the ticket?

[ Does 10 centa seem to be a reasonable guess? Recall that the

t o t a l cost is 110 cen t s . If the tax is 10 cen t s , the cost of t h e t i c k e t is $1.10 - $.LO or $1.00. But t h i s is not cor rec t , I because if the c o s t of the t i cke t is $1.00, then the cost would

[ be only $.go more than the tax.

; Let us again check the clues in the problem. You must; use the

clues correctly if you are to f i n d the correct answer. To help f i n d the amount of the tax, use the clues to wrlte a number sentence

L e t u s use x to represent the number o r cents f o r t he tax. Since

t h e c o s t of the t i c k e t is 100 cen t s ($1.00) more than t h e tax, we must add t h i s amount to x to o b t a i n the c o s t of the t i c k e t .

Thus, the c o s t of t h e t i c k e t may be represented by (x + 100). If we add t h e c o s t of the t i c k e t (x + loo), t o the t ax , x , w e have t h e t o t a l c o s t of the t i c k e t , x + (x + 100) . Thus we obtain the number sentence,

Can you now find the amount of the tax? The correc t answer 1s

5 c e n t s ($.05). The correct price of the ticket is $1.05. Does $1.05 - $.05 = $l.00?

Some questions about numbers may be answered easily with a l i t t l e knowledge of arithmetic. Some d i f f i c u l t problems, however,

are more e a s i l y solved by first writing a number sentence stating the condition of the problem, I 3 the two problems above, we used

the c l u e s to write number sentences. Each clue was a statement about numbers, Some of these numbers were known and some unhown, We then seek to f i n d what number x satisfies the condition

x + (x 4- 100) = 110.

In thf s chapter you w i l l first learn t o write number sentences about problems. Later in t h e chapter you w i l l learn t o use various properties about numbers in solving more difficult number sentences.

A sentence about numbers may be w r i t t e n in t h l a form:

'This sentence about numbers says,

"If seven is added t o a cer ta in number x the resul t

is nine ."

The ' '9" is a part of the number sentence above. Another part of

the sentence is "x f 7 . " mese expressions, x + 7 and 9, are

not sentences. They are only parts of sentences and are called phrases.

A phrase does not make a statement, In a sentence about numbers a phrase represents a number. A phrase that describes or represents a number i s ca l led a number phrase,

Some number phrases represent epeci f lc numbers , For example,

the number phrases ( 3 + 51, 9, T, l6 I11 + V, and 10 represent spec i f i c numbers. In each of these examples, t he value of t h e

number phrase is known, or it can be determined. A number phrase

which represents a spec i f i c number is called a closed nwnber phrase,

or more simply a closed phrase. What number Is represented by x - 4 ? We cannot determine t h e

number unless we know the value of x. Thus, x - 4 may have many

di f f erent values. Munber phrases which do n o t represent a spec i f i c number are called open number phrases, or more s i m p l y open phrases.

We may t h l n k of an open phrase as one whose value is "open" t o

many possibilities. Ikamples of open phrases are, (x - 41, 7y, B

(2 + z), T, and ( 3 r + 2x). To so lve problems by uslng number sentences you m u s t be able

, t o t ranslate the clues glven in the problem into an open sentence. To do t h i s you muat express the numbers in the problem as n m b e r

phrases . Earlier in this sectfon, we used the number sentence, t i x + 5 = 7 .

1s the value of 7 h o r n ? What about x 4- 5? Is 7 an open phraae? What about x + 55'

To work w i t h number phrases you must also be able to translate

the phrase i n t o words. The phrase x + 5 may be translated as

"the number x increased by five." Can 7x be translated as 11 seven times t h e number x " ?

Sometimes puplla are confused because an open phrase such as

x + 7 may have many different translations. For example, o t h e r t rans la t ions are :

" ~ h e number seven added to X, "

or "the nwnber x increased by seven,

or " the sum of x and seven, " or "seven more than the number x. "

However, all of the translations have the same mathematical

meaning. Furthermore, a l l of the B ~ g l l s h t r ans la t ions mean the

aame as "x + 7. " With practice you will learn to understand t h e

di f ferent ways of expressing a number phrase.

[sec. 2-11

Class mercises - 2-1

1. Translate each of the following number phrases:

(a) The sum of x and 5

( b ) The number x decreased by 3

( c > The product of 8 and x

( d ) One f o u r t h of the number x

( e ) n e number x Increased by 10

(f) The number 7 multiplied by x

(g) The number which is 11 subtracted from x

(h) The number x divided by 2

(i) The number which is 6 less than x

(j) The number x decreased by 9

2. F o r each one of the number phrases in Problem 1, find t he

number represented by t h e phrase if x = 12 i n each part.

3 , Translate each of t h e following number phrases into words:

( a ) x + 1 18 ( d l

( b ) x - 3 (4 4x -

(4 2x (f) 6 + x

4 . Find the number represented by each of the number phrases in Problem 3 if x = 6.

5. Find the number represented by each of the number phrases

in Problem 3 if x = -2,

6. Assume t h a t durlng January you saved d dol lars . In February you saved 5 dollars more than you saved in

January,

(a) Write an open phrase which repreaenta the number of dollars you saved in February.

( b ) Write an open phrase which represents the t o t a l number of d o l l a r s you have saved.

[sec. 2-11

7. Write open phrases representing each of the following:

(a) The number of cents in d dimes

(b) The number of ga l lons i n q quarts

( c ) The number of feet in y yards

( d ) The number of cents in one more t h a n n n i c k e l s

( e ) The number of inches in three less than f feet

8. idrite open phrases representing each of the following:

(a) A number plus f o u r

(b) The sum of a number and twice the number

( c ) A number increased by seven

( d ) Five subtracted from a number

(e ) A number subtracted from f i v e

(f) The product of n ine and a number

( g ) The quotient of a number divided by ten

( h ) The quotient of ten divided by a number

(i) A number subtracted from twice the number

(j) Three times a number divided by two times the number

lkercises 2-1 - The unlaown number i s not always represented a s x.

Translate each of t he rollowing number phrases i n t o symbols us ing the letter of each p a r t as the unlmown number. For

example, i n P a r t (a ) use "a" as the unlmown number,

(a ) Tfie sum of six and a number

(b) Eight times a number

(c ) Eight t imes a number and t h a t amount increased by 1

( d ) Three subtracted from eight times a number

Class Ecercises 2-1 - - 1. Translate each of the following number phrases:

(a) The sum of x and 5

(b) The nwnber x decreased by 3

( c ) The product of 8 and x

(d) One f o u r t h of the number x

( e ) The number x increased by 10

(f) The number 7 multiplfed by x

(g) The nwnber which is 11 subtracted from x

(h) The n m k r x divided by 2

(i) The nwnber which is 6 less than x

(j) The number x decreased by 9

2. For each one of t he number phrases in Problem 1, find the number represented by the phrase if x = 12 in each part .

3, Translate each of the following number phrases into words:

(a ) x + 1 18 (d l y-

(b) x - 3 (4 4x

(4 4. Find the number represented by each of the number phrases

in Problem 3 if x = 6.

5. Find the number represented by each of t h e number phrases in Problem 3 if x = 2.

6. Assume that during January you saved d d o l l a r s . b February you aaved 5 dol lars more than you saved in January,

(a) Write an open phrase which represents the number of do l l a r s you aaved in February.

(b) Write an open phrase which represents the t o t a l number of dollars you have saved.

[sec. 2-11


(a) me number of cen t s in d dimes

( b ) The number of gallons in q q u a r t s

( c ) The number of feet in y yards

I d ) me number of cents in one more than n n i c k e l s

( e ) me number of inches in three less than f feet


(a) A number plus four

(b) The sum of a number and twice the number

( c ) A number increased by seven

Id) Five subtracted from a number

( e ) A number subtracted from five

(f) The product of nine and a number

(g) The quot ien t of a number divided by ten

( h ) The q u o t i e n t of ten divided by a nwnber

(i) A number subtracted from twice the number

(j) Three times a number divided by two times the number

Jkerciaes 2-1 - 1, The u n b o r n number is n o t always represented as x.

Translate each of the following number phrases Into symbols

using the letter of each part as t he unlmown number. For

example, in Part ( a ) use "a" as the u n b o r n number.

( a ) The sum of six and a number

(b) Ef ght times a nwnber

( c ) Eight times a number and t h a t amount increased by 1

( d ) Three subtracted from e i g h t times a number

( e ) me amount represented by e i g h t times a number divided

by 4

(f) Two times a number and that amount Increased by 3

(g) Five multiplied by the sum of a number and 2

(h) Ten l e s s than seven times a number

(i) Twelve divided by the sum of a number and 1

(j) The product of two factors, one of which is the swn of

3 and a certain number and the o the r of which I s t h e

sum of 4 and the same number '

2, Find the number represented by each of the number phrases In Problem 1 if the unlmown number is 3 .

3. Translate each of the following open number phrases i n t o words, Write the word "number" to represent the u n b o r n number in each phrase,

m p l e : y + 3 A number increased by three.

(a> 2n + 5

(b) 6 - 3q

(c> 7 ( b - 1) 5 - d ( d l 7

( e ) 15 + 2w

4. Ffnd the number represented by the open phrase 2n + 5 for

each of the following values:

(a) n = 5

( b ) n = -5

( c ) n = o -

( d ) n = 1

5 . Find the number represented by t h e open phrase 6 - 3q for

each of the following values: +

( a ) s = Q (4 4 = 1

(b) q = -1 ( d l q = 5

6 , Write open number phrases to represent each of t h e following:

(a) Ann's age if Ann is three years o lde r than h e r brother, and he is x years o l d

(b) The c o s t of ten pencils a t y cen t s pe r pencil

( c ) The number of c e n t s in q quarters

( d ) A boyts age f i v e years from now if he is y years old now

( e ) A girl's age six years ago if she is z years old now

(f) The total number of p o i n t s Bob scores in two basketball games, if he scored g p o i n t s in the first game and

twice that number i n t h e second game

( g ) The number of dollars Mary has af ter spending 6 d o l l a r s , if she had n dollars at first

( n ) The sum of the three ages, if Kathyts f a t h e r ' s age

is four times her age, and her mother's age Is twenty

more than Kathy ' s age

(i) Tne t o t a l number of cen t s Mike has, if he has k

nickels and t h e number of dimes he has exceeds the

number of nickels. by 1

7. The closed number phrases, 1 + 2, 2 + 3 , 3 + 4, and

11 4- 5 , represent the sums of pairs of consecutive numbers.

\/rite an open phrase represent ing the sum of any two consecutive numbers. Hint: if n represents the smaller

of t h e two consecutive numbers, can the Larger number be

represented by ( n + 1) ?

8. Mrite an open phrase representing the sum of any two consecu t ive odd n w b e r s .

9. Write an open phrase representing the sum of any three

consecutive even numbers,

10, Write an open phrase representing any two consecut ive multiples of t en ,

2-2. Writing Number Sentences

A l l of us use sentences every day. We use aentences i n h l k i n g . When we read, we read sentences, Ijl mathernatics we need to deal w i t h many kinds of sentences. We use sentences t o explain mathematics and to discuss mathematics. Some mathematical sentences make statements about numbers, and it l a t h i s par t i cu la r kind of aentence which we are studying i n t h i s chapter. Consider t h e

f ollowlng sentences :

"Tne sum of 8 and 7 is 15."

"x -t- 3 = 8. I f Five I s greater than the sum of one and t w o . " 'I3 < 2 + 4." "The sum of 3 and 4 1s not equal to the product of 3 and 4."

Each of these sentences consists of two number phrases connected

by a verb or verb phrase, In number sentences verbs or verb phrases are represented by the symbols "=, " I ' < * I t 11>, " and "$, "

A number sentence using the symbol "=I' indicates equality.

The sentence, x + 3 = 8, makes t h e statement t h a t "x + 3 " and "8" are d i f f e r e n t namee f o r the same nwnber, We ca l l such a sentence an equat lon. When x = 5, the statement, x + 3 = 8, i s true and when x is n o t 5 , t he statement is false.

Conslder the sentence, "x - 4 ) 7." Is this sentence an equation? What does the symbol ' represent? Does it have the same meaning as "="? This sentence indicates t h a t the number

represented by "x - 4 " - is greater than the number represented by "7.'' Such a sentence is called an inequality. Inequality

means "not equal, I' Other examples of inequalities are

"x f 1 < 15" and "2x # 18." Some sentences are t rue . For example, "4 + 5 = 3 3 " and

"The sun sets in the west, I t are t r u e sentences. Sentences may n o t be true, however, "3 ) 2 + 4 " and "Abraham Lincoln was the first

president of the United States" are not true sentences.

Consider the sentences,

"Jimmy was at Camp Holly all day yesterday, I'

and "x + 3 = 8. "

Are they true? Are they false? You may answer, "I don't h o w . Which Jimmy do you mean? To what number does "x" refer?'' These

sentences are n e i t h e r true nor false, because they contain words or symbols which do n o t refer to only one th ing . "Jimmy1' can mean any

boy with that name, and "xl' can stand for any number. You might

look at t he camp records and say that, if "~irnrny" means Jimmy Mills

of Denver, the f lrst sentence is true; but , if I t means Jinmy Shultz of Cincinnati, then it is false. The second sentence is t r u e if

but it I s false ff x = 6 or if x is any number o t h e r

that can you say about the truth of the three following

sentences ?

E "George was the first president of the United Sta tes . " "3 + x = x + 3."

These sentences are similar i n that each conta ins a word o r symbol

which can refer t o any one of many objects. Do you see any

I difference between the f irst two sentences and the t h i r d ? Can t h e

I first two sentences be true? Can the f irst tv,o sentences be false? Can t he l a s t sentence ever be f a l se?

I Suppose a number sentence involves a s y m b o l l i k e "x" or

I "Y." If the symbol can refer t o any one of many numbers t h e

sentence is ca l l ed an open sentence. It is not necessarily a t r u e

sentence. It l a not necessarily a false sentence. It leaves the

: mat te r open for further cons idera t ion . Look a t t h i s equa tlon :

This equation is composed of t h r e e parts : a verb 11=, I' and two lo 'I me equation states t h a t f a r open phrases, "x + 7" and "- x - 2'

a certain number x these two open phrases represent the same

number, Can you d i s cove r such a number x? Can you find more than one? Try some numbers. After working f o r a while you m i g h t - say, "me sentence is true if x = 3 or x = 8, but it is fal,se - if x is any o t h e r number." The numbers 3 and 8 are cal led

s o l u t i o n s 3f the open sentence, The set 13,-8) is called the set of s o l u t i o n s of t he open sentence. --

When we find t h e en t i re s e t of s o l u t i o n s of an open sentence, we say that we have solved the sentence. An equation is a p a r t i c - ular kind of number sentence which involves the verb "= . " To solve an equa t ion means to find i t s en t i r e s e t of so lu t ions . The

s e t of s o l u t i o n s of an equation rnay con ta in one member or 1 t may

conta in several members. It might even be t he empty set.

You may have already used a special kind of equation. For

example, to f i n d the number of square units of area in a rectangle you used the fo l lowing :

This is an abbrevfatfon of a rule. In words, t h i s rule is:

"me number of square units of area in a rectangle

is ( o r , l a equal t o ) t he product of the number of units

i n the length and t h e number of l i k e units in the width,"

\hen such a r u l e i s abbreviated and w r i t t e n i n t h e form of an equation it is called a fo rmula , If the length and width of a

rectangle are h o w , then t h i s formula may be used to f lnd the area of that rectangle.

Can you determine t he s e t of s o l u t i o n s f o r the inequality

x - 4 > 7? How large must the number x be in order for the

inequality to be true? Is 5 - 4 > 7 ? I s 7 - 4 > 7 ? Is 12 - 4 ) 7? Do .you see that "x - 4 ) 7" is true if x is any

number greater than 11? Also, "x - 4 > 7" is false f o r any o t h e r va lue of x. Thus t h e set of s o l u t i o n s of this Inequality is t he s e t of a l l numbers which are greater than 11,

Class Exercises 2-2a - In Problems 1-4 below use your knowledge of arithmetic to find

the s o l u t i o n s e t for each of the number sentences. Remember t h a t

each member of the s o l u t i o n s e t must, when It is used as a replacement f o r the unknown, r e s u l t in a true sentence.

1. (a) x + 3 = 5 ( d ) m + 2 5 = 3 1

(b) Y + 3 > 5 -

( c ) k + 13 = 15

( e ) s + 25 < 31 (f) t + 10 # 5

2. (a) x + ( - 7 ) = 2 ( x + (-3) = 6

( b ) Y + ( -7 ) > 2 (e) p + (-15) = -1

( c ) n + ( -9) = -2 ( 5 ) x I- (-15) < -1

3 . (a) 4b = 12

(b) 4a # 12 ( c ) 5w = 35

n 4. ( a ) ~ = 2

( d l gm < 35

( e ) 13x = -13

(f) 7Y = -56

( e j - = 5 -3

- (f) - = 7

- 3 5 . A formula f o r f i n d i n g the perimeter of a rectangle is

p - 2 y + 2 U,, Find th@ perimeter of a rectangle whose

length is 7 feet and whose width is 4 feet, 1

6. Use t he formula a = Tbh t o f i n d the

number of square u n i t s of area in the 1 ha 7" triangle shorn at the right. A

b =14"

Equations are used in many ways in many different fields. We

solve equatlons to find the currents in an electr ical network when we hiow the voltages and tne resistances, to design af rplanes o r

space ships , t o f i n d out what is happenfng in a cancer cell.

We also u s e equa t ions to pred ic t t h e weather. We now know

methods f o r predicting tomorrowls weather very accurately. The

only t rouble is that these methods r e q u i r e the s o l u t i o n of about a thousand equa t ions w i t h the same number of unknowns, Ehen w i t h the

b e s t of t h e modern high speed computers, it would take t w o weeks t o compute t h e prediction of tomorrow ' s weather. meref ore, t h e

meteorologists ( look up t h i s word) make many approximations. They

simpllfy the equations in such a way t h a t they can compute the

prediction in a shor t enough time. They will be able to make

better predictions when we h o w more efficient ways to solve many equat ions with many unknorms ,

Many leading mathematicians are working on such problems and

cont inue to seek new methods f o r so lv ing equat ions. In unfversitles,

government and i n d u s t r i a l laboratories there are actually t n ~ u s a n d s of mathematicians who are working every day at solving equations.

Some use b i g new computing clachines. Others use t he kind of method

you are now learning and work w i t h pencil and paper as you do,

Exercises 2-2a

1, Translate each of the following number sentences into symbols.

(a) The number x increased by 5 is equal to 13.

(b) The number 3 subtracted from x is equal to 7 .

( c ) The product of 8 and x is equal t o 24-

( d ) Ifhen x is div ided by 8 t he quotient is 9 .

( e ) Ten more than t h e number x is 21.

( P ) The number x multiplied by 7 is equal t o -35 .

(g) The number 11 subtracted from x is - 5 .

( h ) The number 6 Less than x is 15.

(i) The number x d i v i d e d by 2 Is equal to -7 .

For each of t h e equations you wrote i n Problem 1, f i n d the

s e t of solutions by using your howledge of arithmetic,

Translate each of the fol lowing number sentences i n t o symbols.

(a) The number x increased by 2 is greater than 4.

(b) The number x multiplied by 5 is less than 10.

( The r e su l t of dividing x by 7 is greater than 2.

( d ) Three less t han the number x i e greater than 6,

( e ) The number x decreased by 5 is less than 13. -

(f ) The product of 3 and t h e number x is greater than 9.

For each of the inequalities you wrote In Problem 2 , use your

howledge of arithmetic to find t h e s e t of s o l u t i o n s .

Translate each bf the f o l l o w i n g number sentences i n t o words, Use the term "a number" or "a certain number" to represent

t he unhown number.

(a ) Y + 2 = 5

(b) z + ( - 3 ) = 7 -

(c) 2a = 10

(4 h + (-5) < 9

(4 Tm < 15

(f) 7 -I- (-k) = 2

(g) d + (-3) < 4 w 0-4 7 > 9

(i) k + ( -7 ) = -2

C (j) - = 6 '30

Using your lmowledge of arithmetic, find the set of so lu t i ons for each of the number sentences in Problem 5 .

What is the area of a square whose l eng th is 15 inches? Use A = s2 aa the formula for the area.

A formula used in f i n d l n g s imple interest is written i = p r t ,

where

i Ls the in te res t In d o l l a r s , p is the p r i n c i p a l (or amount borrowed),

r is t h e ra te ( o r per c e n t ) of Interest per year, t is the time i n years.

Find the in te res t f o r a bank loan of $750 for 3 years a t 6 % interest.

9. To find t h e cfrcumference of a c i r c l e , t h e formula c = 2m may be used, where r s t a n d s for t h e radius. Find t h e circum-

22 ference of a circle whose radius i s 10 inches. (Use - 7 or 3.14 a s an approximation to T , )

10, The formula: d = rt may be used t o f i n d the t o t a l distance traveled if the rate of travel does not vary, where d is

t h e measure of t h e distance, t is the measure of the time,

an6 r is the rate. For example if d is measured in m i l e s and t in hours, r w i l l be in miles p e r hour, Find the

distance traveled by an automobile moving at a rate of 45 miles per hour (m.p,h.) for 13 hou r s ,

1 To f 2nd 19 % of $750 you may use the percentage formula,

where p is the percentage; r is the rate ( o r percent) ;

and b is the base, In t h i s problem, r = 19% or 0.19 and b - $750. Find t h e value of p f o r t h i s problem.

*12. Find t h e area of the floor of a c i r c u l a r room whose radius is 2 22 or 3 . 1 4 for 13 feet. The fornula is A = m . (use - 7

Find the volume of the cylindrical tank pictured a t t h e right if the

radius Is 1 foot and the height 1 is 3- feet. (use 2

22 for a.) 7- Find the capac i ty of the tank In gallons, One cubic foot holds 1 72 gallons.

9 414, The formula F = 4 + 32 may be used ta convert a temperature 5 reading on a Centigrade thermometer to a temperature reading on a Fahrenheit thermometer. Find the correct Fahrenheit temperature reading f o r each of the f o l l o w i n g readings on a

d

Centigrade thermorne t e r ,

( 1 oD ( b ) 100' ( c ) 3 7 O

Graphs - of - Solu t ion S e t s

We can draw a p i c t u r e to represent a s e t of numbers by assoc i - a t i n g the numbers with p o i n t s on a line. Consider t h e s e t

(0 , 3, 6 ) . Each element f s a number associated w i t h a p o i n t on t h e

number l i n e . We draw a p ic tu re t o represent t h i s particular set

by marking a heavy d o t on t h e number line as shorn below:

This drawing is cal led a graph of the set 10, 3 , 63. It is some-

times u s e f u l to draw graphs of s o l u t i o n sets of open sentences. The open sentence, "x + 3 = 8," has the solution set (53.

The graph of t h i s s e t shows a heavy d o t only on t h e point which

corresponds with 5 .

What is t he solution s e t of "x + 3 = 3 + x " ? This sentence

is t rue f o r any number we choose as a replacement for x. The s o l u t i o n s e t for t h i s e q u a t i o n is the s e t of a l l numbers. The

graph of t h i s set is nade by drawing a heavy shaded line along the en t i r e number l i n e as shown below.

Consider the I n e q u a l i t y x - 4 ) 7 . The s e t of s o l u t i o n s f o r t h i s inequality is the s e t of all numbers which are greater than

11. This s e t of s o l u t i o n s is represented on t h e number line a s

shown below :

The number "11" is not in the s e t . We indicate t h i s by drawing

a circle around the point corresponding to 11 on the number line. The part of the number line to the r i g h t of the 11 is shaded

showing that a l l points to the r igh t of 11 are In the set of so lu t iona .

What is the solution s e t f o r the inequality shown below?

You should f i n d t h a t t he s e t of solutions is the set of all numbers less than ' 5 . On the number line t h i s is represented as a circle

4- around the po in t corresponding to 5 and a heavy shaded line drawn along a l l points of the number line which l i e to the left of

' 5 . Here is the drawing :

We have found t ha t the se ta of solutions f o r d l f f erent number sentences may be different. Some of the solut ion sets contaln only one member. Such sets may be represented by a single large dot on a drawing of the number line. The d o t is drawn at the point which corresponds to the number in the set of solutions. IP the set of solutions is the set of all numbers, we may draw a heavy, shaded line along t h e en t i r e number l ine . In t h i s case, the aolution set

is represented by the en t i r e number line. Tfie sets of solutions f o r lnequallties are represented by a part of the number line. A l l of the inequalities we discussed were represented by half-lines on the number line. A clrcle was used to indicate a point not included in the s e t of solutiona.

Claaa merciaes 2-2b - - What l a the solution s e t pictured in each of the following

graphs?

You are n o t y e t ready to solve very complicated equa t ions or inequa l i t i e s . For example, I t is much more difficult to find the

s e t of s o l u t i o n s f o r t h i s number sentence,

than it was to find the s e t of s o l u t i o n s f o r the equations and inequalities we dl scussed earl ier . Other number sentences may be

even more complicated. You will l e a r n much more about these compli-

cated number sentences later in this chapter and again when you

s tudy algebra next year .

[sec. 2-21

Exercises 2-2b

1. Using your lolowledge of arithmetic, f i n d the se t of so lu t ions for each of the following number sentences,

(a) x + 2 = 5 (e l x + -4 > 1 ( b ) 4 + x = 0

( c ) = 6

( d l 3x ( 3

(f) * = -1

(h) 4 - x > l

2 For each of the number sentences in Problem 1 represent the

se t of s o l u t i o n s on a number line.

3, Using your howledge of arithmetic, f i n d t he s e t of solutions for each of the following number sentences,

(a) x + l = l + x ( e ) 3w = -15

(b) Y + -1 > 0 (f) 14 + x = 13

( c ) 1 - b > O ( g ) 13 - x = 14

( d ) a + 2 = 1 1 - a 2 (h) x = -1

4, For each of the number sentences In Problem 3 show t h e set

of s o l u t i o n s on a number l i n e .

5. SometFmes an equat ion or an i n e q u a l i t y is only part of a

sentence. Just as you can buiLd longer sen tences o u t of shor ter ones by using such words as "and, " "or, If and "but," you can join number sentences together to make longer ones. Such sentences are called compound sentences.

Consider the compound number sentence

"x + -4 < 7 and x + -1 > 0." In order to be a so lu t ion of this sentence, a number x must be a solutlon of both the sentence "x + -4 < 7" and

the sentence "x + -1 > 0 .I1

The elements of the solution set of the sentence are the numbers which are in both t h e so lu t i on set of "x + -4 < 7"

tt and the so lu t ion s e t of x + -1 > 0." The s e t of so lu t ions of " x + -4 < 7" is the s e t of all

numbers less than 11.

The s e t of solutions of' "x + -1 > 0'' is the s e t of all numbers greater than 1.

What is the s e t of s o l u t i o n s of t he compound sentence?

Show t h i s set on a number line,

6. Is the set of so lu t ions f o r the compound sentence in Problem 5 the intersection of the sets of solu t ions for the two inequalities or is it the union of t h e two sets

of solu t ions ?

7, For each of t h e following compound sentences find the s e t

of s o l u t i o n s .

(a) x + - 2 < 7 and x + 4 > 6

( b ) x + -3 = 6 and x + -3 > 6

( c ) 2x > 6 and $ < 3

8. For each of the compound sentences in Problem 7 represent

the set of solutions on the number line. h

2 *9. (a) Find the set of 80lutiOnS for x = 9. (There are two poss ib l e solut ions. )

(b) Represent the set of so lu t ions f o r Part (a) on the

number line.

2 , *lo. (a) Find the set of solutions for x < 9.

(b) Represent the set of solutions for Part ( a ) on t he

number line.

1 (a) Find the s e t of solutlona for the following compound sentence :

x + 7 - 6 or 2 x + - i = 5 .

(Eote : ln mathematics "or" means either the f i rs t or the second or both the f i rs t and t he second.)

( b ) Represent t he s e t of s o l u t i o n s for Part ( a ) on the number line,

1 (a) Find the s e t of so lu t ions f o r x + '1 = 4 or x + -1) 4. (!Ibis compound sentence is sometimes abbreviated

x + -1 > 4.) - (b) Show the s e t of s o l u t i o n s for Part (a) on t he

number line.

- *13, Find the s e t of so lu t ions for x ( 10 or x + 9 > 0,

In so lv ing problems, tihe t r a n s l a t i o n of number sentences f r o m

the mglish language, or words, i n t o mathematical language, or

symbols, is often the most important part of t h e task, A l s o it is

often t h e most d l f f i c u l t part of the task. When you are faced w i t h

problems such as those above, t h i n k about them carefully. Often

you w i l l find that the problem is only asking, In a complicated

* way, f o r the s o l u t i o n s of a nwnber sentence, By wr i t i ng t h e

sentence in symbols t h e s e t of solutions may be easier t o find.

It is Important t h a t you understand that the open sentences you write are always about numbers, The problems may r e f e r t o

inches or pounds or years or dollars, but your open sent&ces must be about numbers only.

Consider the problem, "The sum of a certain number and e i g h t

La equal t o two more than the product of f o u r and the nwnber.

What Is the number?" The problem asks us to f i n d a certain number, We represent the number with a letter such as x, We must de f ine

the meaning of x, That is, we must s t a t e what it is t h a t the

l e t t e r x represents. This is t h e f i rs t s tep in wr l t i ng an open

sentence, and it is a very important part of the work done in

f ind ing the solutLon. In t h i s case t he meaning of x Is obvious :

"Let x represent the unhown number. I' ( ~ n more difficult problems the def i n i t l o n of the le t ter used may be more complex .)

The next step Is to describe the numbers in t he sentence, using the symbol f o r the unimown nwnber where necessary in writing phrases. The phrases f o r t h i s problem are:

" the sum of the number and eightt' - - - - - x $ 8

"two more than the product of

f o u r a n d thenumber" - - - - - - - - - - - 4x 4- 2

We now have two open phrases. For o t h e r problems t he re may be more than two phrases. Using these phrases we write the open sentence:

Is t h l s sentence an equation or an inequality? Can you guess what

number will make t h e sentence true? The solution is 2,

Many of the problems in t h i s section may be solved easi ly by

uaing arithmetic, and you may question why it is necessary to write

open sentences expressing the condition of such easy problems.

Remember t h a t in t h i s s ec t ion we are n o t concerned p r i m a r i l y w i t h

the so lu t ions . You should concentrate on the writing of the open

sentences. Later in t h i s chapter you will learn to use prope r t i e s of numbers in finding the s e t of s o l u t i o n s for more d i f f i c u l t

problems ,

Class Exercises 2-2c

For each of t he folLowlng problems wri-te an open sentence

s t a t i n g the condi t ion of the problem.

1. A t r a fn travels at 80 miles per hour, How long does it

take for t h i s t r a i n to make a 6 mile t r i p ?

' F i r s t : Let "t" represent the number of hours t he t r a i n travels.

( ~ o t e t h a t we speci fy that t h e " t " represents a number. In this case it represents t h e number of hours t he train travels. It is n o t s u f f i c i e n t t o say, " ~ e t t represent

the time.")

Second: Write a p h r a s e representing the number of miles the t ra in travels in t hours.

Third: Write an equation stating the condition of the problem,

Mary is fourteen years old. She is five years older than her brother. How old I s the bro ther?

A boy is f o u r years younger than hls aister. If the boy is ten years old, how o ld is the sister?

A boy bought a number of model plane k i t s costing 25 cents each. He spent 75 cents. How many k i t s d i d he buy?

A boy's age

boy now?

seven years from now will be 20, How old is the

How many f e e t are there In a board having a length of 72

Inches ?

HOW many feet are there in a board 5 yards long?

Ann was 3 years old ten years ago. How old is Ann at the present time?

How many dollars may be obtained in exchange for a total of 450 pennies?

If three d o l l a r s is added to twice the money Dick has, the

r e s u l t is less than twenty-three dol lars . How much money does Dick have? (~111 your open sentence f o r the condition in t h i s problem be an equation or an inequality?)

At a cer ta in speed a plane w i l l travel more than 500 miles

in two hours. For what speeds is t h i s true?

If one Is added to twice a girlra age the result is nineteen. What is the girlis age?

A man drove a t o t a l distance of 240 mfles at an average speed of 4 0 miles per hour. How long dld it take for the d r i v e ?

If a baby sitter earns 65 cents per hour, how much w i l l she

earn in 5 hours?

merci se s 2- 2c - For Problems 1, 2, and 3 , write open sentences f o r each

condition stated . 1. A boy is 7 years older than his sister.

L e t x represent t h e number of years in t he boy's age. The sister's age is represented as x - 7.

Write equations showing t h a t :

(a) The sum of their ages Is 21.

(b) The boy's age is equal to two times the sister 's age.

( c ) mree times the sls ter ts age equals seven more than

the boy's age.

(d) The product of 2 and the boy I s age equals t h e p roduc t

of 4 and the girl 1s age,

2. Joan has twice as much money as does C a t h y .

L e t rn represent the number of dollars that Cathy has;

then, 2m represents t he number of dollars Joan has.

Write equations showing t h a t :

( a ) Together they have $15.

( b ) If Joan spends $5, she and Cathy w i l l have the same

amount of money.

( c ) If Cathy spends th ree dollars, Joan w i l l have five times as much money as does Cathy.

3. The length of a rectangle is four feet more than the width.

Let w be the number of feet in the width.

Then w + 4 is the number of feet in t h e length.

Write equations showing that:

(a) The wldth when doubled is the same as the length

increased by three,

{b) Assume t h e l eng th i s doubled. To have t h e same measure, the product of 3 and the w i d t h must be increased by 1.

( c ) Twice t h e w i d t h added to twice the length is equal to 36. (~fi is is the perimeter of the rectangle,)

For Problems 4-14 write open sentences expressing the condition of the problem.

4. In t en more years M r . Smith w i l l be f o r t y years o l d . How old Is he now?

5. If 73111 earns five d o l l a r s more, he will have earned a t o t a l of twe lve d o l l a r s . How much has he already earned?

6 , A girl is two times as tall as her brother. If t h e girl is

64 i nches t a l l , how tall is her brother?

7 . Pau l was 1 4 years o ld in 1958. In what year was he born?

8. Twenty p e r cen t of a number Is 10, What is the number? 1 ( ~ l n t : 2 0 % ~ ?.)

9. A carpenter saws a 50-inch board i n t o two pieces. One piece

is 10 inches longer than the other piece . Find the length

of the shor ter piece.

10. DI a class election Marge received 5 v o t e s more than M c e , How many v o t e s did Eruce receive if there were 35 vo te r s in t he class? (~asume that each v o t e r voted f o r either Marge

o r Rmce but not f o r both.)

11. If a number is added to twice the number, the sum is less

than 27. For what numbers is t h i s true?

12. The populat ion of Minneapolis and St, Paul combined is more

than twice the populat ion of S t . Pau l alone. What is t h e

p o p u l a t i o n of St, Paul if the combined populat ion is one million?

13, Pat and Mike are t w i n s but they do n o t weigh the same amount. If Pat weight 105 pounds, how heavy 1s hake?

14, last year a boy earned more than one hundred twenty d o l l a r s on h i s paper route, What were his average monthly earnings?

1 ~15. M r . Smith is 4 times as old as his son. b 16 years he w i l l be only twice as old , What are t h e i r ages now? ( ~ i n t :

If the son is x years o ld now, how old k r i l l he be in another 16 ears?)

"16, J.n a particular t r fangle one angle is twice as large as another. The t h i r d angle is three times as large as the

i smaller of the two other angles. How many degrees are there i n t h e measure of each angle? (~int: How many degrees are there in t h e sum of the measures of t he angles of a triangle?)

"17. The sum of t h ree consecutive whole numbera is 123. Wnat are the three numbers?

+is. Bob has' $1.25 in n i c k e l s a n d dimes. He has three times as

many n i c k e l s as dimea. Find how many of each he has,

2-3. Finding Solut ion S e t s

We know t h a t there are many different ways to express any

number. For instance 25 = 0 = 9 = 30 - 5 = 52 = (31 - 6). Of 2

all these ways of expressing the number twenty-five, 25 is the

simplest. Consider the following equations:

Each of these equations has the solution x = 5; t h a t is, if in

each equation we replace x by 5 we have a true sentence and If

we replace x by a n y number d i f f e r e n t from 5, t he seitence is

fal.se, These equations are called equivalent because a l l the s o l u t i o n sets are t h e same, In fact, t h e e q u a t i o n x = 5 could

also be Inc luded in the list, J u s t a s 25 is the simplest way

to express the various numbers i nd i ca t ed at t he beginning of t h i s

sec t ion , so x = 5 is the simplest equation equiva len t to the

list of equations glven above.

Class Exercises 2-3a - 1, Find six equations equivalent t o the equa t ion x + 1 = 4.

2. Find six equat ions equivalent to the equation x = 3.

3. Find six equations equ iva l en t to the equation 2x = 12,

4. What methods can you discover to g e t from a given equation one or more equations equivalent to it?

5 . If one equation is equivalent to a second equation, is the

second equivalent to t h e f f r a t ? Why?

* 6 , Are the following two equations equiva'lent : x = x f 3,

x = x - l ?

Consider, f o r instance, the equation

If x is replaced by a so lu t ion of this equation, x 4- 3 and 7 are two different names for the same number. Hence, for instance, (x + 3 ) + 4 and 7 + 4 must be names f o r the same number, whenever x is replaced by a so lu t ion of the given equation. Since it ia

probably easier to see t h i s in terms of numbers f i r s t , consider t h i s :

10 = (15 - 5 ) . Hence,

l o + 5 = (15 - 5 ) + 5.

We have added the same number t o a given number, 10, expressed

i n two ways, These are examples of the

Additfon Property - of Equal i tx . If two numbers, a and b , -- are equal (that is, if a and b are two different - -- - - -- names for the same number), then if you add t he same ---- ------ number to each of them, the two sums w i l l be equal. ---- -----

if a = b , then a f c z b f C.

Graphfcally, we would have the fol lowing figures.

b b+c b+c b To see how this a p p l l e s t o the equat ion: x + 3 = 7, notice

t h a t if x denotes a s o l u t i o n of this equation, x + 3 and 7 are two names f o r the same number. Hence in t h e additfve property,

x + 3 plays the role of the letter a, 7 plays the r o l e of the let ter b ,

if x is a solution of the equat ion. Hence, if x f s a solution

of

x + 3 = 7 it is also a solution of

( x + 3 ) + c = 7 + c ,

no matter what number c is. A brief way of describing this is to

say t h a t we get the second equation from the first by "adding the

same number, c, to both sides of t he equation. I' " me most useful c h o k e of c in the above is 3, for

Then, by the associative property f o r addition, this is equiva len t to -

x + (3 + 3) = 7 + -3.

Hence

Thus, by the addition proper ty of equality, if any number is

a solution of x + 3 - 7, it is a so lu t ion of x = 4. Conversely,

if any number is a so lu t ion of x = 4, we may add 3 to both

s i d e s of the equation to get

and see that if any number is a solu t ion of x = 4, it l a also a solut ion of x + 3 = 7. Hence the equations:

x = 4 and x + 3 = 7

are equivalent equations; t ha t is, their solution aets are the same. By t h i s means we can show tha t :

a = b and a + c = b + c

are equivalent equat ions; that is, if a = b, then

a - k c - b + c , a n d i f a + c = b + c , then a = b .

Uslng thia r e s u l t It also follows that; x = 3 and x f 7 = 10 are equivalent equations since we g o t the second equation from the f irst by adding 7 to both sides of the equation x = 3.

Class mercisea 2-3b

1. Find the so lu t ion s e t of each of the fol lowing equations and check your so lu t lona:

(a) x + 4 = lo ( c ) 3 = x + 4 -

(b) x + 2 = 5 ( d ) -2 = x + -3

2, Use the d i s t r i b u t i v e property to sirnplffy each of the

f ollowlng : -

( a ) 2x + 3x = ? (d) x + 7x = ? -

(b) x + % = ? ( e ) x + (-2)x = ?

Suppose instead of adding a h o w n number t o both aides of the

equation x + 7 = 10 we add an unknown, 2x. Then we would have

2x + (x + 7 ) = 2x + lo.

We can write the left s ide , using t h e associative property, as

(a + x ) 1- 7.

Now, x = 1 x and hence 2x + x = 2x + 1 x = ( 2 4- 1)x = 3x,

using the distributive property. Thus, if we add 2x to both sides of the equation x + 7 = 10, we have

This ia all very well, but suppose we were given the last equation. How could we g e t back to the first equation? Since we

got the last equation by adding 2x t o both sides of the f irs t ,

we should be able to get the first equation by adding -(a) to both s i d e s of the l a s t . Let us see if t h i s is so. Then

-(a) + 3x + 7 = -(a) + 2x + 10. -

Now -(a) + 2x = 0 ,since (2x1 is the add i t i ve inverse of 2x.

Wlt what i a -(a) + 3x equal to? We find it this way:

-(a) + 3x = (-2)x + 3x = (-2 + 3)x = 1 x = x,

using the d i s t r i b u t i v e property. Thus, by adding -(a) to both sides of the equation 3x + 7 = 2x + 10 we ge t x + 7 = 10 aa

an equivalent equation. Since the solution of t h i s equation is 3~ the solution of 3x + 7 = 2x + 10 is also 3 . You should check thia t o show tha t we have made no mistake.

Now we can find the solutfon of an equation like

We wish to f i n d an equivalent equation in which x occurs on one s i d e only. To do this we can add - ( 3 x ) to both sides to g e t

7 3 x ) + 4x + 5 = - ( 3 x ) + 3x + 2.

?his resu l t s in the sentence

(YOU should fill i n the steps needed t o show t h i s . ) Then, if we - add 5 to both s i d e a of this equation (since we want t o have x by itself on one s ide ) we have

- Thus x = 3 Is equivalent to the equation 4x + 5 = 3x -+ 2,

- which shows that 3 is its solution. -

The solutf on of the equation x = 3 is obvious and, since

I t is equivalent to the equation bx + 5 = 3x -k 2, t h i s equation - a l s o has the so lu t ion 3 . f a c t , a method of solving an equa- t f o n is t o f i n d a n equiva len t equation which has an obvious solu- t i on - - tha t i s , of t h e form x = some number. Let us go back over

the process w e used, We f i r s t added -(3x) t o both sides of the equat ion i n order t o g e t an equivalent equation in which x occurred on only one s i d e : x f 5 = 2, Then, since we wanted an - equation of the form, x = some number, we added 5 t o both sides,

To make sure that we have made no mistake, let us check to see - that 3 is really a so lu t ion of the equation:

If x = - 3 , the l e f t ' s l d e of the equation becomes

('3) ' 4 + 5 = -12 + 5 = -7.

- If x = 3, the right s i d e of the equation becomes

3 ( - 3 ) + 2 = -9 + 2 = -7.

Hence f o r t h i s value of x, the number on the left side is equal to that on t h e r l g h t , This is our check.

Of course there are equations which have no solutions, One

such equation is x I- 3 = x. This may be considered to be obvious since no number can be 3 greater than i t se l f . &t let us find - what this equation is equivalent to. We may add x to both sides

and have - - x + ( x + 3 ) = x + x

- emem ember t h a t j u s t as 3 is the number wlth the property - - t h a t 3 + 3 = 0, so x is the number with the property that - x + x = 0,)

So the given equation is equivalent to 3 = 0, This has no

solution and hence the given equation has no s o l u t i o n . Other equations ,have many solutions. Consider 2x = x + x.

Thls is true f o r all va lues of x. You might like to show t h a t

t h i s is equivalent to 0 = 0.

a e r c 1 se s 2- 3a

1. Using the methods of t h e p rev ious s ec t ion , f i n d f o u r equations

equ iva l en t to each of t h e following equa t ions :

(a) x + 7 = 13

(b) 17 = x + - 3

2. Use the add i t i on property to solve t h e following equa t ions . Check your results.

Examgle: x + ( - 3 ) = 11, First use t h e addition proper ty and

add 3 to both sides of the equation, T n i s g ives

(x + -3) + 3 = 11 + 3. By t he associative proper ty of addition t h i s is equivalent to

x + ( - 3 + 3 ) = 14, x + O = l 4

x = 1 4 . -

To check this see that if x = 14, x + 3 is 14 + ' 3 w h i c h

is e q u a l to 11,

(a ) x + 5 = 6

(b) x + S = 5

- (g) -2 = 4 -k x

10 (h) x + - = 2

( c ) x 4- -7 = 7 . (1) Y + "($1 = 5 ( d ) x f -7 = -7 9

( j u + 1 4 = - 5

(e) t + 6 = -13 13 ( k ) - = l + x 7

3 . Apply the addition proper ty to these equations, adding the

indicated number, and write the resu l t ing equation,

Emmple: 3x + 4 5 (add -4) (3x + 4 ) + '4 = 5 + '4 by the addition property.

3x + ( 4 + -4) = 1 by the associative property.

The resulting equation is: 3x = 1.

( a ) 2x + 5 = 10 (add -5)

( b ) 3x + 10 = 5 (add -10)

( c ) 5x + 2 = -2 (add -2)

( d ) l D ~ + - l = g (add 1)

(e ) 2 u + 1 = 1 1 (add -1)

(f) 2 x + - 3 = g (add 3) - 9 ( g ) 4y + -3 = (add 3 )

4. (a) What number do you add (using the addition property) to solve x + 3 = 2?

,(b) What number do you add (using the add i t ion property) t o solve x + ( - 7 ) = 4 ?

- ( c ) What is the r e l a t i o n between 3 and 3 r e l a t i ve

to add i t ion?

(d) What is the re la t ion between 7 and -7 relative to addition?

- 5. (a) If x = 3 , what is x?

- what is x? ( b ) If x = p,

- - ( c ) If x = - 3 , what is x? Answer: By x, wemean t h a t -

number with the proper ty t h a t x + x = 0. So for - x = 3, we have

- 3 + (-x) = 0.

- mt we know t h a t -3 + 3 = 0. Hence x must be equal t o 3, In o t h e r words, - ( - 3 ) = 3.

- 1 - ( d ) If x = ( p ) . what Is x?

- ( e ) If x = - 7 , what is x ?

6 . Earlier in this sect ion we solved the equa t ion

4 x + 5 = 3 x + 2 -

by adding (3x1 t o both sides of the equation. Suppose

we had begun by adding -(4x) to both sides. Could t h e

so lu t ion be found this way? Do you t h i n k this is a aimpler

method of so lu t ion?

7, Slmpl i f y each of the fo l lowing : -

(a) x + 3x = ?

( b ) 3x + -x = ?

8. Solve the equation: 2x + 7 = x ,

9. Solve the equatlon: 2x + 3 = x + 2.

LO, Solve the equation : x = 2x + 6 .

11. Solve the equa t ion : 3x 4- 5 = 2x.

Adding the same number to both s i d e s of an equation is not

the o n l y way to get an equ iva l en t one, lie may a lso multiply both sides by the same number. For example:

I s a true sentence. If we multiply both s i d e s by 3, we get

This l a also true. This is an example of the multiplication property or equality.

The Multiplication Proper ty of EqualLty: If a - and b - are two equal numbers, then ca = cb. -

La it true t h a t ca = cb implies t h a t a = b ? Before

answering this ques t ion , let us consider the so lu t i on of an equation using the multiplf c a t i o n property, S ince our method of

solution 1 s very much l i k e t h a t f o r an equa t ion Involving addition, i n the le f t -hand column below we s h a l l solve an equation using t h e addition proper ty and in the r ight-hand column a similar

equation us ing t he multiplicatLon proper ty .

Problem: Solve 3 + x = 6 I Problem: Solve 3x = 6

F i r s t add the a d d i t i v e inverse of

3 , to both s i d e s or t h e equa t ion t o get

- - 3 + ( 3 + x ) = 3 + 6 .

We can use t h i s same parallel treatment to show tha t

F i r s t multiply both sides of the 1 equation by 3, the multlplica-

t i v e inverse of 3, t o get

1 1 $ 3 ~ ) = $ 6 ) .

Use t h e associative property of

addition:

( - 3 + 3 ) + x = -3 + 6 O + x = 3

x = 3

if c a = c b , and c # O , then a = b.

Use the associative proper ty of multiplicatfon:

1 1 ( 3 ) ~ = j 6 , 1 b x = 2

x = 2

Problem: Prove t h a t , Problem : For c f 0 prove I that lf c I- a = c + b, t h e n a = b, i! ca = c b , t h e n a = b .

- F i r s t add c, the additive

inverse of c, t o both sides of

t h e equalfty to g e t

- - c + ( c + a ) = c + ( c + b )

[sec. 2-24

1 First m u l t i p l y both s i d e s by -, C

the multiplicative i n v e r s e of c ,

(note t h a t t h i s inverse exists only if c # 0 ) to ge t

1 1 - . (ca) = - C C (cb)

Using t h e assoc ia t ive proper ty

of' addi t ion , we have Using the assoc ia t ive property of multiplication, we have

Thus we have shown that if c # 0, ca = cb and a = b are

equivalent equations, This means t h a t if c 0 and ca = cb,

then a = b; also if c $ 0 and a = b, then ca = cb,

Now let us apply t h i s result to the s o l u t i o n of t h e equation,

We wish t o have 3x by itself on one side of the equatfon and, a s

i n the f irst part of t h i s sec t ion , we can accomplish this by adding - 1 t o both s i d e s , This gives us

Since we wish to have an equation of the form x = some number,

we can accomplish t h i s by multiplyfng both sides by 7, the reciprocal (that i s , t he multiplicative i nve r se ) of 3 . Thus the equation

3x = 12

is equivalent t o 3 1 . (3x) = 3 • (12) t I

We have shown that the equation 3x + 1 = 13 is equivalent to x = 4. Since x = 4 has the obvious s o l u t i o n 4, so is 4 the solution of 3x + 1 = 13, From what we have done, we can be sure that if we did not make a mistake x = 4 is t h e s o l u t i o n of

the equation 3x + 1 = 13. But it is reassuring and Is also good

p o l i c y f o r us to check t h i s answer to see if it is Indeed a

solution. If we replace x by 4 in 3x + 1 we g e t 3 - 4 + 1 which is equal to 13. This i a the check we wanted.

Class Exercises 2-3c

1, Indicate whlch property, the add i t ion or the multiplication

property of equality, and which number is to be added oruaed as a

multiplie~ in solving the following equations.

( a ) x + 10 = 22 ( h ) 18 -+ y = 8.6

( b ) 6.2 + x = 1.12 ( y + 6 = 5 + 3

( d l Tx = 15 X (4 6 = ig

(f) 1 4 - x = O I

( g ) 2~ = 17

(k) 19 = 6 - Y

2. Find the reciprocals of each of t h e following numbers: 1 1

(a) 7 . Answer: - is the reclprocal since 7(7) = 1. 7 ( b ) 5

(4 -3 Answer: the reciprocal of a number b is defined (d l 5-

1 1 to be t h a t number - such that b(b) = 1. Since b

1 H(2) = 1, 2 is t h e rec iproca l of 1 2'

3. Find the addi t ive Inverse and the multiplicative inverse of

each of the fol lowing, being careful to sta te which is whlch:

(a> 3 1

(b) 2 5 (4 7

Exercises 2-3b

1. Find the a d d i t i v e inverse and the multiplicative inverse of

each of t h e following, being careful to s t a t e which is which:

2. Solve each of the following equations and s t a t e where you use

the addi t ion property and where the multiplication proper ty of

equality.

(a) 3 x + 2 = 1 4

(b) 7 x = 2 -

( c ) 3x + 7 = 22 1

(d ) 3 . = 7

- 1 ( e ) ( a ) ~ = 14.

Let us try our methods on a more complicated equat ion: - 1 1 (Z)~ + 2 = 2x + v.

We wish first to f i n d an equivalent equation in which only one

side has a term in x. Here we use the addition proper ty and add t o both sides : zx

Using t h e a s soc ia t ive property for addi t ion we have :

- 1 - I 1 Now + (T)x=O since (g)x is t h e a d d i t i v e inverse of p. Also by the distributive property,

Hence the equation ( A ) above I s equivalent t o

that is,

Since we wish the term In x to be by itself on one side of - 1 t he equat ion, we can agaln use the addition property and add (q)

to both sides of the equation tb get

Since we want an equat ion of the form, a number = x, we can use the multiplication property and multiply both sides by 2

and get the reciprocal of 2, 5'

We have, f i n a l l y (you will need to write In some steps),

- 1 1 is the solution of the equation ( T ) x + 2 - 2x + H. Thus 5 Now ge t t ing t h i s r e s u l t was rather long and there were many

o p p o r t u n i t i e s for mistakes, We should check our result , 3 - 1 If x = ~ , t h e l e f t s ide of the given equat ion, (2)x + 2, - 1 become s 3 (H) * - + 2 = 20 17

5 -(&) + 2 = - + - = - 10 10

1 the rlght side of the given equation, 2x + 2 rr x = F , #

becomes 3 1 6 1 1 2 5 - 1 7 5 5 + 2 = - + p = - + - - - 10 10 10

This shows that f o r x 1 -($)x + 2 is equal to 2x =S' There are other methods of solving this equation, One would

be t o begin by multiplying both s i d e a of the equation by 2, to

get rid of the f rac t ions . You are asked to try this out in an

exercise below.

Class Exercises 2-3d

1, What property Is used, and how is it used, to get the second

equation from the f irst?

m p l e : (1) 2x + 4 = 7 - ( 2 ) Zx = 3 addi t ion property, adding 4.

( a ) (1) (&I + 1 = 1 (f) (1) 5.x = 10

(2) & = 0 1 (2) -x = 5 5

2. Solve and check each of the following equations:

( a ) 3y + -2 = 7 X (4 2 =

(b) 7 = 3x + 1 (f) 0.14 4- x = 5,28

L - ( d ) Tt - l a 7 = 1.3 (h) x = 7 - a

Exercises 2-3c

1, Solve the following equations by using the propert ies of

"equals. I' Give your reason for each s t ep . (a) 2x + 1 = 7 ( c ) 8 - 3 = -4

(b) y - 2 = 6 (d) 3x - 5 = - 4

2. Solve the following equations.

(a) x + 3 = 5 (el y - 3 = 5 -

( 1 3 + y = 5 -

(f) 3 - u = 5

( c ) 2 ~ - + 3 ~ 5 ( g ) 2~ - 3 = 5

( d ) 3 + 2m = -5 (h) 3 - 2s = -5

3, (a) In solvlng t h e equation 9x = 27 what number would

you use as a multiplier?

= 4 what number would ( b ) In solving the equation ~x you use as a multiplier?

( c ) In solving the equation what number would you use as a multiplier?

TX = 2

' relative ( d ) What is the relationbetween 9 and to multf plication?

relative ( e ) What is the r e l a t i on between 3 and g, to multiplicatlon?

4 '5

relative (f) What is the re la t ion between - and t, t o multiplication?

4. In solving an equation such as 3x + 1 = 9, you have

learned to use the addition property f irst (to f lnd 3x1 and the nu1 tipllcation property second ( to f i n d x ) . Sometimes you will find it best to reverse the order in which you w e these properties . Solve the following equations by uaing the multlplicatlon property first.

(b) 7 ( x - 2) = 13

1 1 5. To solve t h e equation T ( - x ) + 2 = 5 + 2x, beg1 n by multiplying both aides by 2 and get the equivalent equation

-x $ 4 = 1 + 4x.

Then f i n d the so lu t ion of this equation. Does it agree w i t h that in the previous discussion? lh you think th ia method is easier than the one used?

2-4. Solving mequa11 ties

In Sec t ion 2 , we found by inspect ion s o l u t i o n s of var ious inequalities.. The methods which we used to solve equalities in the

previous sec t ion may ge used for ine~ualitiea as well. To show the similarity, l e t us solve an equation and a related inequality.

To solve: x + 4 = 7 I To solve: x + 4 < 7

Add 4 t o both s i d e s of the equation, using the addit ion property of equality:

(x 4- 4) + -4 = 7 + -4

We h o w that each equation on the left is equivalent to a11 the others. We assumed on the right that the same statements could be made about the inequalities. We need to show the addit ion property of inequality:

- Add 4 to both sides of the inequality, using the "addition property of inequality"

(x + 4) + -4 < 7 + -4

Using the associative property of addition,

The Addition Property - of k e q u a l l t y : - If a ( b, then

a + c < b + c .

Using the associative property of addition,

To show t h i s , first suppose c = 5. men a ( b meana that on

t h e horizontal number l i n e , the point which a repreaenta is to the left of the point which b represents. Now the p o i n t uh ich

a + 5 represents is 5 units to the right of the po in t which a represents; the p o i n t which b f 5 represents I s 5 units to

the r fght of the point which b represents. Hence the po in t which

a + 5 represents is to the left of the point which b f 5 represents. This means

- Second, if c were 5, the p o i n t which a + 45 represents is 5 units to the l e f t of the point which a represents, and

s imi lar ly for b + -5. Again

See the f igure below:

! "+"! ! ,

I' b b+5 b +-5 b

In general, if c is a p o s i t i v e number, the point; represented

by a + c is c units to the right of the p o i n t represented by

a and s imi lar ly f o r b + c and b. 'I%is, if c is a posftive number,

and a ( b then a + c ( b f c

Why would the same result ho ld if c. were a negat ive number? Thus

we have shown the addition property of inequality. Just as i n t he case of equality we can show:

E a ( b , then a + c ( b + c and if a + - c ( b + c , then a < b.

This shows tha t if we add the same number t o both sides of an inequality, we have an equivalent inequality. For instance the

inequality

x + - 3 < 8

is equivalent to the inequality ( x + '3) + 3 < 8 -I- 3, that is,

&ercises 2-4

1. Find the set of solutions of each of t h e following inequalities:

(a) x + 5 < 7 ( d ) y + -3 < 10 ( b ) 7 > x f 5 ( e ) 10 < y + -3

( c ) x + -2 < 8 (f) 2 x + 3 ( x + 2

( h ) x + 4 > 5 + x

(i) 3x -t 2 < 2x + "3 2. Show that if c is a negative number and if a ( b, then

a f c < b + c .

3 . Use the addi t ion property of inequality to show that If a + c ( b -I- c, then a < b. (!Phis may he done by t he same

method whf ch we used for equalities. )

4. If a > b , is it t rue that a + c muat be greater than b -k c ? E so, show why. If not , show why not.

5+ If a + b must it be true t h a t a + c # b + c ? Give reasons for your answer.

6 . If a < b must it be t r u e that 2a < 2b? Show why this is

true.

* 7 . If a < b, must it be true that (-2)a < (-2)b? Why or why

not?

2 - ISumber Sentences with Two Unlmowns -- In the previous examples of number phrases and number sentence

there was only one unlmorm number. We could also have more than one unhown number, Look at t h i s sentence:

If x = 3 and y = 5, is the sentence true or false? If x = 7 , what must y be f o r the sentence to be true? If y = 6, then - we have x + 1 = 6 . What must x be in order for t h i s sentence to be true? Row did you learn to solve an equation like x + 1 = -6 in Section 2-31

Each so lu t i on of the equation x + 1 = y is a pair of numbers. We can make a table l i s t i n g some of these pairs:

Table for x + l = y

Before you continue reading, copy t h i s t a b l e and work out the r n i s s i n ~ numbers. For example, to fill in the third Tine, replace x by 2 in the above equation. Aak yourself, "What are t h e

Possible values of y?"

You should read the rest of this chapter with penci l and paper

handy, DO n o t go on to a new paragraph u n t i l you have answered a l l the ques t ions in the paragraph you have Jus t read. In much of t h i s

section you will find it convenient to use graph paper and a ruler, too.

E x = 0 and y = 1, then the equation x 4- 1 = y is true. Hence, we say that the pair (0, 1) i s a so lu t i on of the equation,

Notice that I t makes a difference which number i s named first. The pair (1, 0) l a not a so lu t i on since if x = 1 and y = 0, then

x + l = 1 + 1 = 2 ( n o t 0)

so that the equatlon x + l = y

is not true. You remember from Chapter 1, t h a t a pair in which the objects

are considered i n a definite order 1s called an ordered pair. Tne ordered pair (2, 7) is the same as t h e ordered pair

(x, y) if x - 2 and y = 7 , and only then. Tnis pa ir 2s

different from the ordered pair (7, 2) . The so lu t i on set of t he above sentence

is a s e t of ordered pairs of numbers. For what number y is the

ordered p a i r ( 2 , y) in t he so lu t i on s e t ?

In order to p i c t u r e the so lu t ion s e t on your graph paper, pick

out two l i n e s f o r the X-axis and t he Y-axis as in Chapter 1, and draw them In heavi ly wi th your penc i l . l a b e l t h e vertical and

horizontal lines as shown. V

Mark off an Your graph paper the pofnts ( 0 1 1 , 2 , etc. ,

whose coordinates are in the so lu t ion s e t of x + I = y. What do

you not ice about them? A l l of them l i e on a simple geometric f igure . To what set of po in t s does the so lu t ion s e t correspond? In Chapter 1 you learned to c a l l it the graph of the given number

sentence, or equation. The graph of x + 1 = y is shown here :

Let us try another example. What 1s the s o l u t i o n set of the equation

If we give x a cer ta in va lue , we obtain an equation to solve f o r y. If we t r y a d i f f e r e n t value for x we obta in a different

equation to solve f o r y. Similarly, if we t r y d i f fe ren t values for y, we obtain different equations t o solve f o r x.

For example : k t x = -3. ?Chis gives t he equation 2(-3) + y = "1 - -

6 + y P 1.

We solve t h i s equation by the methods we learned in Section 4.

6 + ('6 -+ y) = 6 + -1 by the addit ion property,

then

SO

by the associative property of addition

Thus, 5 is a solut ion, the only s o l u t i o n , -

Thus, (-3, 5) is a solution of t he equation 2x -k y = 1.

Follow this example to complete the table of solutions of

2x + y = -1 below. Perhaps you can do some of the steps in your head.

When t h i s table 1s completed we have seven ordered pairs of numbers which are in t h e s o l u t i o n set of t he equation 2x + y = -1. Choose an X-ax is and a Y-axis on your graph paper

and locate the points whose coordinates are these ordered pairs. Do a l l of them seem to lie on a l ine? Draw the l i n e . Locate a

point on the l i ne which is not one of the seven which you have p lo t t ed . Can you f ind the coordinates of this point by measuring certain distances? The coordinates form an ordered pai r of numbers. If your drawing and measurement were perfectly accurate, thia ordered pair would also be In the solution set

of the equation. Is it?

From the examples which you have seen in t h i s section and in Chapter 1, you have perhaps guessed that the graph of any

equation of the form

where a, b, and c are hown numbers, liea on a line. This

is t r u e . For that reason we uaually call an equation of t h i a

type a l inear equation.

Exerciaea 2-5a

1. Make up a table showing some of the ordered pairs from the s e t of solutions for each of the f ollowlng equations, On the

same s e t of axes draw graphs of each of the equations. -

y = x + l , y = 2 x + l , y = 3 x + l , y = 2 x f 1 .

- 2 Do the same for y = x -I- 1, y = x + 2, y = x + 3.

- 3 . Do the same f o r .x + y - 0, x -k y = 1, and x 3. y = 1.

X 4. Do the s a m e for 5 4-5 = 1 and - - $ = 1. 3

5. Do the same for y = x i -1 and x + y = 1.

6. D o t h e s a m e f o r y = 2 x + 3 and y = - l x ) + 3 . (I

kt ua try another problem in which two unknowns are involved. What are the p o s s i b l e lengths of the s i d e s of a rectangle if the perimeter is 16 inches? If we l e t the length8 of two adjacent

a i d e s be called x inches and y inches theri we must have

We must be careful, however! Thia equation is not a complete translation of the real situation into mathematical language. Can the length of a side of a rectangle be a negative number of inches? Can it be zero Inches? We muat have x > 0 and y > 0, and the number sentence which really describes the situation is t h i s :

2x + 5 = 16 and x ) 0, y > 0. We can find several ordered pairs in the aolu t ion set . Which

of the following pairs are solutions?

Remember that to be a solution, an ordered pair must make the entire number sentence,

2x + 2y = 16 and x > 0, y > 0, true.

The graph of the equation part of the above number sentence lies on a line. Sketch it. The second part of the sentence, namely, "x > 0, y > 0, " says that the point corresponding to conditions stated above muat lie In which quadrant? The graph of

the number sentence 1s the part of the graph of a + 2y = 16 that Ilea in the first quadrant.

The graph is a segment with I t s endpoints removed aa shown

here :

Which of the above so lu t lons of the number sentence have been plot ted on the graph in the figure?

In this example we had a number sentence which was b u i l t of

several shorter number sentences, one of which was an equation. L e t us try another number sentence of this type.

Bonnie has in her purse 3 dollars in dimes and quarters. What possible combinations can she have?

Let d be the number of dimes and q be the number of

quarters. Just as the value of 3 dimea 18 10 ' 3 centa, so

the value of d dimea f a 10 ' d cents. The total value of these coins is (10d + 25q) cents, and t h l e must be equal to 3 do l lara . But, wait a minute! We muat make up our minda

whether we want t o measure our money in cents or dollars. Le t us use cents throughout. Then, 3 dol lars is 300 cents. Therefore, the pai r (d,q) I s a solution of the equation

Again we must be careful! Bonnie cannot have twenty-seven - and one-half dimes, nor can she have 3 dimes. The unlmown numbers in this problem muat be non-negative integers. The number

sentence which really describes thfs situation is:

10d + 25q = 300, and d and q are non-negative integers.

[aec. 2-51

The solution set of this number.sentence is made up of the seven ordered pairs :

(0, 121, (5, 101, (10, 81, (15, 61,

(20, 41, (25, 21, (30, 0).

The graph of this number sentence consists of only seven points.

You may disagree with the above s o l u t i o n on one poin t , s ince there is another way to in terpre t t h e statement, "~onnie has in

her purse 3 dollars in dimes and quarters.'' Does t h i s include the possibility of her having no dimes and 25 quarters? Some

people would say "yes" and others would say "no. " If your answer is "no," two solutions would be excluded: no dimes and 25

qua r t e r s , and 30 dimes and no quarters. These are represented by the ordered pairs: (0, 12), and ( 3 0 , 0).

We have found t h e graphs of cer ta in equations. Now l e t us see what the graph of an inequality looks l i k e . To do t h i s ,

first compare the table of solu t ions of y = x + 1, which we have already found with t h e table of solutions of y > x + 1.

This shows that for x = 1, f o r instance, y can be any number greater than 2; not only can y be 3, 4, 5 , and so fo r th ,

but alsa 2 , 2 , 2 . . n u s the graph of y > x + 1 is as

shown in the f igure. The graph of the inequality does not include the line i t s e l f . In such cases the line is shown as a broken or "dotted" line.

(The dotted line is the graph of y = x + 1. No points from t h i s

line are in the graph of y > x + 1.)

What is the graph of the following inequality w i t h two unknown numbers ?

The inequality says that t he product of x and y must be posltive. What do you know about two numbers whose product is a posi-

tive number? Can either one be zero? Can x be p o s i t i v e and y

negat ive I n a pair (x, y) which is a solution? The pair (x, y)

i s a so lu t ion i f both x and y are or if both x and

y are . You f i l l in the blanks. The graph of t h i s

i n e q u a l i t y is, then, the ent i re and quadrants, Y

A

Let us consider one equation which is not a l i nea r equat ion. Consider

2 y = x .

If we take a h o r n value of x in t h i s equation, t he r e s u l t i n g equat ion in the unhown y is n o t hard to solve. Fill in the

t a b l e of values.

Plot these po in t s on your graph paper. Then sketch the graph of

the equation.

This curve is called a parabola. It is a very important curve which occurs In many ways i n problems about natural events.

The p o i n t s on thla curve are t he points whose coordinates (x, b y )

2 are s o l u t i o n s of y = x . All points w i t h coordinates (x, x2), where x stands f o r any number, lle on the parabola. Where will a p o i n t with coordinates (x, y) lle if y ) x2? If a p o l n t (x, y) lies above t h e parabola what can you say about y and x2?

Which must be greater? The so lut ion set of the nwnber sentence

2 Is the s e t of all ordered pairs (x, y) f o r which y )x , and

the graph of t h i s open sentence is contained I n the region of the plane above the parabola which we sketched above, It does not i n c lude the curve itself. me fol lowing figure is the graph of

2 Y > x .

mercl ses 2- 5b - 1. (a) Draw the grapha of t he following equations on the same

2 set of axes: y = x2 and y = -(x ) . 2 2 (b) Do the same f o r y = x and y = x .

- (c) Do the same for xy = 1, xy = I, and xy = 0.

2. Sketch the graphs of the following:

(a) x + y = 1 and x > 0, y > 0. (b) x + y = 10 and x and y are positive integers.

- ( c ) y = x" and x < 1.

( d ) y = 1. (~int: t h i s is the same as y = 1 + (0 . x). 2

( e ) y =I.

( f ) x = 1.

(9) x2 = 0.

(h) x = 0 and y = 0.

* ( J ) y e the larger of the numbers x + 1 and 2 - x. -

*(k) Y = x when x 2 0 and y = x when x < 0. 3. Consider the number sentence

11 , and x and y are non-negative integers,"

with each of the fol lowing equations filling in the blank. W s t the solution s e t In each case, and write the number of solutions which t h e sentence has,

(a) x + y = 1.

( b ) x f y = 2.

( c ) x + y - 2 0 .

( d ) x f 2y = 0.

( e ) x + 2y = 1.

(f) x + 2 y x 2 .

( g ) x + 2 y = 3 .

(h) x I - 2 y = 4 .

(i) x + 2y = 25.

( J ) 5 X + 7 y z 3 5 .

(k) 3 + 7y = 36.

(1) S x + 7 ~ ' 3 7 .

4 . A chain store has 5 tons of coffee in its warehouse in New Orleans. It sends 5 tons to San Francisco and n tons

to New h r k . Not a11 the coffee is sent to either place. The

t o t a l amount shipped is the ent i re warehouse supply. Write a

number aentence in eymbola which describea the relation between a and n. Gn a pair of axes labeled "s l ' and "n"

draw the graph of th i s number sentence.

5. D r a w the graphs of the following inequalities:

( a ) Y < x 2 (4 Y < -(x2)

(b) Y > 4x2 (d l y2 , x.

9 6. In an earlier problem you used the relationship F - 9 + 32

between the temperature reading of a Fahrenheit thermometer

and the reading of a Centigrade thermometer placed in the aame spot. Draw a pair of axes w i t h t he vertical one labeled

[sec. 2-51

F and the horizontal one labeled C. Choose a mall enough unLt distance and use a large enough piece of paper so that - each axis has points 50 and 50. Make a careful drawing of the graph of the above equation. Then answer questions (a) and (b) by measuring certain distances on your drawing.

(a) What is the temperature reading on the Fahrenheit

thermometer if the reading on the Centigrade thermometer d -

is 25 degrees? 15 degrees? 0 degrees? 4 degrees?

(b) What is the temperature reading on the Centigrade thermometer if the reading on the' Fahrenheit thermometer - - ia 30 degrees? 15 degrees? 0 degrees? 50

degrees ?

( c ) Check your answers by solving the appropriate equations. Remember that It is impossible to make a perfectly accurate drawing or measurement. how much did your

anawers In parts (a) and (b) differ from your answers in this part?

'7. The rate for first-class mail in t h e United States (according

t o the act of Congreaa in 1958) is f o u r cents per ounce or f r a c t i o n thereof. That is, a letter weighing not more t h a n

one ounce cos ts four cents, a letter weighing make than one ounce but no t more than t w o ounces costs eight cents, and so

f o r t h . On a pair of axes w i t h the ver t lca l one labeled c,

f o r the c o s t in cents, and with the hor izonta l one labeled w, for t he weight in ounces, draw the graph of the cost of

first-class mail of all weights up to 6 ounces. The first part of the t ab l e is as fol lows:

*w may be 1 or any number less than 1, but it must be grea te r than zero, Shilarly, w is greater than 1 and any number less than or equal t o two,

W

w > O and w j l

l < w < * 2 < w i 3

C

4 8 12

8. aRAINBUSTER. Carolyn asked Edward what the temperature i n the

freezer was. Edward told her, and she asked, "~ahrenheit or centigrade?" He answered, "Both! The readings are the same. "

What was the temperature In the freezer?

9. B;RAiXBUSTER. The income tax l a w of a cer ta in country can be summarized as fo l lows:

A person's b x 1s either (a) $400 less than 20% of his income or (b) zero dollars , whichever is greater.

( a ) L e t T be the tax in thousands of dollars on an income of I thousand dollars. Write the number sentence whfcn expreeses the re la t ion between T and I.

(b) Draw two axes with the vertical one labeled T and t h e

horizontal one labeled I. Use $1000 as your unit so

that, for example, a distance of 4.850 represents $4850. Draw the graph of the number sentence In ( a )

( c ) By measuring the distance on your graph, answer the

following questions:

What is the tax on an income of $10,000?



If a man pays a tax of $1,500 what is h i s Income?

( d ) Check your answers in part ( c ) by us ing the number

sentence in part (a).

Review &ercises

Write open sentences stating the condition for each of t he

following problems. Then f i n d t h e s e t of solutions for each.

1. A board 45 inches long i s t o be cut into two pieces so that one piece is 3 inches longer than the-other. Find the

length of each piece .

2. The width of a rectangle i s 10 u n i t s less than the length. If the perimeter of the rec tangle Is 68 u n i t s , what are t he dimensions of the rectangle?

3. A man left a t o t a l of $10,500 f o r hLs w i f e , son, and

daughter. The wlfels share i s $6,000; The daughter received tw ice as much as the son. How much does the son receive?

4. On Thursday a boy read an add i t i ona l 15 pages of a novel. A t this point he found that he had read less than f i f t y - o n e

pages. How many pages could he have read p r f o r to Thursday

of t h a t week?

5 The manager of a hobby store bought 500 k i t s of model planes t o se l l i n his store. After selling some of the k i t s , he took

an inventory and learned he had fewer than 100 k i t s remaining. How many of the 500 kits could have been sold?

6 A student checked the enrollment at h e r school and learned that during the previous- year the number of g i r l s was never t h e same as the number of boys. How many girls could have been enrolled in the school If the number of boys enrolled

was always 176?

7. me number is three times a s large as another number. The i r

difference is 12. What is the smaller number?

8, Dick has $2.73 i n pennies, nicke ls , dimes, quarters and

half-dol lars , He has the same number of each of the different

kinds of coins. How many of each kind of coins does he have?

9. Alice and Don ran f o r class president. Alice got 30 votes more t h a n Don, but not all 316 s tudents voted. How many votes did Don g e t ?

A s e t of twins f i n d that five years ago their combined ages

t o t a l e d 18. How o l d are they now?

The number of $1 b i l l s is f l v e times t h a t of the $5

b i l l s , and the number of $10 b i l l s is twice t h a t of the $1 b i l l s . How many $5 bills are there if the t o t a l

number of b i l l s is 48?

Let x represent the first of two numbers and l e t y repre-

sent the second number. Write an equatfon expressing t h e

following condit ions.

( a ) The sum of the numbers is 7.

(b) If the second number is subtracted from the f irs t

number the result is 5,

(a) Make a table showing some of the ordered pairs from t h e s e t of aolu t lons for x + y = 7.

(b) Make a table showing some of the ordered pairs from the set of solutions f o r x - y = 5.

( c ) Draw a graph of each equation in parts (a) and (b), using the aame s e t of axes,

(d) Note where the l i n e s intersect, What ordered pair

is associated with t h i s point?

(a) Can you f ind a member of the set of solutions f o r

the compound aentence "x 4- y = 7 and x - y = 5?"

(b) Is there more than one ordered pa i r in the s e t of

so lu t ions f o r the compound sentence in part ( a ) ? &plain.

Find the s e t of solutions for the compound sentence

" x + y = O and x - y = 0.''

Find the s e t of solut ions for the compound sentence "x i- 1 = y and x - 1 = y, I I

Find three consecutive integers such that the sum of the f i rs t and the t h i r d is 192.

*18. The sum of the number of degrees in the measures of two

congruent angles of a t r i ang le i s equal to the number of degrees in the measure of the t h i r d angle. What are

the measures of the angles?

*19. The Jones family and the i r neighbors, the Sni th family,

are going on vacations. The two familles w i l l t r a v e l in

opposi te d i r ec t ions , If the Jones family averages 55 miles per hour and the Smith f a n l l y 45 miles per hour,

when will they be 750 miles apart if they start at t he

same time?

'20. The radar operator on an aircraft carrier detects a contact moving directly toward the carrier. He estimates the

distance to the contact a t 400 miles and the speed of

the contact at 350 miles per hour, How long will it

take one of the carrierls planes to intercept the contac t

if it flies d i r e c t l y toward the contact at 450 miles

per hour?

*21. In Problem 20, assume t he contact is observed at a

distance of 10 miles moving d i r ec t l y away from the carrier, A plane from the carrier starts pursuing the contact. Using the speeds given in Problem 20, how long w i l l It take the carrier plane to overtake the contact? How far w i l l t h e carrier plane be from its base a t t h e

time of con tac t?

Chapter 3

SCIENTIFIC NOTATION, DECIMALS, AND THE PBTRIC SYSTF,M

3-1. Large Numbers and S c i e n t i f i c Notation - In this text we use small numbers, whenever poss ib le , to

make the problems easy fo r you t o do, to make the ideas easy to

understand, and to make the homework easy for your teacher to grade! But numbers which arise in everyday situations are of ten very large or very small. Todayt s newspaper ( ~ u n e 28, 1961), f o r example, mentions the 12,500,000 members of a labor federation, $2,484,000,000 in sta te granta under the National Defense Education A c t , and a nat ional debt of $298,000,000,000. You probably enjoy

at least 3,888,000 seconds in school each year. No doubt you can t h i n k of m y other common uses of large numbers. How many times do you think a heart beats in an average lifetime? How far will

a sa te l l i t e t ravel in 10 years, if it moves at a speed of 50,000

miles per hour?

Numbers as large as a million or a b i l l i o n now occur frequently

in such circumstances as those mentioned above, Actually, we have

names f o r numbers larger than one billion, such as trillion and

quadrillion. Consider the numeral 3141592653589793; such a numeral is hard to read when m i t t e n in this form, One way to

make i t easier to read is to place a comma, to the l e f t of every

t h i r d d i g i t counting from the right as follows:

This separates the number of thousands, millions, billions &nd other major units in a natural way. Although we insert commas f r o m right to le f t , we read the number from l e f t t o r i g h t according

t o the diagram which appears on the following page.

Thus, we read this number as follows:

Three quadrillion,

one hundred forty-one trillion, f i v e hundred ninety-two billion,

six hundred fifty-three million,

five hundred eighty-nine thousand, seven hundred ninety-three. In reading such a number we have to be careful not to use

the word ''and." We can see the reason f o r t h i s if we consider the number 593,000 and how it m i g h t be read. If it were read "five hundred - and ninety-three thousand," as it is by many people, there might be some misunderstanding. If "and" is associated w i t h addi t ion , the meaning would be 500 plus 93,000. If "and" i s interpreted as it is in ordinary English, the meaning would be the two separate numbers, 500 and 93,000. Therefore, it is preferable to read 593,000 as " f ive hundred ninety-three thousand. 'I Omitting the "andrt avoids misunderstanding. We usually do use the "and" to

mark the decimal poin t ; e . g . , 563.12 is read "five hundred s i x t y - 11 three - and twelve hundredths. This use of "andtT does not cause

confusion since 563.12 means. 563 + 0.12. Actually, such numbers as these seldom occur. This does not

mean that numbers o f ' t h i s size are not used, The po in t 2s that we rarely can count precisely enough to use such a number, We

would jus t say that the number counted 1s about three quadrillion.

For example, the population of a city of over a million Inhabitants

might have been gfven as 1,576,961, but t h i s just happened t o be

the sum of the varfoua numbers compiled by the census takers. It I s cer ta in tha t the number changed while the census was being

taken, and it is probable that 1,577,000 would be c o r r e c t to the nearest thousand. For thia reason there is no harm In rounding the o r i g i n a l number to 1,577,000, In fact, for most purposes, we would merely say t h a t the population of the c i t y is "about one and one- half mil l ion , " which could be written also:

City Population W 1, 500,000.

The symbol % t i used to mean "is approximately equal t o . "

There are other ways of writing such large numbers. Many times there are de f in i t e advantages in doing thia. A suggestion

of one possible way to write a large number is given by our statement "one and one-half million. " One m i l l i o n can be written

1,000,000 (but there are lots of zeros in this form!) or

10 x 10 x 10 x 10 x 10 x 10 (even worse, l s n l t it?) or 10 6

(there, isntt that neat?), The Indicated product 10 x 10 x 10 x 10 x 10 x 10 is sometlmea read "the product of s ix

6 tens." When we write it i n exponent form, 10 , the number of 101 s used as factors in the product is indicated by the exponent 6. we can also get the exponent 6 by counting the number of zeros in the numeral 1,000,000.

9 In the same way we write one b i l l i o n aa 1,000,000,000 or 10 . Thus, the national debt of 298 billion dollars could be wri t ten

as 298 x 10' dollars. This way of representing large numbers in terms of powers of 10 1s developed fur ther In the following class exercises.

Class Exercises 3-la - 1. Wrlte the following as decimal numerals and also in exponent

6 form. Example: one millfon = 1,000,000 = 10 , (a) one billSon (b) one triilion (c) one quadrillion

2, S h c e 2000 = 2 x 1000, we can write 2000 in exponent form 3 as 2 x 10 . Using an exponent, mlte the following:

( a ) 7000 (d) ~2,500,000

(b) 50,000 ( e ) 2,484,000,000 (c) 3,000,000 (f ) 506, 000, 000

3. A number l i k e 1500 can be expressed in several ways : 150 x 10 3 or 15 x lo2 or 1.5 x 10 , Slmllarly 325 can be written

as 32.5 x 10 or 3.25 x 1+. Also 298 billion is the same as 298 x 10' OF 29.8 x lo1' or 2.98 x l d l . In each of these examples the last expression is of the form

a number between 1 and 10) x (a power of 10). (- --- Write each of the fol lowing in t h i s form.

(a) 76 (a) 8,463,000 (g) 841.2 (b) 859 (e) 76.48 (h) 9783.4 (4 7623 If) 4832.59 (i ) 3,412,789.435 ( J ) f i f ty- thme billion, six hundred forty-two mil l ion ,

f i v e hundred thousand,

A 8 we remarked, we can wrl te 298 b i l l i o n as 298 x 10' or as 2.98 x 10". Tilese are compact ways of writing the number. Also , it is easy to compare several large numbers written in this form. or example, we can t e l l a t a glance that 4.9 x 1013 is

b l g g e ~ than 9.6 x 1o12 without counting decimal places in 4900000#000000 and 9600000000000. We shall see later on that I t

often simplifies c a l c u l a t i o n s with large numbers t o work with them i n such a standard form. This is especially true of computations by slide rule or by logarithms, as you w l l l learn in high school,

For these reasons it is common practice in scientific and engineer- ing work to represent numbers in this w a y , namely in the form

(a number between 1 and 10) x (a power of 10) .

The number is then safd to be written in s c i en t i f i c notation. If a n u m h r I s a power of 10 then the first fac to r is 1 and we

4 4 usually do not write it. Thus 10000 = 1 x 10 = 10 and

7 lO,OOQ,000 = 10 in scientific nota t ion ,

Definition, - A number is said to be expressed in scientific ---- - notation If it is written as the product of a --- -- -- decimal numeral between 1 and 10 and the proper - -- power of 10. If the number is a power of 10, - - - the first factor l a 1 and need not be written. -- - ----

Remark: Sometimes "scientific notation" is referred to as II powers-of-ten" notation. We shall occasionalLy use t h i s term.

Class Exercises 3- lb - - 1. (a) Ie 15 x lo5 in scienti f ic notation? Why, or why not?

(b) Is 3.4 x lo7 in scientific nota t ion? Why, or why not?

(c) Is 0.12 x lo5 in scientific notstion? Why, or why not?

2. Write the following in scientific notation:

(a) 5687 (bl 1 4 1 (c) % million 3, Write the f o l l o w h g in decimal notation:

(a ) 3 . 7 ~ 10 (b) 9 . 7 ~ 1 0 ~ ( c ) 5.721 x lo 6

4. Since the earth does not travel in a c i r c u l a r path, the distance f r o m the earth to the sun varies with the time of the year.

The average distance has been calculated to be about 93,000,000 railes. (a) Write the above number In scientific notation. The

smallest distance from earth to the a m would be about 1$ % leas than the average; the largest distance would

1 be about IF% more than the average.

(b) Find I$% of 93,000,000.

(c) Find approximately the smalleat distance f r o m earth to sun.

(d) Find approximately the largest distance from earth to sun.

[sec. 3-11

(e) Write the numbers found in Parts (c) and ( d ) I n powers- of-ten-notation.

Note that 146,000 = 1.46 x lo5 could also be written as 1.460 x lo5 or 1.4600 x lo5. Each of these represents 146,000 in sc ien t i f i c notation. Although there are situations i n which

we wish t o write one or more zeros a f t e r the tt611 I n 1.46, we

shall no t do it In this chapter. However, as you w i l l learn later, in the chapter on re la t ive error, scientists use the first f a c t o r in this powers-of-ten notation to Indlcate the prec is ion with which a quantity has been measured.

1, Write the following in powera-of-ten no t a t i on :

(a) 1,000 (a) 10% 107 (b) 10' x lo1 x lo1 x 10 7 ( e ) lo x 105 ( c ) 10 x 10 x 10 x 10 (f) 10,000,000

2. Write the following I n sc ien tLf ic notation:

( a ) 6,000 { e l 78,000 (b) 678 (f) 600 x 10

( c ) 9,ooo,ooo,0Uo (g) 15,600 (d) 459,000,000 (h) 781 x lo7

3, The total number of stars which can be photographed using

preaent telescopes and cameras is estimated t o be about 506,000,000. Write this number in scientific notation.

4. Write a numeral f o r each of the following in ordinary decimal form (which does not use an exponent or indicate a product): (a) lo5 ( e ) 6.3 x lo2 (b) 5 . 8 3 ~ lo2 (f) 8.2001 x lo8 (0) 3 x 10 4

( g ) 436 x 10 6

(d ) 5.00 x lo7 (h) 17.324 x lo5

[sec. 3-11

5 , wri t e the following, uslng words:

{a) 362.362

( e ) 4,000,284,632

(r) 4.2506

6. Round each of the following to the nearest hundred. Express

the rounded number i n scientific notation.

(a) 645 (a) 70,863 (b) 93 (e) 600,000 (c) 1233 (f) 5,362,449

30 7. Thl r ty percent of 500 is equal to x 5.0 x ?

8. The volume of the body of the sun has been estimated a s about

337,000 mil l ion million cubic miles, Write the number of cubic miles In s c i e n t i f i c notation,

3-2, Calculating with large Numbers - Not only is scientific nota t ion shorter in many cases but it

makes cer ta in calculations easier. We shall start with some rather simple ones. Suppose we want to f i n d the value of the pmduct: 100 x 1,000, me first factor I s the product of two tens. The second is a pmduct of three tens, so, we have 100 = 10

2

3 and 1000 = lo . Then,

2 100 x 1000 = 10 x 10 3

Hence lo2 x lo3 = lo5. Notice that the exponent 5 is the sum of the exponents, 2 and 3,

Let US look at another example : 1,000,000 x 100,000. Written 6 in s c i e n t i f i c notation t h i ~ is 10 x lo5 How many times does 10

' 6 appear as a factor in this product? IB 10 x. equal to lo1'?

This is one hundred billlon, but ft is slmpler to leave the number

in the form 1011 than to write a "1" followed by eleven zeros. Notice that again we added the exponents,

Suppose we wish to ffnd the pmduct of 93,000,000 and 10,000. In scientific notation, thia would be:

7 4 (9.3 x lo7) x lo4 = 9.3 x (10 x 10 ) which property?

= 9 . 3 x 10 11

Now t r y a more difficult example:

93,000,000 x 11,000 =

7 = (9 .3x 10 1 x (1.1 x 10 Note: me order ? -the fac to r s

= ( 9 4 3 x 1.1) x (10 x 10 hasbeenchange*

= 10.23 X 10 11 by using the associative and

= (1.023 x 10) x lo1' commutative properties of

11 = 1,023 x (10 x 10 1 multiplication.

In studies of astronomy and space flight, especially, we

encounter very large numbers, The planet Pluto has a mean distance from the a m of about 3666 million miles or 3.666 x lo9 miles. Distances to the stars are usually measured in "light years. '' A

lfght year is the distance that l igh t travels in one year, This

i e a good way to measure such distances. If we expressed them In miles, the numbers would be so large that it would be difficult to write them, much less understand what they mean. But suppose we wish to estimate the number of miles there are in a light year, This will be done in the following class exercise.

Class Exercises 3-2 - 1, It has been determined that light travels about 186,000 miles

per second. In parts {a) to (d) below, do n o t perform the

multiplication, just indicate the product (an example of an 4 indfcated product is 2.4 x 10 x 56 x l o ).

Using 186,000 miles per second as the speed of light,

(a) How far would light travel in 1 minute?

(b) How far would light travel In 1 hour?

( c ) How far would light travel in 1 day?

(d) How far would light travel in 1 year?

( e ) Find the number written In Part (d) and show that 12 when "rounded" I t i a 5.9 x 10 .

(f) The number written in P a r t ( e ) 18 about 6 ?

(g) Why is the number written in Part (d) not the exact number of miles that light travels in one

year? T r y to give two reaaona.

Exercises

1. Mult ip ly , and express your answer in scientific n o t a t i o n :

(a) 6 x lo7 x l o 3 ( e ) l o2 x lo5 x 7.63 (b) 1013 x 12 x lo5 (f) 60 x 60 x 60

4 ( e ) 10 x 3.5 r ( g ) ~ 3 ~ 1 0 ~

( d ) 300 x lo5 x 20 (h) 9 . 3 l o 7 10 lo6 2. Multiply, and write your answer in scientific no ta t i on :

( a ) 9,000,000 x 70,000 (c) 25,000 x 166,000 (b) 125 x 8,000,000 ( d ) 1100 x 5 x 200,000

3. Sound t ravels in air about one-f if th of a mile In one second .

Answer the following questions, assuming t h a t a sFace ship

t r a v e l s at a ra te of speed f i v e times the speed of sound.

I n d i c a t e your answers in s c i e n t i f i c no t a t i on . ( a ) How f a r w i l l sound travel in a l r in one day?

(b) How far w i l l the space s h i p t r a v e l in 20'hours?

( c ) How fa r w i l l the space s h i p t r a v e l in TO days?

( d ) How far w i l l t h e space s h i p travel I n 2 years?

Is t h i s f a r enough to reach the sun?

[aec . 3-21

4. The distance f r o m the North Pole to the equator ia about 10,000,000 meters.

(a) Expresa, in meters, the diatance around the earth through the Poles in scientific notation.

(b) A meter is equal to one thousand millimeters. Express in scientific notation the distance in millimeters from the North Pole to the South Pole.

( c ) One inch le about the same length as 2: centimeters. About how many centimeters w i l l equal a distance of 40,000 feet?

5. The distance around the ear th at the equator is about 25,000 miles. In one second, electricity travels a distance equal to about 8 times tha t around the earth at the equator. About how far will electricity travel in 10 hours?

6 . Suppose you had the task of making ten million marks on paper and you made two marks each second. Could you have made 10,000,000 marks in one year? (One year I s

60 x 60 x 24 x 6 seconds.)

7. The earth's speed in i t s orbit around the aun 1s a little less than seventy thousand miles per hour. About how far does the earth travel in i t s yearly journey around the sun?

3-3, Calculating with Small Numbers -- We have been dealing almost entirely with large number~,

But we also deal with many very small numbers, The mass of the electron and the mass of the proton are typical of such very small quantities, How can we conveniently represent these exceptionally small quantities?

Suppose we start with a power of 10, say lo4, and divide by

3 lo4= 10 10 x 10 x 10 = 3 , 10, We get 10 , f a r =

3 2 Now d i v i d e - 10 by 10, obtaining lo2, Divide 10 by 10, obtaining 4

the resul t 10. Now, starting with 10 and div id ing by 10 t h ~ e

times, we obtained

Notice that the exponents decrease by one each time. Now divide

lo1 by 10. We h o w t h a t the r e s u l t i s 1. Also, we see that if

the exponents are t o contlnue the pattern of decreasing by one at each stage, the next exponent should be 0. For this reason,

0 it is convenient - to define 10 as 1--that is, 10' = 1. Now we - -- have obtained the following:

1 10 lo2 = 10 , , = lo0, and 10' = 1.

The exponent in each answer above is 1 less than the exponent immediately preceding it. This is reaaonable since each time we

divide by LO we remove one factor of 10 from the numerator. lo0 1 Again divide by 10: As the next number we ge t =

If the pattern of exponents i s t o continue, we should expect the next exponent to be 1 less than 0. Thfs is the number which we - write 1; f t I s a negative number, Hence, it seems reasonable to

define lo-' as meaning m. Now divide 10-I by 10. The number 1 1 1 obtained is m + 10 = x m = If the pattern of exponents

- is to continue, the new exponent should be 1 less than 1, or -2.

2 Accordingly, we w a n t t o define 10 as meaning i. It is important -n 10 c t o no t i ce t h a t the number 10 i s not a negative number, It is

9

a positive number, namely the number &. For any pos l t ive integer n, then, we make the following

[aec. 3-31

Definition, n is a positive inte~er, - we define

10" = (t& product - of n - tens), arid

- 10 " = 1 4 10" = 1 4 (the - product - of n - tens).

For n = 0, we define - -

These definitions enable us to write powers of 10 as i l l u s - trated:

Each number indicated on the first line I s equal t o the number Immediately below it.

Class mercises 3-3 - - I, Express each of the followfng using negative exponents:

1 1 1 1 1

1 Tupm ~ , ~ O # 000 7 P m*

2. Expreaa each of the following in fractional form: -3

10 , 10-5, 10-7, -6 10 .

3 . Expreae each of the following using negative exponents: 1 1 -2 Example: one-hundredth = = 2 = 10 ,

10

(a) one -thousandth (c) one-billionth

(b) one-millionth ( d) one -tri lllonth

Recall now the meaning of a number written In decimal form.

We lalow that .4 = 4 4 -1 . Hence . 4 , = m = 4 x + = 4 x l O ,

Also ,005 = = % = 5 x 5 x loe3. his expreasea 10

each of theae numbera in scienti f ic notation.

[aec. 3-33

Now consider .42 = 42 JW. This is true since

4 2 40 2 42 -42 = , + , = , + , = ,. m put t h i s in scientific

notation we write

4 I We can now write 0.16 x 10 in scientific notation:

We f i r e t used scientific notation t o represent very large numbers. We have jus t seen that the use of negative exponents makes it poasible to express very small numbers also in scientific notation. Thus,

When a positive number less than 1 is written in ~ c i e n t i f i c notation we see that the exponent is always a negative Integer,

-1 f o r example, .63 = 6 . 3 x 10 . Now look back at the examples of scientific notd t lon in

Section 3-1. When 1, or any number between 1 and 10, is writ ten in scientific notation the exponent is zero. Thua, 1 - lo0,

0 3 = 3 x loo, 6.79 = 6.79 x 10 . When a number greater than or equal to 10 is written In

scientific notation the exponent will be a pos i t i ve inte er. 1 8 Thus, 27 = 2.7 x 10 . 10 = lo1, 4,680,000 = 4.68 x 10 .

for a number written in scientific notation: In s-ry, - - If 0 < the number < 1, - - the exponent is a n e ~ a t i v e integer; e . g . : - --

-2 . 0 3 = 3 X 1 0 .

If 1 < the number < 10, - - the exponent is zero; e . g . : 7 -

If the number 10.

the exponent is a posi t fve int;'eger; e . g . : - 137 = 1.37 X 10'.

Mote that we are not now trying t o represent negative numbers

in scientific notation. After a l i t t - l e more work with negative

numbers, you will see that It is easy enough to do.

Exercises 3-3 - 1. Write each of the following in s c i e n t i f i c notation.

(a) 0.093

(b) 0.0001

(4 1 (e) 0.00621

(h) 0.000000907

2. Write each of the following in decimal notation.

(a) 9.3 x lo-s 7.065 x l i 3

(b) 1.07 x 10-I (f) 10-l

( d ) 5 (h) 385.76 x

3. Write each of the following in scientific notation.

(a ) 63 x lo4 (e) 362.35

(b) 0.157 x (f) x 432

( g ) 0.00000000305

(h) 69.5 x l p - I

*4. Fill in the blanks in the fol lowing to make a true sentence. Notice that in some parts scientific nota t ion ia NOT used.

(a 0

0.006 - 6 x 10

0 ( c ) 0.0004015 = 4015 x 10

(a> 0

6000.0 = 0.06 x 10

( e > 0.213 = 2.13 x 10 a

+5. Can you thfnk of the one non-negative number that we cannot

express in scientific nota t ion?

3-4. Multiplication: Large and Small Numbers -- You have already multiplied numbers such as lo3 and l o5*

Recall that lo5 x lo3 = 10 8 + = 10 . Often, we need to multiply numbers where negative exponentts appear in t he scientific notation.

[aec . 3-41

- According t o t h e above, x low3 - 1 1 l = 10 -8 3 X ~ = 3

What is the r e s u l t of adding -5 and -3? Do you recal l that

-5 + -3 = -8? so we have shown that

Show as above that lom2 x = 10 ('2 + -4 ) . Is t he same

procedure followed when the exponents are negat ive as when they are p o s i t i v e ; t h a t is, do you add the exponents in each instance? Be sure you see why the answer is yes.

- 3 NOW multiply 4.3 x 105 by 2 x 10 ,

= 8.6 x lo2. - It has been shown above that lo5 X 10 = lo2. However - 5 + 3 = 2 so t h a t

Show as above tha t lod4 x lo3 = 10 ( -4 + 3 ) , ~e can now see

that when we multiply 10' by lob the result is 10 ( a + b)

matter whether a and b are p o s i t i v e or negative. This

illustrates the following

General Properly, If a and b are any p o s i t i v e or - - -- - negative integers then

Of course, the property i s v a l i d i f e i t he r o r both the 3 3 exponents a and b are zero. For example 10 x 10' = 10 ,

0 loo x lo0 x 10 = 1.

There is another idea which is involved in some problems. T h l s idea appears in the following where the answer is desired

in s c i e n t i f i c notation:

In this problem the numbers 4.7 and 5.4 were multiplLed t o

obtaln 25.38 and then this number was written in s c i e n t i f i c nota t ion as 2.538 x 10. Then, 2.538 x 10 was multiplied by

4 10 ,

Exercises 3-4

1. Write the - f ~llowing p r o d u c t s in s c l e n t i f i c notation.

( a ) 10 5 x low2 ( e ) 0.0001 x 0,007 - ( b ) 0.3 - x lom2 (f) (5.7 1013) 10 7 ( c ) 10 7 X 10-6 (g) 1od3 1015

( d ) 0.04 x 0.002 (h) 1012 x 10 x 10 -8

2. Write the following products In s c i e n t i f i c n o t a t i o n .

(a) 0.0012 x 0.000024 ( d ) 3 X 1i6 x 10 -4 -

(b) 6 x 10 x g x 10 - 3 ( e ) 38 x l o w 3 x 0.00012 -

( c ) 14 x lom3 x 10 (f) o.00~896 x 0.00635 3. Using powers-of-ten notation find the products of the following:

(a) 10,000 x 9.01 ( b ) 0.00001 x 10,,000,000

4. Multiply for ty -nfne thousandths by t h e number seven and s i x hundredths us lng s c i e n t i f i c no t a t i on . Express your answer in s c i e n t i f i c no t a t i on .

5. A large corporation decided t o invest so,Te o? f t s su-rplus

money in bonds. If 11 mi l l ion d o l l a r s was invested at

an average annual r a t e oi' $$ , what was t h e annual income

from this investment? Use s c i e n t i f i c notation in the computation, and also express your answer in scientific

notation.

6 . On a certain date t he debt of the U. S. Government, when

rounded t o the nearest 100 billion d o l l a r s , was 300 billion

d o l l a r s . Asslming that the government pays an average r a t e

oi Interest off 3.3135, what is t h e number of dollars

interest paid each year? &press t h e answer i n scientific notation.

7. IF t h e mass of one atom of oxygen is 2.7 x grams, what

is t h e mass of 40 x atoms of oxygen? Express t h e answer in scientific nota t ion .

8 . A chemical mass unit, which is one-sixteenth the atomic mass

of oxygen is approximately equal to 1.66 x lomp4 prams. What is the mass of a billion chemical mass units?

3-5. Division: Larae and Small Numbers

The principles involved in division are suggested by those we have developed for multiplication. We saw how to divide

6 Ir 10 by 10 , that is

Another way to do this is to write

- 1 Also, from our d e f i n i t i o n of 10 " as 7 we see tha t

But, r e c a l l from our study of subtraction f o r negative numbers in Chapter 1 that

Thus, we aee that

106 + 10 - - - 10 6 - ( -4 ) = 10 . 10

-4

These examples suggest the following

General Property. - If a and b - are any positive - or

negative integers then

loa ;. lob = lo8 b

Let us i l lus t ra te the use of this property In the d iv i s ion of two very small numbers given in scientific notat ion. Suppose

- 7 we wish to divide 8 x by 2 x 10 ,

Always remember, however, tha t we c a n check our work in such problems by dealing only with positive exponents, making use of - the d e f i n i t i o n of 10 n, Thus we may proceed as follows.

Be sure to juatify each of the operat ions in this development.

Class Exercises 3-5 - 1, Uaing the general property, perfom the following d i v i s i o n s

and check your work as we have done above:

( a ) l o m 4 10 5 (b) lom2 + 10 -7 .

2. (a) Divide lo7 by lo2+

( b ) ~ n y is equal to 1 0 5 ~

( c ) Are lo'+ lo2 and 10 '7 - ') numerals f o r tne same number or d i f f e r e n t numbers? -

3 . (a) Find b - 3.

( b ) Find 10 (6 - - 3 )

( c ) Use the Illustrative example above t o detelmlne whether 6

10 + 10 -3 and 10'~ - -3) are numerals f o ~ t h e same number.

4. (a) Find 10 ( -4 - 5 )

( b ) IS + lo5 equal to 10 ('4 - 5)? Why? (YOU found t h e first number in la.) - -

5. Is 10 * * 10 7 = l o (-2 - ' 7 ) ?

6. Write another rlumeral for lorn + 1fl. 7 . Perform the indicated divisions. -

[a) 12' + 10-5 ( b ) 10-3 + 10 9 c l o m 3 * 1 0 . 9 6 8. IS ( 6 x lo5) + ( 3 x l o2 ) equal t o 3 1 o5

,, ? ? 1s the final answer 2 x lo"?

9. Perform the indicated d L u l s i o n s and express t h e answer i n scientific no ta t fon .

6 (a) (1.2 x 1 0 ~ ~ ) i. ( 4 x 10 ( b ) (6.4 x lom6) + ( 3 . 2 - x loh5) (c) ( 9 X lo4) + ( 0 .3 x 10 2,

Exercises 3 I. Write the answers to the following in s c i e n t l f l c no t a t i on .

( a ) l d + lo2 (el lo1'+ 1013

( b ) lo3 + 10 ( f ) 10 10 , 1 0 2 ~ ( c ) 1ol4 + 10 4 (g) lo6 + 10l2 ( d l 1017 + ( h ) lo3 + lo4

2. Write the answers to the following in scientific n o t a t i o n . - (a) ld + 10 ( e ) 1d1 + 10 -13 - - (b) lo3 + 10 (f) 1o1O + 10 20

- 4 -

( c ) + 10 (9) l o 6 + 1012

( d ) + 10 -12 (h-) lo3 + l aw4

3 , Write the answers to the following in s c i e n t i f i c no ta t ion . - ( a ) 10 5 a 10 2 (4 10

-11 + 1013

(b) + 10 ( f ) + 10 20 -

( c ) 10 l4 1.10 4

- (g) l o w 6 + lo1*

(d) 10 l7 +d2 (h) 1 6 3 + 4

4. Write the answers to the following in scientific no ta t i on . - - (a) 10 5 + 10 ( e l 10-3 + 10-I - - - (b) 10 14+ 10 -4 - -

(f) 10 =7 -+ 10 -

(c ) 10 11* 10 l3 (9) + 10 20

(d ) lom6 + lom1* . ( h ) lo-3 + 10 - 4

5. Write the answers to t h e fo l lowing in scientific n o t a t i o n . - ( a ) ( 6 x + (3 x lom2) ( d ) (2.4 x 10) - . + 10 '

6. Fill in the blank places with the proper symbol.

( a ) 12% = 12 0 l2 = 1 2 x 1 0 0

= G = = 1.2 x 10 .- -

46 - (b) 46% = - a = 46 x 10 3

0 (d) 350%= = 350 x 10

7. A city government has an income of $2,760,000 f o r this

year. The income t h i s year represents 3% of t he t o t a l

value of taxable proper ty . What is the t o t a l value of taxable proper ty? Use s c i e n t i f i c n o t a t i o n in your computations.

8. A commuter pays$+40 per day for h i s fare. Would it be reasonable t o e x p e c t that he w i l l spend one million cents in fares before he retlres? Assume that he travels 250 days per year.

"9 . At the rate of t en d o l l a r s per second, about how many days

would it take to spend a billion dollars? Assume this 4 goes on 24 hours a day. (1 day 2 8.5 X 10 seconds)

*10. The tax raised in a certafn county is$160,000 on an assessed valuation of$d,000,000. If Mr. Smithrs k x is

$400 what 1s the assessed value of h i s property? '11. It costs about $35,000,000 to equip an armored division and

about 14,000,000 to equip an infantry d i v i s i o n . The c o a t

of equipping an i n f a n t r y d ivfs ion is what percent of the c o s t of equipping an -armored d i v i s i o n ?

*32. The mass of the electron is approximately 9.11 x 10 -28

grams and the mass of the proton is approximately 1.67 x 10 -24

grams,

(a) Which l a the greater?

(b) Approximately what is the ratio of the mass of the proton to the mass of the electron?

3-6. -- Use of Exponents Multiplying - and Dlvlding Decimals

You ho,w how to multiply two numbers in decimal form and also how to divide one by another. When you find the product or

quotient of two numbers, it l a usually easy t o decide where to place the decimal point. But, when a number of multiplications and dlvlslona a m required, you may have d i f f i c u l t y in decidlng where the decimal point belongs in the final anawer, By using

powers-of-ten notation we can work ent i re ly wlth whole numbers until the very end of a complex calculation and then fix the location of the decimal point In an easy way, Also, the exponent notation gives an easy way to explain our usual procedure for determining the decimal point loca t ion . We will illustrate these features Fn this section.

Suppose we wlsh to multiply 32.14 by 1.6. Where is the

decimal point t o be located in the product? Of courae, in such a simple example we see at a glance that

the product must be a number greater than 32 but less than 64, and this t e l l s us where to put the decimal point in our anawer. If we use exponent notatlon we proceed as folzowa:

Here we multiply whole numbers only in the product (3214 x 16).

The factor tells us where to place the decimal point . The

factor lom3 t e l l s us there should be 3 decimal places t o the right

of the declmal po in t in the product, or 51424 x lom3 = 51.424. - Also, from the way we arrived at 3 we see the justification

f o r the rule t ha t the number of places to the r i gh t of the decfmal

p o i n t in the product will be the swn of the number of places to the right of the decimal point In the two factors of the product,

If you are In doubt about the loca t ion of the decimal point In a pmduct or in a quotient, the exponent notation will make it

easy f o r you to decide. 'Ihe advantage here I s t h a t we deal only

with whole numbers in our multlpllcation and then later w o ~ r y about the position of the decimal point.

This form of exponent notation is aimilar to ~clentific notation but differs f r o m it in that the first f a c t o r does not have to be a number less than 10.

Class Exercises 3-6 - 1. Use the above procedure to find each of the fo l lowing products:

(a) 6.14 x 0.42 (c ) 649.3 x 14.68

(b) 0,625 x 0.038 (d) 11, 4 x 0.0031

We can use the same scheme in the division of decfmals, As

an example of how to proceed l e t us divide 14.72 by 6.1.

147' is simply an operation with Now the division - 1472 whole numbers and gives - = 24.13, if we carry

out the divis ion, correct to the nearest hundredth.

Hence,

correct to the nearest thousandth.

Here, also, we used powers of t e n in such a fashion that we could divide one whole number by another in perfoming the actual

division. The exponent 1 simply fixed the position of . the

decimal po in t in the answer. This notation is of ten a real advantage when doing more

complicated problems requi r ing a number of operations with I

decimals. For example,

To see how the answer was obtained, fill in all the steps of this calculation.

Exercises 3-6 - 1, Place the decfmal point in the products to make the following

number sentences true.

(a) 6021 x 0.00003 = (6021) x (3 x lom5) = 18063

(b) 3.42 x 0.02 = (342 x lom2) x (2 x = 684 -1

( c ) 2.5 x 3,000 = (25 x 10 ) x (3 x lo3) = 75

(a) 54.73 x 7.3 = (5473 x 10'~) x (73 x 1;') = 399529

(e) 1200 x 0.006 s (12 x lo2) x (6 x lom3) = 72

[aec . 3-61

2. F i l l blanks with proper symbols.

( a ) 4.52 = 45.2 x 10-I = 452 x 10 0

0 (b) 0.012 = 1.2 x lom2 = 12 x 10

13 ( d ) 38.216 = 382.16 x 10-I = 3821.6 x lo-* = 38216 x 10

( e ) 6.37 x lo4 = 63.7 x lo3 = 637 x lo2 = OX 10' a u

( f ) 0.003~ l o 5 = 3 x 10 = 3 0 x 10

3, Place the decimal point in the quotients to make the following

sentences t rue.

4 . I ~ l u l t i p l y us ing exponent n o t a t i o n , (a) 135 x 0.06 ( ) 0.0035 x 16.301 ( b ) 16,000 x 3,000 = (e) 6,000,000 x 0.0275

Hint: (76 x lo3) x ( 3 x lo3) ( c ) 18,000 x 0.0003 (f) 0.07 X 300 X 0.02 X 6,000

5 D i v i d e using exponent no t a t i on . (a) 6 .3 + 0.3 0,1470

(" 0.75 (b) 0.78 + 13 ( e ) 0.27

(f) 1800

*6. Use exponent8 to place the decimal po in t in the answer.

' 7 . How many pieces of popcorn each weighing 0.04 ounces will it take to make enough t o f i l l 840 bags? Each bag will contain 6 ounces of popcorn.

8 , BRAINBUSTER. A flying saucer can t rave l at 100,000 miles a second. About how long (in yeara) w i l l it take it t o

1 visit and return from a star tha t is l ight years away?

3

1 light y e a r z 5.9 x 1012 m i l e a

7 1 year a 3.2 x 10 seconds

3-7. The Metric System; Metric Units of Length -- We have been using powers-of-ten notation i n working w i t h

numbers, especially with very large and very small numbers. As

ai illustration of how powers of ten arise naturally in technical and s c i e n t i f i c work we shall now study the metric system. As

you will see, this system of measures is based upon powers of ten, and therefore t h e scientific no ta t ion we have studied is especially useful in deal ing with metr ic quantit ies .

The system of measures used most widely in t he United S t a t e s

is called the English system. Some of the u n i t s in t h i s system

are the inch, foot , yard, mile, gallon, pound, e t c . As you have found, these u n i t s are a bit difficult to learn and remember

because of the many di f ferent d e f i n i t i o n s required. In t h i s

English eystem you must learn that 12 inches = 1 foot, 3 feet =

1 yard, 5280 feet = 1 mile , 4 quarts = 1 ga l lon , 16 ounces =

I pound. All the r e l a t i ons seem to use d i f f e r e n t numbers and there is no simple, easily remembered basic re la t ionship among the units.

In most other countries, the system of measures is the metric system, i n which the baaic unit of linear measure is the meter. The metric system is a simpli f ied system of weights and measures developed i n 1789 by a group of French mathematicians. They decided that, aince t h e i r system of numeration was a decimal (base 10) system it would be a good idea to have a decimal basis f o r a system of measures. In such a system the units of length would be some power of t e n times a baaic unit of length. Then it

would be easy to convert from one unit to another. It would only require multiplying or dividing by a power of 10. We aha11 see

that this makes it very much simpler to work with quantitfes expressed In metric units.

Metric Units of Length

The French mathematicians began by

N. Pole m calculating the distance n from the North

Pole to the equator on the meridian through Paris. For the basic unit of length they

1 took ,,, ,,, of this distance. By 10,

distance could be measured again if the atandard bar of unit length wem ever lost,

They named this new standard of lewth the meter and a - standard meter bar was carefully preserved to assure uniformity in future meter units. T h i s definition of the meter was used

until October 15, 1960, when a new atandard of the meter was agreed upon by delegates from 32 nations. This defines the meter in terms of the orange-red wave-lengths of laypton gas.

Precisely, one meter i a now defined as:

1 meter = 1,650,763.73 orange-red wave lengths

i n a vacuum of an atom of the gas -ton 86,

This new def in i t ion has the advantage that the unit is easily measured on an interferometer anywhere in the world. Also,

it allows an accuracy of one part In one hundred million In linear meaaumments. Using the o l d atmdard bar of platinum- irldiwn an accuracy of one part i n one million was the best obtainable.

In terms of our English system, a meter i a a little longer than a yard, namely

1 meter = 39.37 in., (approx.)

Smaller u n i t a of length in the metric system are obtained by dividing by 10. Thus we define

1 1 decimeter 3 - tneter 10 1 1 centimeter = -- decimeter = - 10

meter 100

1 1 millimeter = - centimeter = 10

meter. lOOD For longer units of length we aimply m l t l p l y by 10. Thus,

by definition,

1 dekameter - 10 meters

1 hectometer = 10 dekameters = 100 meters

1 kilometer = 10 hectometers = 1000 meters.

To emphasize the simpllcLty of the r e l a t i ons involved, we write these units in terms of meters, using scientific nota t ion . Then the r e l a t i o n ~ l look lib this.

1 millimeter = meters

1 centimeter = lom2 meters

1 decimeter - 10-1 metera

1 meter 0 = 10 meters 1 1 dekameter = 10 meters 2 1 hectometer = 10 meters

I kilometer = lo3 meters

[see. 3-71

Many attempts have .been made to get the United States t o

adopt the metric system f o r general use, Thomas Jefferson in the Continental Congress worked for a decimal system of money and measures but succeeded only in securing a decimal system of coinage. When John Quincy Adams was Secretary of State, he

foresaw world metric standards in his 1821 " ~ e p o r t on Weighta and Measures. " In 1866, Congress authorized the use of the metric system, making it legal f o r those who wished t o use it. Finally,

in 1893, by act o f Congress, the meter was made the standard o f

length i n the United States. The yard and the pound are now off iclally defined in terms of the metric unf ts, the meter and - the

kilogram. A sudden change from our common units (yards, feet, inches,

ounces, pounds) to metric u n l t s would undoubtedly cause confusion for a t i m e . However, many people th ink tha t we wi l l gradually

change over to the metric system. Our scientists already use the

metric aystem and people in most foreign countries use it also . We summarize the definitions and abbreviations f o r the major

metric units of length Fn Table A , In this table, notice how useful scientific notation is in ahowing relationships t o the basic metric un i t of length, the meter.

Table A

Linear Metric Units

Meter equivalent in Scientific Notation

lod3 rn. -2 10 m.

10-I m.

10' m. 1

10 m. 2 10 m. 3 10 m.

4

Name of Unit

1 millimeter

1 centimeter

1 decimeter #

1 meter

1 dekameter

1 hectometer

1 kilometer

Abbreviation

1 mm.

1 cm.

1 dm.

1 rn,

1 d h .

1 hm.

1 h.

Equivalent in Meters 1

T%iT rn' m.

& me 1 m.

10 m.

100 m

1000 m.

Notice that all the other metric units of length use the word "meter" w i t h a prefix. m e s e prefixes are also used to

name other units of measure in khe metric system,

Prefix

milli

cent1

deci

de ka

hecto

kilo

Meaning

In actual practice the hectometer, dekameter and decimeter are seldom used. The meter, centimeter, millimeter and kilometer

are in very common usage and we shall devote most attention t o

i them.

Two other prefixes which we should mention here are mega, ! meaning mLllion and micro, meaning one-millionth. Thus

Prefix Meaning

mega 1,000,000 = lob

micro 1 = 10 -6 t 000

j You of ten hear now of 3 megatons (3 million tons), 1 megacycle

(1 million cycles) . Even the slang term "megabuck" uses this

classical Greek prefix! In these d&ys of nuclear and atomic studies very small

quan t i t i e s are of ten studied and lengths a a small as one-millionth

t of a meter are common, The micron is deflned as: -6 1 micron = one-millionth of a meter = 10 m.

-6 Thus 3 microns = 3 x 10 rn. = 3 x 1c3 millimeters, s ince

1 micron = lom3 rnil l im~ters. The usual symbol f o r micron is the '6 Greek l e t t e r / (read Mu). Hence 1 4 p = l l r microns = 14 x 10 m.

Have you encountered the terms megavolt, megohm, microwat t, microsecond, microfarad? If not , you will soon be discussing them in science class. Can you figure out what a micro-micron l a 7

Exercises 3-7a

I. Complete each of the following:

( a ) 1 kilometer = hectometers

(b) 1 kilometer = dekameters

(c) 1 kilometer - meters

(d) 1 kilometer = decimeters

(e) 1 kilometer = centimeters

(f) 1 kilometer = millimeters

2. Complete each of the following:

(a) 1111,111 meters = kilometers

(b) 5.342 meters = centimeters

(c) 245.36 meters = kilometers

(d) 0.564 m. = mm,

(e) 6043.278 m. = lun.

(f) 2020.202 m. = cm.

( g ) ,015 mm. = microns

3. Fill in each blank with the corgect number.

(a) 5 m. = cm, ( f ) 3.25 m. = cm.

(b) 200 cm. = rnm , (g! 3500 m. = km.

( c ) 500 m. = lan. (h) 474 cm. - m.

(d) 2.54 cm, = * m (i) 5.5 cm. = mrn ,

(e) 1.5 h. = m. (j) 6.25 m. = cm.

1 4. A meter waa originally def ined to be 10,000,000 Of the distance on the earth's surface f r o m the North Pole to the equator. Assuming the earth is a sphere and using sclentlfic notation, find the approximate number of

(a) meters in the ci~cumference of the earth;

(b) kilometers in the circumference of the earth;

(c) millimeters in the circumference of the earth.

*5. Express in scientific notat ion in tems of meters:

(a) -013 millimeters

(b) 2.34 centimeters

(c) 6,730 M l o m e t e r s

(d) 694 microns

( e ) 1 mega-micron

Conversion to mgllsh Units - Since we use both lhgl iah and metric units in t h l s country,

it is often necessary t o convert f r o m one system to another. The standard U,S . inch 1s now defined in terms of the metric standard by the re la t ion

1 inch = 2.54 cm. (definition of inch),

As we have seen, this gives

39.37 in. = 1 meter, (approx. ) . In tems of yards, t h i s aays that

39.37 1 meter 3 - ydB Or

1 meter % 1.1 yds.

For measurement of longer distances, It is often useful to conver t miles t o kllorneters. Fmm

39.37 1 rneter = 39.37 in. = ft,,

we see tha t I meter = 39*37 rnlles. l;ro

Hence, 1 kilometer =

Naturally, we now ask you to verify by actual calculation that thls relation gives

1 kilometer 2 0.62 miles,

Roughly speaking, then, 1 kilometer % .6 of a mile or, f o r an even

better approximation, 1 kilometer 8 of a mile.

Exercises 3-7b

1. Convert each of the following metric measurements to approximately equivalent English measurements.

(a) 100 meters 8 Y ~ S - \ (b) 200 meters % ydsW I (c) 400 meters rt: Y ~ S - A few of the standard

distances for t r ack (d) 800 meters % ~ d s ( and f le ld eventa . ( e l 1500 meters 8 Y ~ S - I (f) 1500 meters a miles)

(g) 10 kilometers % miles

(h) 100 kilometers % miles

2. (a) The height of Mount Everes t , world's highest mountain is 29,003 ft. Approximately what is t h i s height, rounded to 29,000 ft., expressed in rnecers?

(b) One of the greatest sea depths was measured as 34,219 ft.

Express this depth in meters, approximately, after rounding t o the nearest 100 ft.

Csec. 3-73

3. (a) Show from our earlier re la t ion between miles and ki lomete r s that 1 m i l e % 1.61 kilometers.

(b) The mean distance from the earth to the sun is about 7 9.29 x 10 miles. About how many kilometers is this?

4. A common s i z e of t y p e w r i t i n g paper is 8; inches by 11 inches. What are these dimensions in centimeters?

5 . (a) What is your height in centimeters?

(b) In mlllimeters?

(c) III microns?

6 . Which I s the faster speed, a &peed of 100 ft. per second, or

a speed of 3000 cm, per second?

3-8. Metric Units of A r e a --- We have learned how to find the area of the i n t e r i o r of a

simple closed curve f o r a v a r i e t y of simple curves. We chose the

area of a square region as the best unit t o use in measuring the area of the in te r iop of such a closed curve.

The metric u n i t for measuring areas is also a square region.

We use as a basic unit the area of a square region with each edge of l ength one meter. The area of the i n t e r i o r of this square

region is cal led one equare meter ( a b b ~ v i a t i o n sq.m.).

If you have a cen t ime te r rule available you should draw a

square of side 1 cm. in order t o g e t some idea of the size of one square centimeter. F r o m 1 m. = 100 em. = 39.37 inches we

see that 1 cm. c -39 inch,

or

I cm. b . 4 inch.

Compared t o the basic unit of one square meter, the square 1 centimeter 1s small indeed. Remember that 1 cm.= m., hence

1 1 sq, cm, = - 1 sq.m. 100 100 8q*m* = 10,000

Here is another instance in which you may prefer to use exponent no t a t i on and write instead

-4 1 sq , cm. = 10 sq.m.

Since the meter l a the basic unit of length and the squam meter the basic u n i t of area,it Is important to give the va~ious units of area in terms of square meters. The most comonly used

units are l i s t e d in Table B.

Table B

Here again we see that it is convenient to use exponent

Uni t of Length

notation and write,

1 sq. mm. = 10 -6 sq. m. -4 1 sq. cm. = 10 sq. m. 6 1 sq. h. = 10 sq. m.

millimeter sq. mill imeter

centimeter sq . centimeter

kilometer sq. kilometer

Unit of Area

Equivalent Area in Square Meters

Exercf ses 3-8 - 1. Complete each of the following:

2 6 Example: 1 sq, h. = 1000 sq. m. or 10 sq. m.

(a) 1 sq. cm. - ( ) 2 sq. m. or rn sq. m. 1

(b) 1 sq. m. = (=)* sq. m. or sq. m.

( c ) 1 aq. cm. = lo2 sq, m. or sq. rim.

(d) 1 aq. rn, = (100)' sq. em. or sq. cm.

1 ( e ) 1 sq. m. = sq. lm. OP sq. lan.

2. Draw a sketch to illustrate (c) in Problem 1.

3. Find the area of a rectangular closed region with the following dimensions. Ee sure that both dimensions are expressed in the same unit.

Length Width - (a) 35 cm. 9.2 cm.

(c) .97 m. 37 cm.

> 1.25 mm. 1.2 cm,

4. Express the area in Problem 3 (b) In square centimeters.

5. What l a the area of the i n t e r i o r of a circle whose radius is 6 m.? Use 3.14 for T and find the approximate number of square meters in the area.

6. Find the area In square meters of a square closed region 160 cm. on a seda.

7. If the area of the i n t e r io r of a parallelogram is 783 sq. cm. and the baae is 27 cm., what is the altitude?

8. ( a ) . What i a the metric equivalent in sq. cm. of 1 sq. yd? 1 yd. = 91.41 cm, Hence

2 1 sq, yd b (91.11) sq. cm, = ? sq. cm.

(b) Approxtmately how many sq. cm. equal 1 sq. I nch?

[sec. 3-83

9. (a) Ver l fy that 1 sq. h. z 0.386 sq. mile.

(b) Verify that 1 sq. mile 2.59 sqb h.

10. (a) The land area o f the earth is estimated to be about 6 149 x 10 sq. h. Approximately how many sq. mlles

is t h l s ?

(b) The ocean area o f the earth is estimated as 361 x lob sq. hi. Approximately how many aq. miles of ocean are there?

3-9. Metric Units of Volume

The metric unit f o r measuring v o l q e is a cubical s o l i d . The

length of each edge of this cube is 1 meter. The volume of t h i s cube is thus 1 cubic meter (abbreviation 1 cu. m.), --

The cubic meter is a ra the r large un i t of volume. A much

smaller unit is the cubic centimeter (cu. c m . ) . As we have seen, the centimeter is about . 4 in,; hence, the cu. cm. is a cube about the s ize of a small sugar cube. To suggest the size of the

cubic meter, note that 1 1 1 1 cu. cm. = x x CU' m* = 1

CU. m. 9 O ~ ~ , ~ ~ ~

In scientific notation, ' 6 6 1 cu. ern. = 10 cu. m. and 1 cu. in = 10 cu. cm,

By similar ca l cu la t ions we can f ind the commonly used multiples

and ,subdivisions of the cubic meter . These are displayed in Table C .

Table C

Unit of Length

millf meter

centlrneter

Unit of Vo lwne

CU. mm.

CU. cm.

Equivalent Volume in Cubic Meters

1

(1000) CU'

1

kilometer cu. lun.

3 CU. m. (100)

(1000)~ cu. m.

Here, even more than before, we aee the convenience of the

exponent notat ion In writing -3 I cu. m.= 1 i 3 x 10-3x 10 cu. m. = 10-9 cu. m.

1 cu. cm. = low2 x 10-2 x 1.0~~ cu. m. = 1.0~~ cu. m.

look at the calculations above. You see that

aeneral Property: If a and b are any poaitive - - or negative Zntegers, then -

b ab [loa) = 10 .

Exercises 3-9 1. Complete each of the following:

~xample. me= are (10001~ or 1,000,000,000 cu. m.in I cu, h. 3 (a) There are 10 or cu. mm. in a cu, crn. 1 3

(b) mere are (rn) or cu. rn. In a cu. cm.

1 3 (c) There are (=rn) 0' CU. m, in a cu. m,

6 (d) mere are (10 ) or cu. nun. in a cu. hn.

2. A rectangular solid has dimensions of 6 cm., 7 cm., and

8.4 cm, Calculate the volume of the i n t e r i o r of this so l id .

Recall that the volume of the in ter ior of a rectangular solid is equal to the product of the measures of the length,

width, and height, when the measurements are expressed in the same unit.

3. What I s the volume of the interior of a rectangular solid whose height is 1 4 m. and whose base has an area of 36.5 sq. cm.?

3-10. Metr ic U n i t s of Mass and Capacity ---- The metr ic wit f o r the meastire of mass is defined as the

mass of water contained by a vessel with a volume of one cubic centimeter. The mass of one cubic centimeter of water Is cal led ----- --- a gram, This is a very convenient defini t ion, f o r when we b o w - t he volume of a container we immediately mow the maas of water it can contain. For example, if the volume of the I n t e r i o r of a container is 500 cu. cm., then the mass of water it can contain Is 500 grams. me important thlng to note In thls d e f i n i t i o n is t ha t the numerical measures are the same.

Wen we speak of the volwne of a box or other container,

we f requent ly use the term capacity. By the capacity of a container we simply mean the total volume which the container w i l l

hold.

In ta lk ing about the volume of liquld a container w i l l hold, we f requent ly use special units, such as, pint, quart,and gallon, in the English system. Thus, we may say the capacity of a tank

is a cer ta in number of gallons, and i t s volume is so many cubic feet.

In the metric system the usual uni t of capacity is the - l i t e r (abbreviatedy .). One - l f t e r l a defined as the capacity of a cubical box with edge of length 10 cm. (1 decimeter). Thus, one l i t e r means a volume of 1000 cu. cm, We say a cube of edge - 10 em. has a volume of 1000 cu. cm, We say its capacity is one l i t e r . It can contain a mass of 1000 grams (or one kilogram) of water.

One l i t er is approximately one quart, or more pmcise ly ,

1 l i t e r = 1000 cu. cm. = 1.056712 qt.

me other most comon metric measures of capacity are the

and the

milliliter (my.) = .001 l i t er

k i l o l i t e r (ky.) = 1000 liters.

A mass of 1000 kilo rams is called one metric 9. Hence, 8 a metric ton contains 10 grams. The metric ton is the mass of 1 k i l o l i t e r of water.

The most used units of volume, capacity, and mass are summarized in Table D. Note especial ly that 1 z. cm. corresponds - to 1 a. of mass and to one milliliter of capacity. - ----- -

Table D

There are several abbrevlatlons commonly used for the gram.

The abbrevlat lons, g , or gm., are both generally accepted and

the form, gr., is also used. In this t e x t we abbreviate gram

as gm., kilogram as k g m . , and milligram as mgm.

Unit of volume

1 cu. em.

1 cu. dm.

(1000 cu. cm. )

1 cu. m. 11000 cu. dm)

(1,000,000 cu. cm.)

Unit of mss

1 m* 1 kgm.

(1000 gm. )

1 metric ton (1000 kgm. )

(1,000,000 gm. )

Unit of capacity

I mL = . O O l l .

1 1. (1000 rnt. 1

(1000 liters)

Exercises 3-10 - 1. The volume of' a jar is 352.8 cu. cm. What is the mass of

the water it can contain, expressed in:

(a) grams?

(b) kilograms?

2. (a) What is the capacity in milliliters of a rectangular tank of volume 673,5 cue ern?

(b) What is i t s capacity in liters?

3. A cubical tank measures 6 feet 9 Inches each way and is f i l l e d with water,

(a) Find I t s volume in cubic inches.

(b) Find i t s volume in cubic f e e t . Recall t ha t 1728 cubic inches = 1 cubic foot.

(c) Find the weight of the water, Recall that 1 cubic

foot of water weigha 62.4 lb.

4. The dimensions of the tank in Problem 3 are about 2.06 meters each way,

(a ) Find i t s volume in cubic meters. (b) Find i ts contents in l i t e r e . Recall that there are

1000 liters in a cubic meter, (c) What is the mass of the water? Recall that 1 l i t e r

of water has mass of 1 kilogram,

5 . How d i d the t i m e needed t o solve Problem 4 compare w i t h the time to solve Problem 3? What is the main advantage of computing in the metrfc system?

6. A tank has a volume of 2500 cu. cm.

(a) What is the capacity of the tank in milliliters? 1

(b) How many kilograms of water wi l l the tank hold?

( c ) How many metric tons of water w i l l the tank hold?

7. A cubical box has edges of length 30 cm.

(a) What is the volume of the box in cu. cm.?

(b) What I s the capacity In l i t e r s?

(c) How many kilograms of water will the box hold?

(Assume t ha t it is watertight, of course!)

*8. The volwne of the sun is estimated to be about 337,000 million

million cubic miles or 3.37 x 1017 cu. miles.

(a) Using the fact that 1 mile % 1.6 kilometers, express

the volwne of the sun in cubic kilometers. (Sfmply indfcate multlplications in your answer if you wish, )

(b) Express the sun? s volume in cu. cm., leaving your answer in the form of indicated m l t l p l i c a t l o n .

9. The B r i t i s h Imperial gallon, used in Canada and Great B r i t a i n , is equivalent to 1.20094 U. S. gallons,

or, 1 British Imperial gal. 2 1,2 U.S. gal.

( a ) When you buy 5 "gallonst' of gaaol ine in Canada, how many U.S. gal lons do you receive?

(b) How many Imperial gallons are required to flll a barrel which holds 72 U,S . gallons?

Research Problems:

Use the Twentieth Yearbook of the National Council of -- - Teachers - of Mathematfcs as your reference book.

(a) Col l ec t a list of all the various units of measurement f o r length, area, volume, and capacity that you c a n

f ind l i s t e d in your reference book, Bring t h i s list

t o school.

b ) Write a composition on ."Why I Prefer the English System of Measures to the Metric System'' or "Why I Prefer the Metric System of Measures to the h g l i s h System." You

may use your reference book to a i d you b getting information. A goad discussion is given in "The Metric System--Pro and Con,'' by Chawcey D. make and Ralph

M. D r e w ; Popular Mechanics, December 1960.

( c ) Which weighs more, a p o d of feathers or a pound of gold?

B r i e f of Relations among Units - - - The fol lowing table sunrmarizes much of the work in the

m e t r i c system. From i t you can derive a l l the multiples and subdivisions of units of area, length, volume, maas,and capacity,

Table E -- Iength

10 millimeters (rnrn.) = 1 centimeter (cm, )

100 centimeters (cm. ) = 1 meter (m.)

1000 meters (m. ) - - 1 kilometer (ha )

Capacity

1000 milliliters ( m d . ) = 1 l i t e r (1, ) = 1000 cu. cm.

Mass

1000 milligrams (mgm. ) = I gram (gm. )

1000 grams (gm. ) = 1 kilogram (kp. )

1000 kilograms (kgm,) = 1 metric ton

Some important conversion r e l a t i o n s between corresponding English and metric units are listed below for reference.

[aec. 3-10]

Table F

1 inch

1 meter

1 cm,

1 b.

1 mile

1 l i t e r

1 liter

1 gal

Capacity

Z

2.54 crn. (definition of the inch)

39.37 in,

.39 in.

0.62 miles

1.0567 qt.

0.2642 gal,

3,785 l i t e r s

[sec . 3-10]

Chapter 4

CONSTRUCTIONS, CONCIRUrn TRIANGLES,

AND THE PYTHAGOREAN PROPERTY

4-1. Introduction - to Mathematical Drawings and Constructions

Drawing pictures and diagrams helps us s o l v e many problems. Aeronautical engineers draw pictures of each part of a new ai rcraf t to h e l p them study the problems involved. Arch i t ec t s make draw-

ings of f l o o r plans and pictures of how completed buildings will look before the actual building is started. Theatrical directors

sketch the stage and the location of properties to he lp decide

how a certain scene should be staged. A carpenter makes drawings of the object that he is bullding. Elect~icians make diagrams to

show how a machine should be wired. Many of your problems will be.much easier to solve I f you develop the habit of drawing

pictures or diagrams to help you see the re la t ions in your problem. Sometimes silly mistakes are made because students do not take the time to draw a picture of a problem situation.

For some problems a roum sketch of the situation is s u f f i - cient. Rough sketches can be drawn freehand in such cases. Although "roughly" drawn, the sketch may help you "see" the problem There is no sense in wasting time on an accurate drawing I f a rough sketch will serve.

Some problems can be solved by measuring drawings, but when drawings are used this way they

should be accurate representations. Many tools are used to make accurate drawings. A man whose job is to

make accurate drawings is called a draftsman. He uses a compass and a straightedge, but he uses many

other tools . Draftsmen use such

t o o l s as protractors, T-squares, 30-60 triangles, 45-45 t r i ang les ,

rulers, parallel rulers, pantographs , and French curves to help

make drawings accurate. You will be using some of these tools

but o the r s a re too expensive to use at t h i s t ime. Find out what the following tools are and how they are used by draftsmen:

(a) 30-60 triangle

(b) French curves

( c ) Pantograph

Tools alone cannot produce accuracy. You must use them in

a way t h a t gives accurate results. In the seventh grade you

learned t o use a protractor; however, some of you may need to

review t h i s with your t eacher .

In measuring an angle, when a ray of an angle is not long enough to reach the scale of the protractor, extend t h e ray o r lay a s t ra ightedge carefully along the ray. Check t o see t h a t you

use the correct scale on t h e p r o t r a c t o r . Plane figures, you recal l , are figures that lie completely

on a flat surface such as your paper or t h e chalkboard. Lines,

angles, and polygons are examples of plane figures, Lines are t h e s i m p l e s t of these f igures. The r e l a t fon of two or more l l n e s on a plane is of specla l interest. Perpendicular lines are Lines t h a t intersect so as t o form 90' angles. Parallel lines are t w o or more lines t h a t do n o t intersect, or in s e t language, l ines whose intersectfon is the empty set.

Perpendiculars may be drawn accurately with the a i d of a

protractor. A t any point on a line, measure an angle of 90'. The rays may be extended t o form lines. Lines, rays, or segments may be perpendicular t o each o t h e r .

The edges of most r u l e r s are considered to be para l l e l . A

quick way to draw two parallel lines is simply to draw a line on each side of such a r u l e r without moving It, Of c o u r s e , this

method has l i ~ . l t e d use since a l l such pairs of parallel lines are t h e same distance apart . To overcome t h i s d i f f i c u l t y t h e pro-

t r a c t o r is needed.

Pro t r ac to r s can be used to draw parallel lines accurately if you remember that corresponding angles are congruent when formed

by parallel l ines and a transversal. A transversal is a l i n e that in te r sec t s two or more lines in d i s t i nc t points.

To draw the f igure a t t h e

right, use a straightedge to draw

line 1 . A t any point on l i n e -

draw a line t so that a 60' angle /

is formed. Draw a t h i r d line J, , t intersecting line t to form a \ ,300 60° angle. Since the corresponding angles are congruent, line 1 .

is parallel t o line ,f, . 1, I6 l ~ n e J parailel to

llne 1, in each figure? How do you h o w ?

T-squares and triangles also I

1

are useful in drawing parallel lines. The f i g u r e at t h e right

I l lustrates the use of the triangle

allel lines may be drawn,

and the T-square In drawing lines

t h a t a re para l l e l . By moving the

triangle along the T-square or by

-

- moving the T-square, sets of par-

Another type of accurate drawing that w i l l be atudied i n t h i s

c h a p t e r is a compass and straightedge construction. These con- atructions are drawings that are made by using only two t o o l s , a compass and a straightedge. Of course, a pencil (or chalk) can be used. Since "compass and straightedge construction" i s such a long phrase, the sLngle word construction will be used in th i s

chapter t o mean drawings that are made with a compass and straightedge only. A straightedge is a ruler without any units marked on it. It is used to make straight lines--not to measure their lengths .

Exercises 4-1 0

1. Draw an angle of 60 without using your protractor. Now measure t h e .angle with your pro t rac to r . HOW close to 60' is your answer? Draw another t o see If your estimate is

improving.

2,. Sketch the following angles with these measures in degrees.

( a ) 45 (4 30 (4 10 ( g ) 110

(b) 90 (a) 120 (f) 160 ( h ) 65 Measure each wlth your pro t r ac to r to see how well you estimated

the size of each angle. T r y agaln if you aren't sat isf ied w i t h your f i r s t t r i a l .

3 . Which angles in problem 2 are acute angles? Which are obtuse angles?

4. With ruler and pro t r ac to r draw a transversal intersecting one of two parallel lines at an angle of 80'

(a) How many pairs of corresponding angles can you find?

(b) How many pairs of vertical angles are there in your

drawl ng?

( c ) How many degrees are there i n each pair of v e r t i c a l angles? In each p a i r of corresponding angles?

( d ) Can you draw three or more parallel lines each inter-

secting a transversal at an angle of 80°?

5. With ruler and protpactor, draw a triangle that has two angles 1 of 60' with a side + inches long between the vertices of

the angles.

6 . Using your protractor and ruler, draw a triangle w i t h sfdes 4 cm. and corn. The angle formed by these sides is 110'.

What kind of t r i ang l e is th is?

7 . Do you recall t h a t a parallelogram is any quadrilateral whose opposite sides lie on parallel lines? Is a square a parallel-

ogram? Are all rectangles parallelograms?

(a) Draw rectangle ABCD with ruler and protractor. What

size is the angle at ver tex A? at vertex B?

( b ) A r e A and B consecutive vertices of parallelogram ABCD?

( c ) Recall a property of parallelograms: The angles of a

parallelogram at two consecutive v e r t i c e s are supple-

mentary. The sum of the measures of angle A and

angle B is . State t h e sum of the measures of

angle B and angle C. Do the same w i t h each pair of

consecutive angles.

8. In parallelogram RXYZ

z

9 . Using your ruler and protractor, draw a parallelogram with

sides of 1 inch and 2 inches and with one angle of 75'.

10. BRAINBUSTEEI. A farmer plans t o divide h i s land equally among h i s sons. The land is shaped like the diagram. then he had

two sons, he planned to div ide it as shown. It was easy t o

divide when he had three sons. Now he has four sons. How can he divide t h e land so that each son g e t s the same amount in the same shape?

Dlvlded f o r 2 sons Divided f o r 3 sons How should it be

divided for 4 sons?

11-2. Basic Construct ions

In drawing geometric f igures , you have used several mechanical aids so t h a t you could draw these pictures accurately. In the 2000-year period slnce the development of geometry as a science by Euelid and o t h e r Greek scholars, t o place certain r e s t r i c t i o n s on the t o o l s that cou ld be used has been considered useful by many writers of geometry texts. Under these restrictions, a straightedge may be used to make straight line sements. A com-

pass may be used to draw c i r c l e s and arcs. An arc is any connected por t ion of a c i r c l e .

In your constructions you may use a r u l e r as a straightedge.

In g'eometry, however, we think of a straightedge as having no

marks on it with which measurements may be made.

You may observe t h a t we draw, or con~truct, a line segment by tracing with a pencil along an instrument which has a straight edge. On the other hand, our method of drawing a c i r c l e is quite

1 compass. We do not trace w i t h a pencil along an object which has

a circular-shaped edge. Efforts to devise an instrument with i which a straight l i n e segment could be drawn without using a 1 st ralght edge form an interesting chapter in the h i a t o r y of geom-

1 etry. These efforts were successful only about 100 years ago. Oeorne t ric drawl ngs made with compass and straight edge only

are cal led constructions. In this sect ion you will learn several bas ic constmctlons used in geometry.

Follow the directions and complete each construction 1

through 4.

1. Copyiw a segment

(a) Draw a working l i n e a little A * 8

longer than the segment t h a t 4

is to be copied.

(b) Place the point of the compass at one endpoint of the segment Step a that is t o be copied.

( c ) Open. the compass u n t i l the 1

penci l touches the o ther endpoint. he distance

*FB between t h e point of the

compass and the end of the Steps b and c

pencil is the length of the radius of the compass.)

( d ) Without changing the radius - I of the compass, place the * d re' point of the compass on the working line at A ' (any Step d

po in t ) and mark an arc where the pencil crosses the line.

The sement from the po in t A t where t h e compass point is placed, to Br (the intersection of the arc and the line)

is the same length as the o r i g i n a l segment.

A n y time that you need segments of equal length, this construction

is used. I

[ ~ e c . 4-21

2. Bisecting a line segment

The word b i sec t means to d l v i d e i n t o two equal parts.

(a) P l a c e the poin t of t h e

compass at one endpoint

of the segnent. S e t t h e

compass so that its radius is more than half of the distance between the two endpoints.

(b) Draw arcs above and below

the center of t h e segment. Be sure t h e arcs are long enough t o Include p o i n t s

above and below t h e center .

( c ) Without changing t h e

radius of t h e compass,

place the compass point at the o the r endpoint.

D r a w arcs t h a t cross *the

f i r s t two arcs.

(d) Draw a line through the p o i n t s where t h e arcs cross,

This l i n e bisects the

o r i g i n a l segment.

Measure the two parts obtained in ( d) . A r e they t h e same length? What is the

re la t ion between t h e segment and the b i s e c t o r that has been constructed?

[aec. 4-21

3. Copylng; an angle

( 8 ) Draw a base l ine , part of which w i l l be used ae one

ray of the angle.

(b) Place the point of the compass a t the vertex of the angle and draw an arc through both rays of the

angle.

( c ) From a point P, on t h e

base line, draw an arc with the radius used

in (b),

(d) Place the point of the compaaa at the interaec- tion of one ray of the original angle and the arc that crosses it. Place the pencil of the compass on the other intersect ion.

( e ) With the compass set as d e t e d n e d in (d), place the point at the intersec- t i o n of the base line and the arc. Draw an arc that crosses the arc drawn In ( c ) .

(f) Draw a ray from P on the base l ine through the intersection of the arcs.

Use your pro t rac to r to, check th i s construct ion,

4. Bisecting an angle

The four figures illustrate t h e s t e p s in bisecting an angle

with straightedge and compass.

Study these steps in t h e

constructions. Then draw an angle on a piece of paper and

bisec t it,

Can you state in your own words what is to be done in each

step?

Use your pro t r ac to r to

measure t h e two angles t h a t you

have constructed. A r e they

equal in measure?

1. Use your ruler t o draw a hor i zon ta l l i n e 1.3 inches long. Construct on a given vertical l i n e a segment of the same length.

1 2. Use your ruler t o d r a w a vertical segment 2v inches long.

Construct on a given horizontal line a segment of t he same length.

3 . Use your ruler to draw an oblique segment 5 centimeters long.

Construct on a given hor i zon ta l line a segment of t h e same

length.

4. Bisect each segment t h a t you constructed in Problems 1 through 3 . Use only compass and straightedge,

5. Draw an acute angle and an obtuse angle. Copy each angle

using straightedge and compass only.

6 . Draw an acute angle and an obtuse angle. Bisec t each angle using straightedge and compass only.

7 . (a) Draw a large t r i ang l e . Bisect each angle of the

trfangle. Extend the bisectors until they intersect.

( b ) When th ree o r more lines i n t e r s e c t I n one po in t , they

are called concurrent lines. Do the bisectors a p p e a r - t o be concurrent lines?

8. Draw a segment and then divlde it into 4 equal p a r t s . Use

straightedge and compass only.

9 . Draw an obtuse angle and construct rays whfch d i v i d e the

angle into fou r congruent angles.

The bas ic constructions t h a t you have s tud led can be used in many d i f f e r e n t ways. A t t h i s time you will explore only a few. When you study geometry i n high school, you w i l l f i n d many more. T h i s will be a discovery lesson. If necessary, sho r t exp lana t ions w i l l accompany a problem.

Exercises - 4-2b

1, Draw a line segment about as long as t h i s one.

(a) Using this length aa a radius, draw a c i r c l e with one endpoint of the segment as the center.

(b) With the same radius and the o the r endpoint as center, draw another c i r c l e .

( c ) A t how many points do the c i r c l e s Intersect?

(d) Choose one poin t where the two circles intersect and draw segments from this point to each endpoint of the or ig ina l segment.

( e ) Compare the measures of the three segments.

(f) What kind of triangle d i d you construct?

2. Construct a triangle whose sides are the same lengths as the segments given here.

You can use the p lan i n Problem 1, except thac circles in s t eps (a) and (b) w i l l not have the same radius.

3 . (a) Construct a triangle with a base t h e same length as the segment drawn here, and wfth the angles at each end of the base congruent to these angles.

H i n t : Use the base a s one s i d e of each angle.

(b) W i l l a l l triangles constructed w i t h these measures look alike? This construction is used to draw triangles

when two angles and the s ide common t o these angles are mown.

[aec . 4-21

4. (a) Construct a t r i ang le w i t h two sides the same size as

these segments and w i t h the angle formed by t hese segments the same size as the angle drawn here.

(b) Will all triangles constructed with these measures look alike? This construction is used when two sides and

the angle determined by those sides are h o w n .

5. Construct a rlght tr iangle that has one a c u t e angle of 60'. Hint: Can an equilateral triangle be used as t h e basis for

this? How many degrees are there in each angle of' an equi-

la tera l triangle? Two r i gh t triangles can be made from an

equilateral triangle in two ways, using cons t ruc t ion 4 or using construction 2. Try both methods and check your con- s t r u c t i o n with a protractor.

6 . Draw t h e following:

( a ) 3 concurrent llnes (b) 4 concurrent lines ( c ) 5 concurrent lines

7. a ) Draw three rays such that the endpoints of the rays are the only point of intersection.

(b ) How many angles are formed by the rays in part (a)?

8. Construct a segment that has the same length as t h e d i f ference between the lengths of these segments.

9 . ( a ) Draw a triangle, then find, by construction, the mid-

point of each side. Connect each of these midpoints

to the opposite vertex. These segments a re ca l l ed t h e

medians of a t r i ang le . (b) Are the medians concurrent?

A polygon is a simple closed curve formed by straight line segments. A polygon with four s i d e s is a quadrilateral. There

are many kinds of quadrilaterals. Trapezoids, parallelograms,

rectangles, and squares are all quadrilaterals. A polygon with five si'des is a pentagon, a six-sided polygon is a hexagon, and an eight-sided polygon is an octagon. If the sides are all t he

same length and the angles a l l have t h e same measure, it is a

regular polygon.

If every vertex of a polygon is a point on a circle, we say

t h a t t h e polygon is inscribed in the circle. In this sect ion, you will use the constructfons you have learned in order to Inscribe equilateral triangles, squares, hexagons, and octagons.

The problems contain enough clues for you to do all of these

constructions.

Exercises 4-2c

1. h a w a circle of radius 2 inches. With a compass set to the radius of the circle and starting with any point on the circle, mark off an arc on the circle. Move the point of the compass

t o one point where that arc crosses the c i r c l e . Mark another

arc on the c i r c l e . Continue until the arc drawn falls at the

starting point. If you do this carefully you will discover

that the last arc drawn f a l l s exactly on the starting point.

(a ) How many arcs are there?

(b) Connect each intersection of the circle and an arc to the intersection on each side of it.

( c ) What figure do these segnents form?

( d ) How can you use these po in t s to construct an equilateral triangle?

(e) How can you form a six-pointed star?

2 Draw a c i r c l e and one diameter. Use your prot rac tor to draw a dlameter perpendicular to the first diameter. Connect t h e

endpoints of the diameter8 i n order.

(a) What figure doea this form?

(b) How can you form a polygon with twice as many s ides? There are two ways t h a t this can be done. Can you

find both of them?

(Since you used your protractor, the figure you have drawn is not called a construct ion. We will learn in Section 4-5 how to construct a right angle.)

3 . Construct a circle and construct in the circle an inscribed

equilateral triangle, an Inscribed hexagon, and an inscribed

polygon of 12 sides.

4. Many designs can be formed with these basic cons,tructions.

Three are given here, See if you can copy them; then make up

In the last sec t ion you worked with geometric cons t ruc t ions .

In this sec t ion you will explore some of the properties of t h e

figures you constructed. Most of t h e constructions are examples of' symmetry and also examples of congruence.

This sec t ion is developed so that you will be able to discover

f o r yourself what is meant by syrmnetry, and in Sections 4-4 and 4-5 you will study congruence.

Class Exercises 4-3

I. (a) Fold a sheet of

notebook paper (or other paper

with square corners) down t h e

middle. Starting at t h e folded

edge, c u t o r t e a r o f f a right

triangle w i t h the longer leg along the f o l d , as in t h e second

f i g u r e . Cut a right triangle from both s i d e s of t h e folded

paper at t h e same time. The

right-hand figure above is a

"double" sheet. Unfbld the p a r t

which remains. What shape is it?

(b) Label as A the v e r t e x at t h e fold, and the other

vertices as B and C. Label as D t h e intersection of the fold and the side. TQe figure now resembles t h e figure below.

( c ) Refold your triangle

along AD. Do right triangles ABD and ACD exactly f i t over each other? Ke say that triangle ABC has symmetry

with respect t o t h e l i n e Ad because - when fo lded along AD the two halves

exact ly fit. Line is an axis of -- symmetry of the t r i ang le .

( d ) How many axes of symmetry does an isosceles t r i ang l e have? An e q u i l a t e r a l t r i a n g l e ? A scalene triangle?

2. (a) Take another piece of notebook paper and fo ld it

lengthwise down t h e middle.

( b ) Take your folded sheet and fold it crosswise down t h e middle

( so t h a t t h e crease folds along i t s e l f ) .

Cut off the corner where the folds meet, as indicated in t h e f igure a t the right. FOLD Unfold t h e piece y o u cut o f f . What

shape is It?

( c ) b b e l your figure a s in

the f l g u e a t the right. If you fold - along AC, do the two halves exactly ---d ---- . fit? What happens if you f o l d along - DB? Is there an a x l s o f symmetry?

How many?

3. Look at t h e regular hexagon

I n this figure. Does each dotted llne determine an axis o f symmetry?

There are o t h e r axes of symmetry. Can you find them? How many axes

of symmetry does a regular hexagon

have?

4, D r a w a c l r c l e and one of its diameters. Is t h i s diameter on

an axis of symmetry? Does a clrcle have other axes of symmetry? Are t h e r e 5 axes of symmetry? loo? lo5? Are there more than

any number you may name?

5 . Look at t h e e l l i p s e in t h e

figure t o the right. It is a figure

you ge t if you slice of f t h e t i p of --------- a cone bu t do not s l i c e straight across. Is an axis of symmetry?

A r e t he re others? How many axes of

symmetry does an e l l ipse have? The segment is cal led the major axis of the ellipse. On another axis of symmetry I s a minor axis of the ellipse. Why do you t h i n k is called the major axis? Where is t he minor axis?

From these exercises you have learned that many of t h e geo-

metric figures you know are symmetrical w i t h respect to a line.

Many ornamental designs and decorations also have such symmetry.

Definition. A figure - is symmetrical with respect --- to a line 1, if f o r each A on t h e figure, t h e ~ e is a --- -- --- point B on the figure f o r which 1 is a bisector -- -- - -- of AE and is perpendicu la r to AB. - -- -

Exercises t - 3 - l . , D r a w a rectangle and draw i t s axes of symmetry. Label each

a x i s of symmetry. How many axes of symmetry does a rectangle

have?

2. Draw an equilateral t r i a n g l e , and l a b e l each axis of symmetry. How many axes of symmetry a r e there?

3. Draw a square, and l abe l each axis of symmetry, How many axes

of symmetry does a square have?

[aec. 4-31

4. Draw and la5el the axes of symmetry, if t h e r e are any, f o r

each of the figures. How many axes of symmetry does each

i figure have?

5. Fold a piece of paper down the middle and then cut designs

in it. Unfold. Is the design symmetrical with respect t o

t h e fold? Is the fold an axis of syrnrnetry?

6 . Fold a piece of paper down t h e middle, and then fold it again in the m i d d l e , perpendicular t o the f i rs t fold. Cut a design

in it and unfold. Were are t h e axes of symmetry?

7 , We say t h a t a circle I s symmetrical w i t h respect to a p o i n t , its center, and that an ellipse is symmetrical with respect a

pcint, i t s center (the point where i t s major and minor axes intersect). We also say t h a t the f igure below is symmetrical

with respect to t h e point 0. Describe i n your own words what

you think is meant by symmetry with respect t o a po in t .

Which of t h e figures in Problem 4 have symmetry w i t h respect

t o a point?

8. When an orange is cut through the center in such a way that

each section of t h e orange I s c u t in half , we may think of t h e surfaces made by t h e cu t as symmetrical. Symmetry of

this kind I s symmetry with respect t o a plane. Name other objects t h a t are symmetrical with respect t o a plane.

4-4. Congruent Tr iangles

I n Class Exercises 4-3, Problem 1, by paper fo ld ing and

cutting, you produced an isosceles triangle. The axis of symmetry ( t h e fold) divides the isosceles t r i a n g l e into two r i gh t triangles

which have the same size and shape, When two figures have the

same size and shape we say that they are congruent. The two right

tr iangles are congruent triangles. Can you th ink of other congruent figures? Are two circles

congruent, if each has a r ad ius of f ive inches? Are two line segments having the same length congruent?

Since an angle i s a geometric f igwe , may we talk about two

congruent angles? Congruent figures have the same size and shape. Now if two angles have the same measure they have the same, size, and t o say t ha t two angles have the same size mean8 that they have the same measure. Appearance t e l l s us that two angles with equal measures have t h e same shape.

Angles B and F, a s shown, have equal measures. We may sag

LB is congruent t o LF, and we may write ,& 2 LF, where t h e s lpbo i I t 2 $1 - stands f o r t h e phrase "is congruent t o . " Is LF 2 LG?

We know t h a t t w o c i r c l e s are congruent i f they have t h e same

radius. Two squares are congruent If they both have t h e same

measure f o r t h e i r s i d e s . Two line segments are congruent if they

both have t h e aame l e n g t h . Two angles a r e congruent if the i r measures are the same.

Are two rectangles congruent if t he i r bases are equal? No.

If t h e i r bases and heights are equal? Y e s , You can see t h a t the

rectangle requires two conditions f o r congruency. Triangles are so basic I n much of mathematTcs, sc ience , and

engineerfng, t h a t we need t o know conditions under which triangles

are congruent. The situation here involves more conditions than

i n the figures we have already discussed. r F

u

If triangle DEF were traced on paper and the paper cut

along the s ides of t h e triangle, t h e pape r model would represent

a triangle and i t s i n t e r i o r . The paper model could be placed on triangle ABC and t h e two t r i ang les would exactly f i t . The two

triangles are congruent. If poin t D were placed on point A - with along AC, poin t F would f a l l on point C, and po in t E would fall on point B. I n these two triangles there would

[sec. 4-41

be these pairs of congruent segments and congruent angles:

LB 2 LE Use your ru l e r and

p r o t r a c t o r to check LC 2 LF these measures.

Recall that another way of expressing LB 2 LE is m (LB) = rn (LE) . Our choice of expression wlll depend upon whether we wish to

emphasize t h e angles as being congruent figures or the measures as being equal numbers.

If two triangles are congruent, then for each angle OP s ide

of one triangle there is a congruent angle or s i d e in t h e other t r i ang le .

In t h e Class Exercises;you will study with your teacher conditions which wlll make two triangles congruent.

Class Exercises 4-4 - 1. Construct a trlangle which 18 congruent t o triangle ABC;

A

You m i g h t s tart t h i s construction by constructing a segment - - BfCi congruent t o BC.

I

B' c ' Next you m i g h t th ink of constructing an angle congruent to

el ther LB or LC.

B' [aec. 4-41

4

The figure shows the construction of ray BIK so that

As a next step two p o a s i b l l i t l e s might be considered: - (1) Mark A 1 on B I K or (2) Construct ray CIL

so that so that - - ~ 1 ~ 1 2 BA LB'C'L 2 LC

A A L A"

K"

B ' -. c B ' C'

Draw A t C I Mark A " .

You now have a triangle AIB'CI and a triangle A " B ' c ' .

In both cases only three parts of triawle ABC have been --- - -- copied.

Is triangle A t B t C I 2 A N ? Is triangle A " B ' C ' 2 ABC? Ln order to obtain the answers to these queationa you must measure the parts of triangle A l B t C l and of triangle A " B ~ C I t o see if these triangles are congruent to ABC,

If your constructions and measurements are correct you w i l l

f ind that A ABC 2 A A 1 3 f C 1 and A A B C Z A A " B ~ C ~ .

In (I) by copying two s i d e a and the included angle of triangle A X , you have been able to cons t ruc t a triangle congruent to

triangle ABC. In (2 ) by copylng two angles and the included s i d e ,

you have been able to construct a triangle congruent to trlangle

ABC.

In (1) we refer to the included angle and in (2) to the

included s i d e . Do you see why the word "included" is convenient?

In the first case the s i d e s are part of the angle and In the second case the side is part of two angles.

While one construction is not sufficient evidence on which

to base a conclusion, your experience with that of your teacher

and your classmates should convince you of the fallowing properties

of two congruent triangles.

Twc - t r i a n g l e s - are congruent if two sides and t h e included ----- angle -- of one t r i a n g l e - are congruent respectively --- to t w o sides

and t h e included angle o f t he o ther triangle. We will refer t o -- --- t h i s p roper ty as Proper ty S .A .S. ( ~ l d e , Angle, ~ f d e ) .

Two - triangles - are congruent if two angles and the included -- -- side of one t r i a n g l e a r e congruent r e s p e c t i v e l y t o two angles --- - -- and t h e included side of t h e o the r triangle. Me w i l l r e f e r ta -- ---- t h i s p r o p e r t y as P r o p e r t y A . S .A. (Angle, S i d e , Angle ) .

You a r e asked to accept these two properties and a t h i r d

one t o be developed in Problem 2, on t h e basis of experiment.

As you continue you will use these three properties as a means

3 f showing o t h e r properties,

2. C m s t r u c t a t r i a n g l e using E , E, and CA as sides. Your

work s h o u l d look like this figure:

You wzll reca l l t h a t this is the construction of Frob1e:n 2 ,

Zxercises 4-2b.

Is your t r i a n g l e t he same size and shape as triangle AEC?

Use p r o t r a c t o r o r cbrnpass and ruler t o check.

Two triangles - a r e csngruent if t h e three sides of' one t r i a n g l e ------ are congruent r e spec t ive ly to t h e three sides of t h e other triangle. ------- We will re fe r t o this as'property S.S.3, (side, Side, side)

Notrce that t h e congruence s e t s up a one-to-one correspondence

between pairs of s ides of two congruent- triangles, because we can l e t congruent sides correspond t o each o t h e r . That i s , suppose

we c a l l a , b, and c the sides of one triangle and r, s, and

t the s i d e s of a triangle congruent t o it and suppoae sides a

and r, b and s, c and t are congruent. Then we may c a l l a and r corresponding aides, b and s corresponding sides, and c and t corresponding a i d e s . For this one-to-one correspondence it is true that if' two triangles - are conaruenf then t he i r -- correspandinq -- s ides are c o n ~ m e n t . We could s e t up t h e same kind

of correspondence for angles and have: -- If two triangles are con- -- gruent then their correspondiq angles are congruent. The converse

of the first of these two statements 18 a true statement, but the

converse of the second is not a true statement. You w i l l reca l l t h a t the converse of the first statement ib, if the corresponding sides of two triangles are congruent, the two triangles are congruent. (property S.S.S.) See Problem 2 below.

Exercises 4-4 - Use triangle ABC in Problems 1 and 2.

A c -4

1. (a) Construct a t r i ang l e HJK such that m = m, XzBC, and 2 m.

(b) Construct a triangle HJK such that LH 2 LA, 2 LC, and 2 E. - - r u -

( 0 ) construct a triangle HJK such that m = AC, HJ = AEj and fi 2 LA.

(d ) Were the triangles you constructed in (a), (b) and ( c ) congruent to A ABC? W h f l

2. Conatruct a trlangle HJK, whfch has LJ 2 LB, LH 2 LA, and 2 LC, but in which no side is congruent to a s ide of

t r iangle ABC.

Hint: Choose a segment which is not congruent

to any of t h e sldes of t r i a n g l e ABC. Construct LJ and L H .

(a) Why do we not need t o cons t ruc t LK? (b) A r e t r iangles AEC and HJK congruent?

( c ) t h y do we not s t a t e a congruent t r i ang les property like

t hose in this section, which we could r e f e r t o as

Proper ty A . A . A . ( ~ n g l e , Angle, Angle )?

In Problem 3 through 8, t he re are pairs of triangles. Congruent sides a re indicated by the number of s t r o k e s on corre-

sponding sides of two t r i ang les , and congruent angles by the

number of arcs in corresponding angles, The 2 triangles in a

p a i r may appear to be congruent when they are n o t .

Use p r o p e r t i e s S.S.S., S . A . S . , o r A . S . A . t o decide which

pairs are cmgruen t and w!~ich are not. I d e n t i f y the p r o p e r t y you

d z e in showing t h e triangles to be congruent if they are.

Problems g through 17 r e f e r t o parts o f t r iangles AEC and

PC&. Use P r o p e r t i e s S.S.S., S . A . S . , or A . S . A . t o decide which

pairs are congruent and which are not. - - - 9 . LA 2 LP, 2 LQ, AB = PQ

+ - 10. ABzP&, A C ~ P R , B C = Q R

11, LA 2 LB 2 LC 2 LP 2 2 f~

12. LC 2 fi, LB 2 LQ, AB 2 P3 m -

13. LA 2 LP, LB 2 LQ, BC =

I k , A F = 7, Pa = 7 , LA) = 28, ~(LP) = 28, CA = 10, RP = 10

15. A 8 = 3 , BC = 4, CA = 5, a = 4, P & = 3, RP = 5 - b - 16. LA 2 LP, LB 2 LQ, AB = QR

* - ' V - % - 17. AB = AC = BC, = a, LP is not congruent t o LQ. 18. If two t r iangle i in t h e same plane are drawn on opposite

sides of a l i n e and a r e symmetrical with,respect to the line, how can one triangle be superimposed on the other?

19. BRAINBUSTER. A t r i ang le has 3 angles and 3 sides. We have

seen t ha t two triangles are congruent by Property S . S . S . ,

Property S.A.S. , or Property A . S . A . In Problem 2 we saw that two triangles need not necessarily be congruent if the 3 pairs of angles (one from each triangle in a pair) are congruent. Therefore we can - not say that if 3 of the 6 parts (angles and sides) of one triangle are congruent t o 3 of the 6 parts of another t r iangle , t h e triangles are congruent. In Problems 1 and 2 we have considered fou r cases. There are two cases remaining. What are t h e p Construct several triangles to show whether in these cases, two trfangles are necessarily congruent.

4-5. Showing% Wiangles -- To Be C o w ent

If we wish to m o w whether two triangles are congruent, we could measure each side and each angle. This would require 12 measurements , 6 f o r each triangle, One way we could cut down on .

t h e amount of work would be t o measure only two angles In each triangle. Since the sum of t h e measures of the angles of a triangle is 180 we could determine the th i rd angle i n each t r iangle without measurement, This would leave only 10 measurements if we were trying to see if two triangles are congruent. Of course we

could also test for congruence by "cutting and fitting," but thia is a l s o time-consuming and often inconvenient.

We can shorten our use of measurement If' we make use of one of the three properties of congruent t~langles. For example if we find two pairs of sfdes (one from each triangle in a pair) in two triangles are congruent, we would then measure the included angle in each triangle. 3f the included angles are found to be

congruent, then we need take no further measurements. Property

S.A.S. t e l l s us that the two triangles are congruent. In the figure, 2 n. 4

Hence, the triangle is isoscelea. LDAB 2 LDAC. Hence, AD bisects - LBAc. Is D the midpofnt of BC? It looks as if it m i g h t be. We

could make measurements which

would answer this question If we

allow f o r the approxfmate nature of measurement. Would t h e bisector 8 0 C

of the angle determined by the congruent sides of any isosceles tr iangle always intersect the opposite side in the midpoint of that side?

We can find an answer to the question if we make use of one

of our congruent triangles properties. Consider A DAB and A DAC.

[aec . 4-53

ABZAC by construction

LDAB 2 LDAC AD Is on the b t sec to r of LBAC + - AD = AD AD is in both triangles

By Property S,A,S., A D A 3 2 ADAC, since 2 sides and the

included angle of ADAB are congruent t o the corresponding sides

and corresponding angle of A DAC . - BD 2 DC since they are corresponding sides of congruent

triangles. Therefore, D is the midpoint of m. If Property S.A.S.

is true, we now can be sure t h a t :

The bisector of t h e angle determined by the congruent sides

of an isosceles triangle intersects the t h i r d side in the midpoint

of t h a t s i d e .

Exercises 4 - ~ a

1. In the figure the constmr~tion of the bisector of LABC is shown. Two segnents, AD and CD are drawn.

(a) What parts of t r iangle ABD are congruent to correspond-

ing parts of triangle CBD by constructlon?

(b) Is t r i ang le ABD congruent to t r i ang le CBD? Why?

( c ) IS LABD congruent to LCBD? my?

2 In the figures shown here the construction of &JK makes

BJK 2 LEFG. Segments and are drawn.

(a) What parts of triangle EFG are congruent to corresponding parts of triangle HJK by construction?

(b) Is triangle EFG congruent to t r i ang le WJK? Why?

( c ) Is LJ congruent t o LF? Whfl A

3. Use your protractor to find t h e measures in degrees of the

3 angles in each triangle.

(a) A r e there some pairs of congruent angles? If so,

list them. 0 C (b) Could we say that t h e

triangles are c o m u e n t ?

( c ) Suppose that triangle DEF were constructed w l t h

LA 2 LD and LB 2 LE, and w i t h the same length as E. What would be true about the two triangles? Why? E F

4. M r . Thompson wishes t o measure the distance between two posts

on edges of h i s property. A grove of trees between the two

posts (X and Y) makes it impossible to measure the distance XY direc t ly . H e locates point Z such that he can lay out a l i n e from X t o Z and continue it as far as needed.

Poin t Z is also in a position such that Mr. Thompson can lay out a line and con- t lnue it as far a8 needed. M r . Thompson knows t h a t

Ll 2 L2 since they are vertlcal angles, He extends - m SO t ha t &Z 2 75, and

% - XZ so that XZ = RZ.

(a ) Are triangle8 XYZ

and QZR congruent?

mfl (b ) How can M r . Thompson

determine the length of XY?

5 . ( a ) In the figure below are the shaded triangles congruent?

(b) Which properties do you use t o t e s t the congruency of

these triangles?

We w i l l make use of t h i s figure again when we study similar triangles.

6 . Line 1, and line j2 are parallel lines cut by a transversal t .

(a) What do you know about angles 1 and 2?

(b) Are angles 2 and 3 congruent? Whfl

( c ) Show that Ll 2 /3.

7 . In the parallelogram, ABCD, the diagonals AC and BD in tersect at E.

(a ) Is angle 1 (in AABE) congruent to angle 2?

( b ) What kind of angles are 1 2 and L3? Are they

congruent?

( c ) How does the size of Ll compare w i t h the size of L3? ( d ) Show t h a t L6 2 L7 and 2 L4. ( e ) When two t r iangles have three pairs of congruent

angles, are the trLangles always congruent? If not, what else is needed?

(f) Is any side of ABE congruent to the corresponding side of A CDE?

( g ) Show that the diagonals of a parallelogram b i sec t each other.

"8. If two angles of a t ~ i a n g l e are c o n g r ~ e n t , t h e sides opposite them are congruent. Use a property of congruent triangles t o show that this statement is t r u e .

Hint: A triangle can be congruent t o i t se l f . Show ABC 2 BAC, where LA 2 LB.

Erecting - a perpendicular -- from a point on a l l ne --- Follow the four steps in t h e con-

struction as illustrated in the figures,

and construct a perpendicular from a paint on a line. Measure the angles i n the f igu re you construct t o see if the

line you have constructed appears t o be

a perpendicular .

If the segments and HJ are drawn, t w o triangles GPJ and HPJ are formed.

- GP = WP by construction.

a - = HJ by construction.

m z m a common s i d e to both triangles.

A GPJ 2 AHPJ Property S . S . S .

LGPJ 2 LHPJ corresponding angles of two congruent t r iangles.

If a protractor is laid along 8I with the vertex mark at P and 0' mark

+ on PH, then t h e 180' mark will be on

the ray %. This means t h a t the sum

of the measures of t h e angles at P is

180 and, since these measures are equal, each must be 90. Hence, angles JPG

and JPH are right angles. You have now seen how to construct

t h e perpendicular t o a l i n e at a point

on the l i n e , and you have seen why the

construct ion "works. "

Exercises

1. Draw a segment about 4 inches in length, and then construct perpendiculars to t h e segment at each endpoint.

H i n t : Extend the l i n e segment when necessary.

2. Construct a perpendicular to a line from a point not on the line.

Hint: Follow the steps in the constructions given on

the preceding page.

3 . YOUP construction in Problem 2

will look l i k e t h e f igure at the

right. Label t h e intersection

of t h e 2 arcs Q and the i n t e r - s e c t i o n of Pi'& and as C.

(a) Why is t r i a n g l e APQ con-

gruent t o triangle BPQ?

(b) Why are angles 1 and 2

congruent?

( c ) What o t h e r pairs of angles are congruent?

4. Using congruent triangles in t h e figure In Problem 3, show w *

t h a t PQ is perpendicular to AB. Hint: Use Property S . A . S . to show triangle ACP Is

congruent to triangle BCP, Use a p r o t r a c t o r to find t h e sum of t h e measures of LACP and LBCP.

5. The construction of the perpen- 4 - d icu la r b i sec to r of segment CD

is shown, Usually t h e same radius

is used f o r the four arcs. However,

it is only necessary for the two

arcs that intersect on one side . C D , of t h e scgxent to have equal

r ad i i . Thus, t h e arcs drawn from C and D t h a t i n t e r s ec t at E

have equal radii, and the two arcs v

drawn from C and D that intersect at F have equal rad i i .

In the rigure, CE is not congruent to m. By applying some

of the properties about congruent triangles, show why bisects CD and is perpendicular t o m.

6 (a) How many pairs of congrueilt triangles are there In the figure f o r Problem 5? List them by pairs.

(b) List the pairs of correaponding aides f o r each pair of congruent triangles.

( c ) List the pairs of corresponding angles f o r each pair of congruent trianglea.

7. (a) Draw a tr iangle. Then erect perpendiculars from each vertex to t h e oppoalte s i d e . Extend the perpendiculars until they intersect. (1t may be necessary to extend a side of the triangle so that the perpendicular w i l l

meet t h i s s i d e . )

(b) What do you notice in t h i s figure?

8. In what ways are the constructions for bisecting a l i n e segment (sect ion 4-2) and constructing the perpendicular to a l l ne segment al ike , and in what way are they different?

4-6, The - R i g h t Triangle

Triangles may be named according to the measures of the

angles. In the following s e t of triangles each has an angle

whose measure In degrees 1s 90. Trianglea having this property

are called right trlawles.

The ancient Egyptlans a r e said to have used a particular

right triangle t o make corners "square." This trlangle has sides 3 units long, 4 units long, and 5 units long. When such a triangle is made of t a u t l y stretched rope, the angle between

t h e two shor t e r sides is a right angle. While t he Egyptlans are thought to have made use of t h i s

f ac t , it was left to the Greeks to prove t h e r e l a t i o n s h i p involved.

The Greek philosopher and mathematician, Pythagoras, who l i ved about 500 B.C. became interested

In the problem. Pythagoras I s

credlted w i t h the proof of t h e

basic property which we w i l l A I Unit study in t h i s sec t ion; t h i s

property is s t i l l hown by h i s

name, the Pythagorean Proper ty .

It is thought that Pythagoras looked at a mosaic like the one pictured in Flgure 4 - 6 ~ . He noticed that there are many triangles of di f fe ren t s i z e s that can be found in the mosaic. But he not iced more than this. If each s ide of any triangle is used as one side of a square, t h e sum of the areas of the two smaller squares is the same aa the area of the larger square. In Figure 4 - 6 ~ , two triangles of different size are Inked in and the

squares drawn on the aides of the t r i ang l e s shaded.

Figure 4-6A Figure 4-6B

Count the number of the smallest triangles in each square. For each triangle t h a t is inked in, how does the number of small triangles in the two smaller squares compare with the number in

the larger square? If you draw a mosaic like t h i s , you will find t h a t t h i s is t r ue not only f o r the two triangles given here but for even larger triangles in this mosaic.

Pythagoras noticed the same relation in the 3-4-5 tr iangle that the Egyptians had used f o r so long to make a ri&t angle. The small squares are each 1 square unit in size. In the

three squares there are 9, 16 and 25 -11 squares. Notice that

9 + 16 = 25. Pythagoras was able to prove tha t in any right

triangle, the area of the square on the hypotenuse (longest side) - -A - --

is equal t o t h e sum of the areas of the squares on the o t h e r two 7 -------- ---- sides. This is the Pythagorean Theorem or, as we shall call It,

the Pythagorean Property. So far t h i s has been shown only f o r two very special right

triangles. It is true for a l l right trianglee. Some of you

may try to prove it fop yourselves by studying Section 4-7.

Exercises 4-6a

1. Using your straightedge and protractor draw the following:

(a) 30-60 triangle (b) 45-45 triangle ( c ) 70-20 triangle

2. Show for each s e t t h a t t h e square of the first number is

equal t o t h e sum of the squares of the other two numbers.

(a) 5 , 4, 3 ( c ) 2 5 , 7 , 2 4

(b) 13, 5, 12 ( d ) 20, 16, 12

3 . Make a drawing of t h e triangles w i t h the sides of length

given in part (a) of Problem 2. Use your p r o t r a c t o r to show t h a t this t r i ang le is a r i g h t triangle.

4. Draw right triangles, the lengths of whose shorter s i d e s

(in centimeters) are:

( a ) 1 and 2 (b) 4 and 5 ( c ) 2 and 3.

Measure, to the nearest one-tenth of a centimeter if possible, the lengths of the hypotenuses of these triangles.

5* Use the Pythagorean P r o p e r t y to find the area of the square on the hypotenuse fo r each triangle in Problem 4.

In the last set of exercises, you worked w i t h sides of right

triangles and aquares b u i l t on the a i d e s of r i g h t triangles. In Problem 4 you measured the length of the hypotenuse. The hypotenuse is the s i d e opposite the right angle. It is not very useful to h o w t h e area of these squares, but many uses can be made of the

Pythagorean Property if we can use it t o f'lnd the length of the t h i r d side when we know t h e length o f two aides. In mathematical

language the Pythagorean Property is c2 = a' + b2 where c

stands for the measure of the hypotenuse and a and b stand f o r

the measures of the o t h e r two s i d e s . The measures of any two aides can be substituted in the number sentence above, and from t h i s the

t h i r d value can be found. We can use a familiar triangle t o show

this. If the two short aides are 3 units and 4 units, what is

the square of the hypotenuse?

Since c2 is equal t o 25, we will h ~ o w c if we can f i n d

a number whose product is 25 when it is multiplied by i t s e l f . Of course, 5 x 5 = 25, so c = 5; 5 is the p o s i t i v e square root of 2 5 , 13 a number is the product of two equal factors , then each f a c t o r is a square root of the number. The symbol f o r the positive square root is: . The nwneral is placed under the slgn;

f o r example, 6 = 5.

What is f i ? a ? 6 ? f i ? The first three

are easy t o understand since 3 x 3 = 9, 4 x 4 = 16, and 6 x 6 = 36, but there is no integer that can be multiplied by i t s e l f t o give the product 30. I n fact, there i s no rational number whose square is 30!

Can you find a number multiplied by i t s e l f that w i l l g ive a product c lose to 30? Yes, 5 x 5 = 25 which is close to 30.

What is 6 x 6 or 6*? This t e l l s us t h a t 6 is greater than 5 but l e s s than 6 . We m i g h t try to get a closer approximation

by squaring 5.1, 5.2, 5.3, 5.4 and 5.5 Now (5.4)2 = 29.16 2 and (5.5) is equal t o 30.25. Because 30.25 is closer t o 30

than 29,16 we might assume that f i i s nearer 5.5 than 5.4. However we would need to square 5.45 t o get 29.7025 before we

can say t h a t is 5 .5 t o t he nearest t en th . You may want to estimate the square r o o t s of some numbers

this way o r use the table at t h e end of this sec t ion . Note that in the table a t the end of this aection, J30 is

5.1!77 t o t h e nearest thousandth. Rounded t o tenths this would be 5.5. You can see how c lose your estimate is to the number in

t h e table .

The table gives the decimal value (to the nearest thousandth) that is c loses t to the square root of integers from 1 t o 100. You can a l s o use t h e table to f t n d the square roo t of a l l counting

numbers up t o 10,000 t h a t have rational square roots.

[aec. 4-63

Exercises 4-6b

When approximate values are used in these problems, use the

symbol, , in the w o ~ k and answer.

1. Use the table to f ind the approximate value o f :

(a> f i ( c ) fi (4 rn {b) fi (a) JF (f) m

2. Use the Pythagorean Property t o find t he length of the hypot-

enuse f o r each of these triangles. (a) Length of a is l", length of b is 2"

(b) Length of a l a 4', length of b is 5' ( c ) Length of a I s 2", len&th of b is 3"

(d ) Length of a is 5 yd.and the length of b is 6 yd. ( e ) Length of a i s 3 f t .and the length of b is 9 ft. ( f ) Length of a is 1 unit and the length of b is 3 units.

3 . Sometimes the hypotenuse and one of the shorter sides is Icnown. How can you f i n d t h e length of t h e other side? As an example,

use this problem. The hypotenuse of a r igh t t r i ang le is 13 ft.

and one side is 5 ft. Find the length of the t h i r d s i d e . 2 c 2 = a + b 2 2 132 = 5 + b 2 -

13* + -(52) = b2 ( add i t ion proper ty of equality)

169 + -(eg) = b2

144 = b2

a =J'F The t h i r d s ide is 12 feet long. Find the third side of these right triangles. The measurements are i n feet.

(a) c = 15, b = 9 (b) c = 26, a = 24

( c ) c = 39, b = 15

4. A telephone pole is steadied by guy wlres as shown. Each wire is to be fastened 15 St. above the

[aec. 4-61

ground and anchored 8 ft. from the base of the pole. Bow much wire Fs needed t o s t re tch one wire from the ground to the point on the pole at which the wire 1s fastened?

5. A roof on a house is bullt as show.. How long should each rafter be if it extends 18 inches over t he wall of the

house ?

6 A hotel bullds an addit ion across the street from the original building. A pas- sageway l a built between the

two parts at the th i rd - f loo r

level. The beams that sup- p o r t t h i s passage are 48 ft . 1 - above the street . A crane operator is lifting these beams i n t o place w i t h a crane arm t h a t is 50 ft. long. How far down the street from a poin t d i r e c t l y under t h e beam should the crane cab be?

7 . A garden gate is 4 ft. wide and 5 ft . h3gh. How long should the brace that extends from C to D

be?

8. Two streets meet at the angle shown. The streets are 42

ft. wide. Lines f o r a cross- walk are painted so t h a t the

walk runs in the same direction a s t h e s t ree t . If it i a 40 ft. from one end of the cross-walk to the poin t that is on the perpendicular from the o t h e r end poin t of t h e walk, how long is the cross-walk?

9 . How long f s t h e throw from home pla te to 2nd base In a soft-

ball game? The bases are 60 ft. apart , and a softball diamond is square in shape. Give your answer t o t h e nearest whble f o o t .

"10. D r a w a square whose s i d e s are o f length 1 unit. m a t is the l eng th of the diagonal? Check by measurement. NOK draw a

right triangle w i t h the s ides 1 u n i t long. Wnat is t he length

o f the hypotenuse?

"11. IJow draw a right triangle of sides "square root of 2" and 1

units in length as shown In t he f i g u r e . In t h e f l g u r e t h e measure of the length of AB is the square r o o t of 2. What f a the length of the hypotenuse of t h i s new triangle?

T A B L E

SQUARES AND SQUARE ROOTS OF NUMBERS

Square Square Number Squares r o o t s Number Squares r o o t s

1 2 3 4 5

6

6 9

10

11 12 13 14 15

16 17 18 19 20

21 22 23 24 25

26 27 28 29 30

31 32 33 34 35

1,296 1,369 1,4114 1,521 1,600

1,681 i ,7611 1,849 1,936 2,025

2,116 2,209 2,304 2,401 2,500

2,601 2,704 2,809 2,916 3,025

3,136 3,249 3,363 3,481 3,600

3,121 3,8244 3,949 11,096 4,225

4,356 4 , 4 8 9 4,624 4,761 4,900

6 . 000 6.083 6 1614 6.245 6.325

6.403 6.481 6 557 6.633 6 . '708

, 6.782 6.856 6,928 7. 000 1.071

7.141 '7.211 7.280 7.348 7.416

-1.483 'i' ,550 7.616 7.681 -{ ,746

7 -810 7.874 '1 . 93 7 8.000 8.062

8.124 8.185 8.296 8.307 8.367

1 4 9

16 25

36 49 64 81 100

121 1 4 4 169 196 225

2 56 289 324 361 400

411.1 484 52 9 57 6 62 5

676 729 784 841 900

961 1,024 l,c89 1,156 1,225

1.000 1. 411t 1.732 2.000 2 . 2 3 6

2.4119 2 .6b6 2 828 3.000 3.162

3.317 3.464 3.606 3 7'12 3 ,873

4.000 4.123 4.243 '4.359 4.472

4.583 4.690

4n796 4.899 5.000

5.099 5.196 5.292 5 .385 5.477

5.568 5.65'7 5.745 5.831 5.916

36 3 7 38 39 30

41 42 4 3 41: 45

96 4 7 48 4 9 50

51 52 53 5h 55

56

:B 59 60

61 62 63 64 65

66 67 68 69 70

Square Number Square s r o o t s

4-7. "One --- Proof of the Pytha~orean Property

There are many proofs of this property. The one used here l a

not the one used by Pythagoras. You should actually draw and cut the squares called for in the explanation.

Draw two squares the same sfze. Separate the first square into two squares and t w o rectangles as shown here:

Figure 4 - 7 ~

L e t t-he measure of each side of the larger square in Figure 4-7'A be and t h e measure of each s i d e of t h e small square be k. Notice the areas of the small squares and rectangles.

2 One square has an area of measure a . 2 The o the r square has an area of measure b .

Each rectangle has an area of measure ab.

Since the area of Figure 4-7A i a equal t o the sum of the areas of

a l l of i t s parts, the measure of the area of Figure &-?A is

Class Exercises 9 1. L e t a = 4 and b = 3. Show that (a + b12 = a* t 2ab + b 2

f o r these numbers.

L e t a = 2 and b = 6 and Check the same relationship.

Now t u rn t o the second square, U s e the same numbeq a and

b, t h a t were used in t h e f i r s t square.

Mark the lengths of f as shown here and draw the segnents m, - , RS and 5. Tne large square is separated into 4 t r iangles and a quadrilateral t h a t appears t o be a square.

1 The measure of each triangular area is ab. There are four

congruent triangles. The sum of the measures of t h e areas of 1 all f ou r t r i a n g l e s is q7 ab) o r 2 ab

If you look back to Figure 4-7A you will see that 2ab is

t h e measure of the area of the two rectangles. Cut the two

rectangles from the first square. Cut along the diagonal of each rectangle. See if t h e fou r t r iangles you cut are con-

gruent w i t h those In t h e second square. See if you can fol low these s t eps .

*square =a2 + 2ab + b2 (from Figure 4 -7A) 1

*square = 4 ( p ab) + A~~~ (from Figure 4 - 7 ~ )

Isec. 4-73

2 Therefore a2 + 2ab + b = 2ab + ApWS Wny?

a' + bz - - *PQRS (addltion property of

equality. ) 2 T h i s shows t h a t PWS has an area whose measure is a' + b units,

but a2 is the measure of the area of one small square in t h e f lrst f igu re and b2 is the measure of the area of the other square.

From this the area of t h e figure in the center of the second square

is equal to the sum of the areas of the two small squares.

Place the square whose area measure is a* along the s i d e of length a of one triangle in the second square. Place the square whose area measure is b2 along the s ide of length b of the same

triangle. The areas of the squares on the two s i d e s of the triangle are equal t o the area of the figure i n the center of Figure 4-73. A l l we need to do now is to prove that t h i s figure is a square!

What are the properties of a square?

1. The four sides are congruent.

2. Each angle is 90' in measurement.

If we can prove these two conditions f o r the quadrilateral in Figure 4-7f3, t h e Pythagorean Property has been groved.

As has been stated, t h e f o u r triangles are congruent s i n c e f o r each pa i r two corresponding sides and the angles determined by these a i d e s a r e congruent. As a r e s u l t , PQ = QR = RS = SQ because they are measures of corresponding segments of congruent triangles.

So fa r we have shown that the squares in Figure 4 - 7 ~ are congment to the squares on the short sides of anv one of the triangles in Figure 4-78. We have also shown that the sum of the areas of these

squares is equal to t h e area of PWS, and t h a t PQRS has four con-

gruent s i d e s . L e t us prove that the angles are r i g h t angles.

(1) In A PST, rn(L 1) + m ( i 2) = 90 Why?

(2) m ( L 1) = m ( L 4) Why?

(3) Therefore rn(L 4) + m(L 2) = 90 Why?

(4) and m ( L 2) -I- rn (1 3 ) + m(L 4 ) = 180 Why?

(5) m ( i 3 ) + 90 = 180 Wny ?

( 6 ) and m(L 3) = 90 Why?

We can go through the same type of reasoning to show that

angles 5 , 6 and 7 are also r i g h t angles. PQRS has been proved t o be a square and its area has a l s o

been proved equal to the s m of the areas of the squares on the tother two sides. i t

4-8. Quadrilaterals

Symmetry and congruence can be found in some quadrilaterals

o r pa r t s of quadrilaterals. It is a l s o poss ib le to find applications of the Pythagorean Property in quadrilaterals. This section

haa problems based on quadrilaterals that make use of these three ideas. In the Exercises it will be of assistance to keep in mind that :

A trapezoid has only one pair of parallel aides. A parallelogram has two pafrs of parallel sides.

A parallelogram which has four r i gh t angles is a rectangle.

A rectangle w i t h four congruent sides is a square.

Exercises 4-8 - 1. A figure is symmetrical with respect to a line if it has that

line as an axia of symmetry. Which of these figures are always symmetrical w i t h respect to a line? (a) Trapezoid

(b ) Parallelogram

( c ) Rectangle

(d) Square

2. How many axes of symmetry are there in: (a) A rectangle?

(b) A square?

3. Is is possible to draw a trapezoid that 1s symmetrical with

respect to a l ine? If so, draw one.

4. Is it possible to draw a parallelogram that is symmetrical with respect to a line? If so, draw one.

5 . One diagonal of a quadrilateral separates the figure I n t o two triangles.

(a) What i a the sum of t h e measures of the angles of a quadrilateral?

(b) Name the quad~ilaterals t h a t a re separated i n t o congruent triangles by a diagonal .

6 . Rectangle ABCD has a diagonal that is 82 inches in length. The width 2s 18 inches. mat i0 in.

1s the length? 11 A 0

7. The diagonals of this parallelogram a re perpendicular and the sides are equal in measure. The s h o r t e r diagonal is 1 4 fee t in length and the longer one is 48 feet. How Long is each

side? (~int: The diagonals of a parallelogram b i sec t each other. ) IB

/ 8. Two diagonals have been drawn in each of the following figures. i For each fFgure, A , B, C, D, E, and F, answer all questions.

Answer questions (a) and (b) for one figure before you answer any questions f o r t h e next figure.

B Quadrilateral Parallelogram Parallelogram with

a l l sides congruent Figure A Figure B Figure C

Trapezcld Figure D

Rectangle Square Figure E Figure F

[ aec . 4-81

(a) How many triangles are there in the figure?

( b ) Do any pairs of triangles appear to be congruent? It so, name the triangles fn pairs.

( c ) If you found triangles ABC and triangles CDA congruent, Ln one or more figures, choose one figure and show why they are congruent. Use the properties, S.S.S., S . A . S . ,

o r A . S . A . t o show t h l s . If you found triangles ABO and C W congruent do the same f o r these triangles. If you found both s e t s congruent in some f igure, choose

which pair you wish t o show congruent.

You have been drawing figures which are contained within a plane. You are now t o practice drawing on t h e surface of your paper pfctures of figures in space. You have found that it is easy to draw a plane figure on the surface of your paper or on t h e chalkboard. You will f i n d t h a t it I s not so easy t o draw pictures of s o l i d s on paper o r on t h e chalkboard. This is because you must draw the figure on a surface in such a way that it w i l l appear to have depth. In o the r words, you want t o make a drawing on your paper have the appearance of a box. This requires the

use of projec t ion which you have poss ib ly studied In a r t .

1. R i g h t P r i s m s

(1) Rectangular B i g h t Prisms. A good example of a rectangular prism is a cereal box. One way to draw a box is as follows:

(a) Draw a rectangle such as ABCD in the figure

be low.

(b) Now draw a second rectangle RSTV in a posftion similar to the one in the flgure. You may wish

to use dotted line segments in parts of t h i s figure. - - - ( c ) Draw AR, BS, W , and E.

[aec . 4-93

When you look at a sol id you cannot see a l l of the edges, o r

faces, unless t h e solid I s transparent. For this reason we represent the edges, which are not v i s i b l e , by dotted l i ne segnents . This a l s o he lps to give the proper pe r spec t i ve to t h e drawing.

If you wish, t h e dot ted l i n e segments do not have to be drawn.

(2) Triangular Right Prisms. Now t h a t you have drawn a

rectangular prism, a t r i a n g u l a r p r i s m will be easy.

(a) D r a w any triangle ARC.

(b) At p o i n t s A and C draw Lines of equal measure

perpendicular to E . Label the end p o i n t s of these perpendiculars as H and T.

(c) Draw BS parallel t o f i and o f equal measure with m.

( d ) Draw and 3. Then may be drawn w i t h a

dotted l i n e .

(e) Compare your figure w i t h the one below.

(f) How many faces does t h i s s o l i d have?

(g) In what way are t h e faces dif ferent?

(3) Hexa~onal Rimt Prisms

(a) Draw a hexagon, similar to ABCDEF of t he f i gu re

below. Thls is not intended to be a regular hexagon. In order to get the proper perspective it may be necessary to draw some s ides longer t h a n other sides. Thls is the way a hexagon would appear if looked at from an angle.

(b) Draw at A and F perpendiculars to havhg equal rneaswces. Label the end points

R and S. ( c ) Draw and parallel to and of equal

measure w i t h m . If you wish, you may draw dotted lines to represent the edges whlch are not v i s i b l e .

C

( d ) How many faces does this s o l i d have?

( e ) In what way are the races different?

II. Pyramids

Perhaps t h i s is t h e f i r s t time you have heard of the

s e t of s o l i d s , ca l led pyramids, in a mathematics text . You have probably heard of the famous Pyramids of Egypt. A

pyramid has one base, which I s a region formed by a polygon,

and triangular faces which a r e made by joining the vertices of.the polygon to a point which is not in the plane of the polygon. A more accurate deecrlption of pyramids w i l l come in a later chapter. Let us draw one.

( a ) In t h i s drawing let the base repreeent a square with a vertex at the bottom of the drawing. Here again, you

must be careful to get the proper perspective. (b) First, draw only two aides of the square, such as a

and E, as shown In the f lgure below. ( c ) Now select a point P, directly above poin t B, and

draw PA, and ( d ) m, and i% may now be drawn as dotted l i n e s e p

ments intersecting at D, with parallel to BC and CD parallel to m.

(e) How many facea does t h i s pyramid have?

111. Intersecting Planes There are times when it is useful to represent, on a

surface, the intersection of two or more planes. An example i a the intersection of a wall and t he f l oo r of your room. Another example is seen when you open your book and hold up

a page or so. Drawing such a representation I s not too difficult, and it becomes easier w i t h practice. Look at

the figure below and follow the instructions in drawing a

f igure of your own.

(a) Draw a parallelogram ABCD. (b) Select a point R on AD and draw parallel to m.

- ( c ) Draw a perpendicular W to BC and a perpendicular -

RS to AD so that and have equal measure. NOW draw TS.

Id) What kind of a figure is RSTV? - ( e ) It is not necessary that VT and % be perpendicular

as descr ibed above. It is desirable, however, tha t

RSTV be a parallelogram.

. Line Intersecting a Plane This type of drawing is also useful at times. It is

illustrated below. (a) Draw a para1lelogPa.m such as ABCD. (b) Select a point R on the surface of ABCD.

C

A

( c ) Now draw a l i ne through R so that it appears to pass

through the surface of ABCD. This will requi re some

p rac t i ce .

(d) You w i l l have a better picture if t h e line through Fi

is not parallel to a s i d e of t h e parallelogram.

Exercises % 1. Draw a rec t angu la r prism so tha t it will appear to be t a l l and

slender.

2. Draw a triangular prism so that the triangular faces w i l l

appear to be right triangles.

3. Draw a pentagonal prism.

4. Draw a rectangular prism 80 that it w i l l appear t o be sho r t and fat .

5. Draw a pyramid with the base a quadrilateral which does not appear t o be a square, a rectangle or a parallelogram.

6. Draw a pyramid w i t h a t r iangular base.

7. Consider a rectangular prism.

(a) O n which pairs of faces are there congruent rectangles?

*(b) Describe the positions of 3 planes of symmetry of a

rectangular prism. Think of a chalk box and 3 different planes each of which would divide the box into two par ts

which are congruent to each o t h e r . A l s o see Problem 8 of Sect ion 4-3.

8. Consider a triangular prism of which one face l a an equilateral tr iangle.

(a) Describe the congruent triangles or rectangles . *(b) Describe four planes or symmetry.

*9 . Answer the questions in Problem 8 I f the triangular faces of a triangular prism are scalene triangles.

[aec. 4-91

10. Are there any congruent triangles or polygons on the faces of a pyramid w i t h a square base?

11. Draw a picture of a book ahowlng two pages a t dlfferent angles,

12. Draw a picture of a flat target with an arrow through it.

Chapter 5

RELATIVE ERROR I

5-1. Greatest Possible Error

The process of measurement plays such an impor tan t pa r t in contemporary l i f e that everyone should have a c l e a r understanding of its nature. A substantial part of the arithmetic taught in the

elementary school r e l a t e s to measurement. Most of t he early work in measurement is designed t o familiarize you with common units of measurement and their use, and with the ra t ios between t h e i r

measures. The bas ic concept t o be developed in this chapter is t h a t the

j process of measurement of a single thing y i e l d s a number which

1 r epresents t h e approximate number of u n i t s . This i s in c o n t r a s t ) t o the process of counting separate obJects, which y i e l d s an exact number. When the number of separate objects is rounded or esti-

mated, the resulting number is treated as an approximation in the ! same sense a s a measurement. Since measurements are approximate, I I calculations made with their measures, such as sums or products, / y i e l d resul ts which are also approximate. ! ! When you use numbers to count separate objec ts you need only

coun t ing numbers. In counting you s e t up a one-to-one correspond-

: ence between the objects counted and the members of t h e s e t of

counting numbers. When you count the number of people i n a class-

room you. know the result w i l l be a counting number; there may be ! exactly 11, but t he re cannot be 11; or 1 . If there are a grea t many people, o r if you are not sure you have counted correct-

ly, you may say there are "about 300," rounding the number t o t h e

nearest hundred . However, if you d i d count careful ly you could

deterniine exactly the number of people in t h e room.

When you measure something, the situation I s d i f f e r e n t . When

you have measured the length of a l i n e segment with a ruler divided

i n t o quarter-Inches, the end of the segment probably f e l l between

two quarter-inch marks, and you had to judge which mark appeared closer. men if the end seemed t o fall almost exactly on a quarter-inch mark, If you had looked at it through a magnifying

glass you would probably have found t h a t there was a di f ference .

And if you had then changed to a ruler with the inches divided into sixteenths, you might have decided that the end of the seg-

ment was nearer' to one of the new slxteenth-inch marks than t o a quar te r - inch mark.

In any d i s c u s s i o n of measurement we assume proper use of

instruments. Improper use of instruments can occur through

ignorance, or careleasness. These mistakes can be corrected by learning how to use the Znatrument and by caref'ul inspection during the measurement process. But, even w i t h the best instruments

and techniques, scientists agree t ha t measurement cannot be

considered exact, but only approximate. The important t h i n g to

know is 5ust how inexact a measurement may be, and to state c l e a r l y how inexact it may be.

Look at the l i n e above, which p i c t u r e s a scale d i v i d e d into

one-inch units, ( n o t drawn to scale) . The zero po in t is l a b e l e d ?? , and po in t B I s between the 2-inch mark and the 3-inch mark.

Since B is clearly closer t o the two-inch mark, we may say that - the measurement of segment AB is 2 inches . However, any point

1 1 which is more than IT inches from A and less than 2? inches

from A would be the endpoint of a segment whose length, to the nearest inch, is also 2 inches. The mark below the l i n e shows t h e range w i t h i n which the endpoint of a l i n e segment 2 inches

long (to the nearest inch) might fall. The length of such a 1 segment might be almost inch less than 2 inches, or almost

1 - inch more than 2 inches. We therefore say t h a t , when a line 2 segment is measured t o the nezrest inch, the "greatest p o s s i b l e

1 error" is inch. This does not mean that you have made a

m i a t a k e ( o r t h a t you have not). It simply means t h a t , if you

measure properly t o the nearest inch, any measurement more than 1; inches and less than 2$ inches will be correctly reported

in the same way, as 2 inches. Conseq~ently such measurements 1 are sometimes stated as ( 2 2 inches . he symbol "+" - is

read "plus or minustf. ) By t h i s we mean that the greatest possible

inch. To say I t in another wgy, e r r o r in the measurement is 2 the measurement 2 Inches is correct to the nearest inch .

Suppose we see a s ign at a milepost which says "~hicago 73 miles.tt What unit of measurement are we to assume and what e r ro r possible? We do no t r e a l l y know, although a reasonable fnterpre- t a t i o n is t h a t the mileage is correct to t h e neares t mile, and

1 t ha t an e r ro r o f no more than 7 mfle is t o be expected.

B u t what a b o ~ t the milepost which indicates a distance of 1

mile t o t h e next milepost? Are we to assume that the u n i t is 1

mile and t h a t this measured mile indicates a d i s t a n c e l y i n g

between 0.5 miles and 1.5 ~ ~ i l e s ? C lea r l y , t h i s is n o t a

! reasonable interpretation in t h i s case, s ince we expect t h i s

measurement $0 be much more prec i se . To s t a t e what the g r e a t e s t p o s s i b l e e r r o r in a measurement is,

we need to know how the measurement was made and how accurate the instrument of measurement was. Ordfnarily, we do n o t know g h l s background. In f a c t , we r e a l l y do not know j u s t what is

meant when someone says, an ob jec t is "two inches long," o r "the distance is 1 mile.'' Usually it does not particularly matter

1 $1 whether the 2 i n c h measurement is correct t o or to . Ti7 o r t h e mile measurement is correct t o . I mfle o r .01 mile.

Sometimes, though, it 2 important t o i nd ica te what the e r r o r may

be. In scientific and technical work the greatest possible error 1s specifically s ta ted . For example, a length would be sta ted aa

( 2 2 &) inches o r possibly ( 4 2 0.005) inches, n o t simply as

2 inches or 4 inches. Often In business and industry the tern1 "tolerance" i s used.

By tolerance we mean - the greatest error which I s allowed. The --- tolerance might be s e t by the person who is purchasing a cer ta in manufactured product o r by the opera t ion of a machine. For

instance, an automobile manufacturer mlght specim that the

cylinders of an engine should have a diameter of 5 inches with

a tolerance of one-thousandth of an inch. This means the diameter

cannot vary more than 0.001 inch from 5 inches; the dZmension

would be g i v e n as ( 5 4 0.001) inches. On the o ther hand, a

producer of water pumps might demand a tolerance di f f eren t from

0.001 inch. Laws of t en s p e c i e tolerance for instruments i n commercial use l i k e scales f o r weighing. Scales are allowed to vary within certain limits. Court cases are sometimes declded

on the basis of tolerances allowed in the calibration of police car speedometers.

Class Exercises 5-1 - 1 1. When you measure to the nearest P inch, what is the greatest

possible error?

2. A meter stick is divided into centimeters and tenths of a centimeter ( m l l l h e tera) . A line segment was measured,wlth such a scale and stated to be 3.7 em.

(a) What was the u n i t of measurement? (b) The measurement is (3.7 -, ?) cm,

( c ) What was the greatest possible error? Give your answer In cm. and also in m.

1 3. S c i e n t i s t s f requent ly measure t o the nearest of a centfmeter. The greatest possible e r r o r f o r such a unit is cm. or m.

4. The greatest possible error in a measurement is always what f r ac t iona l part of the u n i t used?

5. . A tolerance of -0005 in. is specified for a metal sheet of thickness 0.350 inches. Allowable thickness f o r the sheet

will be between In. and in.

5-2. Prec i s ion - and Sianificant D i g i t s

1 Consider the two measurements, 14 ifiches and 12T inches.

As commonly used , these measurements do not i n d i c a t e what u n i t of measurement was uaed. Suppose t h a t the u n i t for the first measure-

1 1 rnent I s 8 inch, and the unit f o r the second measurement is 2 inch . Then we say t h a t the first measurement is more precise than the second, or has greater precision. Notice that the precision

of a measurement depends upon the smallest unit used I n t h e

measurement. The greatest possible error of the first measurement 1 I 1 1 1 l a of 8 inch, o r inch, and of the second I s 2 of

inch, o r $ inch. The greatest possible e r r o r is l e s s for the I first measurement than f o r the second measurement. Hence the

j more precise of two measurements is the one made w l t h the smaller

I unit, and f o r which the greatest possible error is therefore the i smaller. 1 TO summarize: The grea tes t possible e r r o r in a measurement 1s I . - h ik of the smallest unit of measure used in t h e measurement. The

tz

more premse of two measurements is the one f o r whleh t h e greatest - ! possible error is smaller.

For example, a measurement made using a metric scale may use

centimeter and millimeter d i v i s i o n s (units) of the scale. A

: measurement of 37.6 cm., made to the neares t .1 cm. (or nearest 1 1

' m m . ] has a greatest possible error of (1 m.) or (.1 em.)

. = .05 cm. This says tha t the actual l eng th l i e s between (37.6 - ' .05) cm. and {37,6 + .05) cm. We i n d i c a t e these limitations by

writing 37.6 2 .05. This is a convenient way of saying two th ings at once, by use of the - + symbol.

It is very Important that measurements be stated In a way

which shows clearly how precise they are. When we use the + - no ta t i on there is no doubt about what we mean. Another way t o

show clear ly what we mean is to make certain agreements about what

is meant when we w r l t e a number in decimal fom--the form which most of ten occurs in s c i e n t i f i c and technical measurement. When we

write tha t a l eng th has been measured as 7 6 inches we under-

stand t h a t the measurement has been made with an error no greater

[aec. 5-21

t h a n .005 in. Thus the measure 17.62 I s correct to the second decimal place to the right of the decimal p o i n t . In t h e - + notation this would be eqtl ivalent t o wr i t ing (17.6% 2 .005)

inches f o r the measurement. this agreement, each of the four

d i g i t s in 1 7 2 serves a real purpose, or I s " s i g n i f i c a n t . I 1

in measures l i k e 1462, 3.1 and .29637 all the d i g i t s are

understood to be s i g n i f i c a n t . But i n a numeral l i k e 0.008 t h e

three zeros simply serve t o fix t h e decimal po in t . In this case

we say only the 8 is a s ign i f ' i can t d i g i t .

In the numeral 2.008, a l l f o u r d i g i t s (2, 0 , 0 , 8) are

s i g t ~ i f i c a n t . In a numeral like 0.0207 the first two zeros are n o t s i g n i f i c a n t but the t h h - d is. Thus, 0.0207 has th ree - signiricant digits 2, 0 , 7.

When we write 2960 ft. or g3,000,000 mlles it is not clear

which, if any, of t h e zeros are significant. We agree t ha t they are - not significant, since they serve to fix the location of the

decimal p o i n t . Thus 2960 ft. has three significant d i g l t s

( 2 , 9 , 6 ) in its measure. The measurement is precise to the

nearest 10 ft. and the greatest possible e r r o r is 5 ft?

Whenever we wish any of the zeros a t t he end of a number l i k e

28000 or 2960 to be significant, we agree to i n d i c a t e the

final zero which is s i g n i f i c a n t . Thus 2960 ft. i nd fca t e s a - measurement correct to the foot. The measure has four significant d i g i t s 2 , 9, 6 , 0 The measurement 93,000,000 milea is - correct to the heares t 100,000 miles. The numeral has three

significant d i g i t s 9 , 3, 0.

Def in i t ion . A digit i n a decimal numeral is said t o be a - ----- "significant digit" 9f it serves a purpose o the r -- - than simply t o loca te (or emphasize) the decimal - - poin t .

Some fu r the r examples:

Significant Digi ts (in -- orde r )

3, 9, 0, 6 7, 3, 4 , 0

6 , 9, 2

I n 39060, t h e "0" between 9 and 6 is significant, the o the r "0" is not; it simply locates the decfmal point (under-

stood). In the nwera l 73.40, the "0" is significant because

it is not necessary t o have it to locate the decimal point, In

0.00523, all the " 0 ~ 8 ~ ~ are used simply to l o c a t e or emphasize t h e

decimal po in t . We understand that the lef t -most zero may o r may

not be written, and, if w r i t t e n , is slrnply for clarity in

l o c a t i n g the declmal po in t and r e a d i n g the number. When a number is written in scientific n o t a t i o n we agree t h a t

a l l of the d i g i t s in the first factor are significant, thus : - 4

73,000 ft. = 7 . 3 x 10 f t . 4 73,000 ft. = 7.30 x 10 ft . 4

73,000 ft, = 7.3000 x 10 ft.

Also, the measurement 2.99776 x lo1' crn./sec. , f o r the velocity

of ilght, ha8 6 significant d i g i t s ; the nieasurement 2.57 x l oM9 cm. f o r ,the radius of the hydrogen atom, has 3 significant

diglts; the measurement for the natlonal debt in 1957, 2.8 x 10" 8 dollars, has 2 significant d i g i t s ; 4.800 x 10 has 4 signif-

l c a n t d i g i t s . In the 1as.t case, the two f i n a l zeros are sLgnif- icant. Were they not , the number should have been written as

8 11.8 x 10 . It is this poss ibl i ty of indicating significant digits

In scientific nota t fon which is one of the principal advantages of the no ta t i on .

Exercises

Suppose you measured a line to the nearest hundredth of an

inch. Which of the following states the measurement best?

3.2 inches 3.20 inches 3.200'inches

Suppose you measured t o the nearest t e n t h of an inch. Which of the following could you use t o state the result?

4 inches 4.0 inches 4.00 inches (4.0 2 0.05) inches

3. Tell which measurement in each pair has the greater precision.

( a ) 5.2 feet, (42;) feet (b) 0.68 f e e t , (23.5 2 .05) feet

( c ) 0.235 inches, 0.146 inches

4. What is your age to the nearest year--that is, a t your nearest

b i r t h d a y w i l l you be t e n , eleven, twelve, t h i r t e e n , ... ?

A l l of you vrho say "13" must be between and 1%

years old.

5. ( A ) For each measurenient below tell the place value of t h e

last s i g n i f i c a n t d i g i t .

(B) T e l l the greatest poss ib le e r r o r of the measurements. ( a ) 52700 feet ( d ) 52.7 feet ( b ) 5272 f ee t ( e ) 0.5270 feet ( c ) 52700 - feet (f) 527.0 feet

6 (a) Which of the measurements in Problem 5 is the most

precise?

(b) Which is the l e a s t precise?

( c ) Do any two measurements have the same precision?

7 . Show by underlining a zero the precision of the following measurements :

(a ) 4200 f e e t measu-red to the nearest foot (b) 23,000 miles, measured to the nearest hundred miles

( c) 48,000,000 people, reported t o the nearest ten-thousand

8. Tell the number of s i g n i f i c a n t digits in each measurement: ( a ) 520 feet ( e ) 25,800 ft. (b) 32.46 in. (f) 0.0015 i n . ( c ) 0.002 in. ( g ) 38.90 ft.

( d ) 403.6 ft. (h) 0.0603 in.

9 . How many significant d i g i t s are in each of the following:

(a) 4.700 x 10 5 ( a ) 6.70 x (b) 4.700 l o4 ( e ) 4.7000 x 10 ( c ) b.7 x 10 15 ( r ) 2.8 x 109

5-3. Re l a t i ve Er ror , Accuracy, and Percent of Error

While two measurements may be made w i t h the same precision

( t h a t is, with the same unit) and the re fo re with the sane greatest

p a s s i b l e e r ro r , t h i s errcr is more Lrnportant i n some cases t h a n in 1 o t h e r s . An error o f 5 Inch in measuring your height would n o t

inch in measuring your be very misleading, b u t an error of 5 nose would be mTsleading. We can g e t a measure of the importance o f the greatest possible e r r o r by comparing it w i t h the measure-

m e n t , Cons ide r these measurements and t h e i r g rea tes t poss ib le

error's :

4 in. - + 0.5 in. 58 i n . - + 0.5 i n . SZnce these measurements are bo th made to the nea re s t inch, the

p e a t e s t poss ib le error in each case is 0.5 inch. I f we d lv ide

t h e measure of the greatest possible e r r o r by the number of units i n the measurement we g e t these resu l t s . ( ~ o t e tha t the measures are numbers and the measurements are not . We shall r e f e r to the

number of units in the measurement as the measure,)

The quotients 0.125 and 0.0086 are called relative e r r o r s . The relative e r r o r o f a measurement I s d e f i n e d as t he quotient of t h e measure of the greatest posslble error by the measure.

Relat ive e m o r r~easuwe of the greatest possible e r r o r the measure

Percent o f error is the r e l a t i v e error expressed as a percent.

I n the above two examples the relative e r r o r expressed as a percent I s 32.5% and 0,86% . When written in thls form it is called the percent of e r r o r , --

The measurement w i t h a re la t ive e r ror of 0.0086 (0.86% ) i s more accurate than the measurement with a r e l a t i ve e m o r o f 0.125 ( 1 2 . ) By d e f i n i t i o n a measurement with a smaller r e l a t i v e error is said t o be more accurate than one with a larger

r e l a t i v e error.

The terms and precision a r e u s e d in i n d u s t r i a l and

[see . 5-31

Try a ~imllar eomparison f o r the two measurements 93,000,000

miles and ( .03 - + ,005) In.

scientific work in a apeclal technical sense even though they are often used loose ly and as synonyms in everyday conversation. Pre-

c i s i o n depends upon the size of the unit of measurement, which l a twice the greatest possible e r r o r , while accuracy is the relative

error o r percent of error. For example, 12.5 pounds and 360.7 pounds are equally precise; t h a t is, precise to the nearest 0.1

of a pound (greatest possible error in each case is 0.05 pound) . The two measurements do not possess the same accuracy. The second

measurement is more accurate . You should verif'y the last statement by computing the relative errors in each case and comparing them.

An astronomer, for example, making a measurement of the distance t o a galaxy may have an e r r o r of a trilllon miles (1,000,

000,000,000 miles) yet be far more accurate than a machinist

measuring the diameter of a s t e e l pin to the nearest 0.001 inch. Again, a measurement ind icated as 3.5 inches and another as

3 . 5 feet are equally accurate but the f i r s t measurement is

probably more preclae. Why?

Suppose we have two measurements of the same quantity, say

3.5 in. and 3.500 in. There are two significant digits in the first measure and four significant digits in the aecond measure. What does the number of ~lgnificant d i g i t s in the measure tell' us

about the accuracy (relative error) of the measurement. Clearly, the greater the number of significant digits in the measure, the

greater the accuracy of the measure. To illustrate this we write

the following:

3.5 in.

two significant digits [3,5] Relative error = 3

or Accuracy r .01

Prec is ion = .1 in.

3.500 in.

four significant d i g i t s (3,5,0,0]

Relative er ror s or

$3 Accuracy z .0001 Precision = ,001 in.

93,000,000 miles

Two s i g n i f i c a n t digits { 5 , 3 ) 00 000 Relat ive e r r o r = 93~00; ,000

o r

Accuracy = -005

Prec i s ion 1,000,000 mi.

(-03 2 .005) in.

One significant digit 13)

Relative error =

o r %

Accuracy sr .2

Precision -01 in.

Exercises 5-3 In a l l computation express your answer so that it Includes t w o

significant digits.

1. State the g r e a t ~ s t possible er ro r f o r each oi' these measure-

ment s.

(a) (52 - c 0 . 5 ) ft. ( e ) 7.03 in.

(b) (4.1 2 0.05) in. (f) 0.006 ft.

( c ) 2580 mi. 4 (8) 5 .4 X 10 mi.

( d ) 362 ft. (h) 54,000 - mi.

2. Find the r e l a t i ve error of each measurement in Problem 1.

3. Find the greatest possible error and the pe rcen t o f e r ro r f o r each of the fo l lowing measurements. ( a ) (9 .3 2 0.05) ft. ( 0 ) 9.30 x lo2 ft. ( b ) 0.093 ft.

4 ( d ) 9.30 x 10 ft.

4. What do you observe about your answers f o r Problem 3? Can

you explain w h y the percents of error should be the same

f o r a l l of these measurements?

5. F i n d t h e precision or the following measurements. ( a ) 26.3 ft. ( d ) 51,000 mi. (b) 0.263 ft. ( e ) 5.1 ft. ( c ) 2630 ft. (f) 0.051 i n .

6 How many s i g n i f i c a n t d i g i t s are there in each o f the following? (a) 52.1 in. ( c ) 3.68 in.

( b ) 52.10 in. { d ) 368.0 in.

Find the r e l a t i ve error of each of the measurements in Problem 6 .

From your answers f o r Problems 6 and 7, can you see any

relation between the number of s i g n i f i c a n t digits in a

measure and the r e l a t i v e error in the measurement? What is

the re la t ion between the number of signlficant digits in a

measure and the accuracy of the measurement?

Without computing, can you tell which of the measurements

below has the greatest accuracy? Which is the leaat accurate? 23.6 in. 0,043 in. 7812 in. 0.2 in.

Arrange the following measurements in the order of t he l r

precision (from least t o greatest) : 1 3 1

( a ) ( 3 $ + $ ) in., ( ~ 7 2 ~ ) 1n.j ( 3 % ~ ~ ) in.,

2 1 (4% 2 I n , (22.25 2 .125) In.

(b) 4.62 in., 3.041 In., 3 in., 82.4 in., 0.3762 in.

Arrange the following measurements in order of t h e i r accuracy

(from leas t t o greatest): 1

( 6 - + F) ft. (3.2 - + 0.005) i n . ( 7 . 2 2 0.05) miles

(4 i 9) in. 3 yd. ( 4 - ++) in. . Count the number of significant digfts in each of the follow-

i n g measures:

( a ) 43.26 (e ) 0.6070 (1) 76,002

( b ) 4,607 (f) 0.0030 ( J ) 43,000

( c ) 32.004 ( g ) 4.0030 (k) 0.036 I d ) 0.0052 (h) 0.03624 (1) 200.00004.

Express the following measures i n scientific no ta t i on : ( a ) 463,000,000 ( d ) 32.004 ( 9 ) 36.8 x lo5 (b) 327,000 ( e l 2 (h) 0.80 x 10-7 ( c ) 0, 000462 (f) 0,0000400 (I) 72 billion.

[aec . 5-31

1 4 By inspec t ion arrange the following numbers in orde r o f their magnitude, from l e a s t t o greatest. L i s t by letter only. ( a ) 3.6 x l o5 (f) 4 . 1 x 10

6

(b) 3.5 X 10 8

(8) 3.527 x l o 2 (c) 4 x (h ) 3.55 x 10 8

( d ) 3.527 X 10 8 (i) 3.4 x lo-T

(el 3.5 x ( J ) 3.39 x lom8

1 15. BRAINBUSTER. A master niachinlst measures a 3$ inch piston

head t o the nearest 0.0001 inch while an astronomer measures by the parallax, the distance t o Canls Major ( the star ~irius)

correct t o the nearest 10,000,000 miles. The dis tance t o

Sirius is 8.6 light years (1 light year r; 6 x lo1* miles).

Which measurement is more accurate? I

5-4. Adding and Subtractkna Measures

S i n c e measurements are never exact, the answers t o any ques- t i o n s which depend on those measurements are also approximate. For instance, suppose you measured t h e ,length of a room by making two marks on a wall , which you called A and By and then measur- i n g the d i s t ances from the corner t o A, from A t o B, and

from I3 to the o the r corner . Measurements such as these whose

measures are t o be added, should a l l b e made wi th the same precision. Suppose, t o the nearest f o u r t h of an inch, the measurements

1 3 were 72 inches, 4% inches, 2% inches. You would add the

2 * Therefore t he measurement i s 13% inches . measures to get 135 F .

Of course, the d i s t ances might have been shorter in each case. The

1 measures could have been almost as small as 728, 44, and 223 i n

which case the dis tance would have been almost as small as 13%

inches, which is three-eighths of an inch less than 13% Inches.

A l s o , each d i s t ance might have been longer by nearly one-eighth

of an inch, in whlch case the t o t a l length might have been almost

three-eighths of an inch longer than 13$ inches . The greatest

possible e r r o r of a sum is the sum of the greatest possible e r r o r s .

If w e were atidfng measures of 37.6, 3.5, and 178.6, the great-

e s t poss ib l e e r r o r oi' the s;un would be .O5 +.05 + .05 or .15.

The r e s u l t of this a d d i t i o n would be shown as 219.7 f .15. Computation involving measures is very important in today's

w o r l d . Many rmles have been l a i d down giv ing the accuracy or precision o f t h e results obtained f rom computation w l t h approxi-

mate measures. Too many rules, however, might create confus ion and would never replace ba s i c knowledge of approximate data . If the meaning of greatest possible error and of relative e r ro r is understood, t h e p,recislon and accuracy of t h e result of computa-

t i o n with approximate data can u sua l l y be found by applylng common

sense and judgment. Common sense would t e l l us t h a t with a large

number of measurements the errors will, to a c e r t a i n extent, cancel

each o the r . The general principle is t h a t t he sum or dirference of - ----- -

measures canno t -- be more precise than the l e a s t prec ise measure --- i n v o l v e d . Therefore t o a d d or subtract numbers a r i s i n g from approximations, first round each number to the u n i t of t h e least

p r ec i s e number and then perform the operation.

As we have seen, the g rea tes t possible error of a s u m ( o r d i f fe rence) of several measures is the sum o f the greates t

possible er rors of the measures involved. (TO estimate the expected e r r o r of a sum, t a k i n g into account the way t he e r ro r s would o f t e n cancel each other , we n e e d t o use some ideas of

p r o b a b l i l i t y , not y e t a t our d i sposa l . )

Exercises 5-4 1. Find the greatest possible e r r o r for the sums of the

measurements in each o f the following. (when a measurement is 1 g iven as % in., yold may assume t h a t the unl t of ' measurement

1 was 2 in. and correspondingly for o the r fractions.)

1 1 0 ( a ) 3 in., $ in., 9 in.

( c ) 4.2 in., 5.03 in, I t ( d ) 42.5 in., 36.0 in., 49.8 in. i ( e ) 0.004 in., 2.1 in., 6.135 in.

(f) $ in., 1 i n , g i n . 2. Add the following measures:

( a ) 42.36, 578.1, 73.4, 37.285, 0.62 (b) 85.42, 7.301, 16.015, 36.4 ( c ) 9.36, 0,345, 1713.06, 35.27'

3. Subtract the following measures:

( a ) 7 . 3 - 6.28 (b) 735 - 0.73 ( c ) 5430 - 647

5-5. Multiplying D i v i d i n g Meaaures

You mow tha t the number of u n i t s i n the area of a rec tangle 1s found by multiplying the number of units in the l e n g t h by t h e

nunlber of the same u n i t s in the width. Suppose t h a t the dimensions

of a rectangle are 30 Inches and 20 inches . Since the measur- 2ng was done to the nearest inch, the measures can be sta ted as

(30 2 .5) and (20 2 .5] . This means that the length might be almost as small as 29.5 inches and the width almost as small as

19.5 inches. . The length might be almost as large as 30.5 inches

and the width almost as large as 20.5 Inches.

20 inches

Y

30 inches

Look a t t h e sketch to see what t h i s means. The outside l.1nes show how the rectangle would look if the dimensions were as large

as p o s s i b l e . The i n n e r lines show how 1t would look I f the length

and w i d t h were as small as possible. The shaded part shows t h e

d i f f e r ence between the largest possible area and the smallest p o s s i b l e area with the given measurements.

Let u s see what the di f fe rences a r e . The gfven measurements

are 20 f ' t . x 30 f t . The smallest poss ib le dimensions are (20 - .5) ft. x (30 - . 5 ) ft. and the largest possible dimensions

are (20 + .5) ft. x (30 + .5) ft. Least Poss ib le - Given Greatest P o s s i b l e

Area - Area I -

Hence there is a difference of (625.25 - 575.25) or 50

s4. ft. in t he two possible errors. The computed area of 20 x 30 sq. ft. or 600 sq. ft. is a b o u t 25 sq. ft. greater than

the smallest area and about 25 sq. St. less than the largest area possible.

(20 - . 5 ) x ( 3 0 - . 5 ) sq. ft.

or (600 - 10 - 15 +.25)sq.ft.

or

575.25 sq. ft.

20 x 30 sq, ft.

o r

600 sq. ft.

(20 +.5)x(30 +.5) sq.ft.

or (600 + 10 + 15 +.25)sq.ft

or 625.25 sq. ft.

Therefore, if we wish t o be very careful about our statements

we r ~ r i s t make clear what is meant when we say t h e area of t h e rectarlgle is 600 sq. ft. As we have seen, this answer is n o t

correct to 1 square foo t , b u t it is correct to less than 100

sq. ft. If we wish to indicate the situation as best as we k n m

it we may write the area as (600 - + 25) sq . ft. (we have chosen

here t o round the grea tes t possible area 625.25 to 625 sq , ft. You might pre fe r to write 575.25 sq. ft. - < t h e a r ea - < 425.25 sq.

ft.) If we choose t o write the area as 600 sq . ft. we mus t

interpret the numeral as one with one signiffcant digit. This

says t he area is g i v e n to w i t h i n 100 sq. ft. and hence lies

between 550 s q . ft. and 650 sq. f t . T h i s i s co r rec t , but not

q u i t e so good a r e s u l t as our answer (600 2 25 sq. ft. )

It is r e a l l y fmpossible to give a satisfactory rclc f o r the

t d ~ s l t i p l i c a t i o n of approximate measures in d e c i ~ a l or f r a c t i o ~ a . 1

fo rm. However, when data are expressed in decimal form a rough

~ u i & can b e suggested f o r f i n d i n g a satiafactorjr product. - The

' nu.mber o f significant digits in the product of two numbers is no t - -- -- -- m o m than the number of significant digits in t h e l ess accura te --- --- f a c t o r .

Uote t h a t this says the number of significant d i g i t s Is not more than the nunlber oS significant digits in the l e s s accurate -- f a c t o r - - i t does not assure you tha t there will be tha t many!

As an i l l u s t r a t i o n of t h i s principle consider t he following

problem; What is the area o f a rectangle w i t h sides measured as

1 10.4 cm. and 4 ,7 cm.?

To f i n d the area we might multiply 10.4 by 4.7 t o o b t a h 48.88. Now there are three significant digits in 10.4

; and on ly two significant digits in 4.7 Hence, the product cannot

; have more t ha f i two significant d i g i t s and we round t h e a r e a to 49 sq. cm. Hence, the area of the rectangle is c: 49 sq. cm.

If we wish to f i n d a be t t er est imate of the poss ib le error

we m ~ s t use the "2" scheme we used earlier. Since

(10.4 + .05)(4.7 + .05) = 48.88 + .52 + .235 -+ .0025

= 43.6375

232

and

(10.4 - .05)(4.7 - - 0 5 ) = 48.88 - .52 - ,235 + .0025 = 48.1275,

we see that 48.1 sq. cm. < the area of the rectangle < 49.7. If we use only two s i g n i f i c a n t figures we see tha t the area lies between 48 and 50 sq. cm. Hence (49 - + 1) sq. cm. is a good answer.

You might ask, why not round the numeral 10.4 to 10 and

work only with two significant digits in each fac tor? ~hen'we

would get

10 x 4.7 sq. cm. = 47 sq. cm, f o r the area, and we see that th i s is - n o t cor rec t to two s i g n l f -

icant figures.

For such reaaons as th i s , we ordinarily agree to the follow- i n g general procedure when multiplying two factors which do not have the same number of s i g n i f i c a n t d i g i t s .

If one of the two factors contains more signfficant d i g i t a ----- than the other, round off the factor which has more significant --- --- --- d i g i t s so that it contains only one more sianificant d i g i t than

CC-

the other factor. -- Suppose we wish to find the circumference of a circle with

the diameter D equal to 5.1 mm. The circumference C = TD.

What value of R shall we use? Since the diameter 5.1 is given to two significant digita we use three significant digits for T

or g 2 3.14. Then

which we round to 16, aince only two digits are s ign i f i can t in the product . Hence, the circumference is approximately 16 m.

If we were dealing with a large c irc le with diameter D

measured as 1012 inches, then we would use ~w 3.1416 and round

the result of the multiplication C = (3.1416) (1012) to four significant digita.

Div i s ion is deflned by means of multiplication. 'Therefore

it is reasonable t o follow the procedure used f o r multiplication

in doing d i v i s i o n s involving approximate data.

When a multiplication o r division involves an exac t number

such as 2 in the formula for the circumference of a c i r c l e

(C = 2ar), the approximate number determines t h e number of signif-

icant digits in the answer. We ignore the exact number in de te r -

mining the significant digits in the answer. An exact number is a nurn3er that is n o t found by measuring.

Exercises 5-5 - 1 1 1. Suppose a rectangle is % inches long and lT inches wide .

Make a drawing of the rectangle. Show on the drawing t ha t the 1 1 1 l eng th is (% - + $) inches and the width ( I T 2 Inches.

Then find the largest area possible and the smallest area possible, and f i n d the d t f f e r ence , or uncertain p a r t . Then

find the area with the measured dimension, and f i n d the r e s u - l t 1 to the nearest square inch.

2 . Nultiply the following approximate numbers :

( a ) 4 . 1 x 36.3 ( b ) 3 . 6 x 4673

4 ( c ) 3.76 x ( 2 . 9 x 10 )

3 . Div ide the fol lowing approximate numbers:

( a ) 3.632 $ 0.83

(b) 0.000344 0.000301

[c) (3.14 x l o 6 ) 4 8.00s

4. Find the area of a rectzingular f i e l d which is 825.5 r o d s

long and 305 rods w i d e .

5. The circumference of a c i r c l e is s t z t e d C = ~ d , in which 3

is the diameter of the c i r c l e . If i-r is given as 3.141593, f f n d t h e circumferences of c i rc les whose diameters have the following measurements:

( a ) 3.5 in.

(b) 46.36 ft.

(c ) 6 miles.

6 . A machine stamps out parts each weighing 0.625 ~ b . How much weight is there i n 75 o f these parts?

7 . Assuming t h a t water weighs 62.5 lb. per cu. ft., what is the

volume of 15,610 ~bs.?

There are many rough rules fo r computing w i t h approximate

data but they have to be used w l t h a great deal of common sense. They don't work in a l l cases. The modern high speed computing

machine,,which adds or multiplies thousands of numbers per second, has to have special rules applied to t h e data whfch are f e d to it.

Er ro r s Involved In rounding numbera add up or disappear In a very unpredictable fashion in these machines. As a matter of fact "error theory" as applied to computers is an ac t ive field of research today f o r mathematiclans.

Chapter 6

REAL lWN3MS

Review Numbers

In your study of mathematics you have used several number

systems. You began with the count lng numbers, and you may have

known a good d e a l about these numbers before you entered the first grade I n school. These numbers are so farnlliar t h a t it is easy to

overlook some of t h e ways in which the sys tem of countlng numbers

d i f f e r s from other systems. Consider the fo l lowing questions:

(a) Thlnk of a particular counting number. What is t he next

smaller counting number? the next la rger? If n represents a

: counting number, what represents t h e next smaller counting number? i the next larger?

( b ) Is there a counting number which cannot be used as a

replacement for n in your answer to question ( a ) ? Why?

(c) Is there a smallest counting number? a l a rges t? If s o , I what a re they?

( d ) Is the s e t of counting numbers closed under the opera t ion

of 1) 2ddi t i on?

(2) s l ~ b t r a c t i o n ?

( 3 ) multiplication?

( 4 ) d i v i s i o n ? ( e ) How many counting numbers a r e t h e r e between 8 and ll?

between 3002 and 4002? between 168 and 163? Between any two counting numbers is there always another c o u n t i n g number?

In Chap te r 1 you studied posltive and nega t ive rational

I numbers. The s e t o f integers c o n t a i n s the s e t of counting numbers

( ca l l ed p o s i t i v e integers). For each positive integer a t h e r e is - an opposite nwnbep a. The opposites of positive in tegers are

I c a l l e d negatlve integers. If a is a countlng number, t h e n -

a + a = 0 . What integer is n e i t h e r positive n o r negative?

The r;et of i n t ege r s i s contained in a n o t h e r s e t of nwnbers

whlch we c a l l the s e t of r a t i o n a l numbers. A s you know, t h e

se t of integers is adequate f o r many purposes, such as r epo r t i ng

t h e p o p u l a t i o n of 3 c o u n t r y , the number of d o l i a r s you have ( o r

owe), t h e number of vertlccs In a t r i a n g l e , and so on. The

i n t e g e r s a l o n e a m n o t s u i t a b l e for many purposes, p a r t i c u l a r l y

f o r the process of measurement. If we had o n l y t h e i n t e g e r s to

use for measuring we would have to i n v e n t names f o r subdivisions

of units. We do this t o some extent; ins tead of saying 1 f e e t we sometimes say _C. f e e t 4 inches. But we do n o t use a differ

1 ent name f o r n s u b d i v i s i o n oP an inch. Instead, we speak of 7T inches, or 7.25 i n c h e s , w i n g r a t i o n a l numbers which a re n o t

i n t e g e r s . If we had o n l y t h e i n t ege r s , we could never speak of 1 32 quarts, or 2.3 miles, o r 0.001 I n c h .

R e c a l l that a r a t i o n a l number may be named by t he fraction

symbol " ~ " where p and q are i n t ege r s , and q # 0 . 4 '

J u s t as there is a n e g a t i v e i n t ege r which cor re sponds to each

p o s i t i v e i n t e g e r (or coun t ing number ) , there is a negative rational number w h i c h corresponds to each positive r a t i o n a l number.

You a l r e a d y may have observed t he f a m i l i a r p r o p e r t i e s f o r

r n t i o n a l numbers, w h l c h may D e summarlzed as follows:

Closure: If a and b a r e r a t i o n a l nwnbers, then a + b is a r a t i o n a l number, a - b (more c o m m o r ~ l y w r i t t e n ab) is a

a i s a r a t i o n a l number, a - b i s a rational number, and

r a t i o n a l number if b # 0.

Cornrnutativlty: If a and b are rational numbers,

then a -I- b = b -!- a, and a . b = b . a , ( a b = b a ) .

Associativity: If a , b, and c are r a t i o n a l nwnbers,

then a + ( b + c ) = ( a + b ) + c, and a ( b c ) = ( a b ) c . Identities: There is o r a t i o n a l nwnber zero such t h a t

if a is a r a t i o n a l nwnber, t h e n a f 0 = a, There is a rational

nwnber L such t h a t a . 1 = a .

Distributivity: If a, b, and c a r e r a t i o n a l numbers,

then a(b + c ) = ab 4- ac.

Adaltive i n v e r s e : If a is a r a t l ~ n a l number, t h e n there IS a ra t ional number (-a) such that a + (-a) = 0.

Multiplicative inverse: If a is a r a t i o n a l number and a f 0 , then there is a rational number b such that ab = 1.

Order: If a and b a re d i f f e r e n t r a t i o n a l numbers, then

either a > b, or a < b.

Class Exercise B-1 1. Is there a smal les t nega t ive Integer? A largest one?

2. If n represents a negative integer, w h a t represents the

next la rger one? the next smaller one?

3 Is t he set of negative Integers c losed under the operation

of

(a) addition? ( c ) multiplication?

(b) sub t r ac t i on? ( d ) d i v i s i o n ?

4 . -press each of the following in the form 2 or *(El, 9

where p and q are counting numbers.

5. Which of the prope r t i e s for r a t i o n a l numbers does each s t a t e -

ment i l l u s t r a t e ? ( a ) $ + Q =

2 7

( b ) ( I + = and 5 8 ia a rational number.

- 1 2 2 ( . = ) and ( ) is a r a t i o n a l number. 4

- 7 What i s the add i t i ve i n v e r s e of (g)? - ' I What is the multiplicative inverse of

What is ano the r name for "rnult ; ipl icat ive i n v e r s e " ?

If , o r ( , is the simplest fractional form f o r a

r a t i o n a l number which is an Integer, what must q be?

How can you tell whether two fractions represent the same

r a t i o n a l number? r,

What are three o t h e r names for the rational number '? 7

Exercises 6-1

Look a t each statement below and tell whlch of t he properties

l i s t e d f o r r a t i o n a l numbers it I l l u s t r a t e s .

- 3 ". 1 is a r a t i o n a l number. ( a ) ( , ) + g = p and

r; '= (b) g + 0 = 3

- +a md ( d ) - ( = 3 2 'g 1s B rational number.

2 1 1 2 2 1 ( e ) ( T + - P ) = (- 3 - L) 3 + (7

- 1 1

II 3 7 11 7 (9) m + (W + m) = (m+ + m

- 2 . Express each of the foilowlng In t h e Iorm - o r (i), whera

q p and q are c o u n t i n g numbers.

- 1 ( c ) ! l + T

3 . Write each of these in simplest P r a c t i o r ~ a l form.

(d') c$

( c ) -0.62 ( f ) 12.5

)I. What is the zdditive i nve r se of each of t h e following?

( a ) '28 -t. 1 (4 37

5 . Complete the s ta tement , h he simplest name for c ra t ion t : l a number written in t h e form is the one in which a a n d

b hzve no common f a c t o r except I T

6. A rational number w h i c h does n o t have a r e c i p r o c a l is t h e P number - when p is 9

7 A r ~ a n g e the following rational numbers in o ~ ~ d e r . L i s t t h e

g r e a t e s t one last.

*8. F i n d the averEge of t h e two rational numbers -8 a n d ' ) I .

*9. Is it always possible to find t h e average o f two integers and

have t h e average Ge an i n t ege r? E x p l a l n .

10. Multiply each of the fo l lowing by 10.

( a ) 0.73333 ( d ) 0.1142142

(b) 0.090Q09 (4 13.46333

( c ) 16.31212 ( f ) 846.46b6

11. Mul t ip ly each number in Problem 10 by 100.

1 2 Multiply each number In Problem 10 by 1000.

6 - Density - of Ratlonal Numbers

One o f t h e o b s e r v a t i o n s you have made about t h e integers is t h a t every integer is preceded by a p a r t i c u l a r l n t ege r , and is followed by a p a r t i c u l a r integer. The Integer which precedes - 8 4s -9 , and t h e i n t e g e r whlch f o l l o w s 1005 is 1006. In

o t h e r words , I f n is an integer, then its predecessor is

( n - l), and i t s successor is ( n + 1).

This means t h a t on t h e number l i n e t h e r e are wide gaps between

the p o i n t s which correspond t o t h e integers.

Figure 6 - 2 ~

Now consider a l l the r a t l o n a l numbers, and the p o i n t s on the

number l i n e which correspond to-them, Such p o i n t s are c a l l e d

r a t i o n a l points. On the nwnber line below a r e shown the r a t i o n a l

points between -3 and h which may be named by the fractions

w i t h denominators 2 , 3 , I ! , and 6 .

Figure 6 - 2 ~

Exercises 6-2a

1. Nalce a drawing of a number l i n e similar t o t h e one I n Figure 6 - 2 ~ . 14ark t h e points which correspond t o t he se numbers:

2 . On the number l l n e which you drew in Problem 1, f i n d the p o i n t s

that correspond t o t h e s e numbers:

3 , Were any of t h e points in Problems 1 and 2 the same point?

If so, which ones?

I , Suppose y o u have a l ready located p o i n t s f o r t h e rational

numbers which are represented by Prac t i o n s with denominators

2 , 3, Ii, 5 y and 6 . You then l oca t e points represented by

fractions w i t h denominator 7 . Row many new points ( n o t already located) f o r sevenths w i l l t h e r e be between the p o l n t s

f o r t h e in tegers 1 a n d 2? Between t h e p o i n t s f o r and

" ?

5 . Suppose tha t you t h e n loca te p o i n t s f o r f r a c t i o n s wi th denom-

i n a t o r 8 . How many new p o i n t s w i l l t h e r e be betr:?en t h e

points f o r any two consecutive i n t ege r s?

6 . Conslder a l l r a t i o n a l p o i n t s f r o m 0 t o 1 which a r e named

by fractions w i t h denornlnators 1 t o 8 i n c l u s i v e . These

points a r e named on the fo l lowing page. The f i r s t row shows Lhe fractlons w i t h denominator 1, the second row the frac- t i o n for t h e new p o i n t w i t h denominator 2 , t he third row

the fractions f o r the new po in t s w i t h denominator 3 , and so

on.

( a ) m y is O omitted from the row f o r th i rds? 3

(b) Why is $ omitted from the row for f o u r t h s ?

( c ) Why are there more new polnts named i n t he row f o r f i f t h s

and in the row f o r sevenths than in the row f o r sixths?

7. The r a t i o n a l numbers named in Problem 6 are combined in one

row below, and llsted in order from smallest to largest.

Explain why the f i r s t six f r a c t i o n s should be In t he order

shown; the l a s t six fractions.

8. I n Problem 7 more numbers could be o b t a i n e d I n the row of f r a c t l o n s and more polnts among t h e corresponding set of

rational points by Inserting next the f r a c t l o n s w i t h denomina-

t o r 9, then the fractions with denominator 10, and so on. How many new polnts would correspond to fractions w i t h denominator 9? Wfth denominator lo? With denominator ll?

9. In Problems 6 and 8, which denominator accounts for the largest number of points not already named? What kind of a

number seems t o account f o r t h e l a rges t number of new points when it is used a s a denominator? Why?

We may fo l l ow a d i f f e r e n t method f o r naming and locating new r a t i o n a l p o i n t s . Consider two p o s i t l v e r a t i o n a l numbers r and

s, with r < s . Then consider what happens when we add r and s to each of these nunbers. Let us consu l t the number l i n e .

r : - -a - adding r to r b 0 r 2 r

(r+s) r

I 4 - I 1 oddinq r to s 0 s r ts

S 4 : I f C( adding s to s

0 s 2's

We see that 2r < r + s < 2s. Taking ha l f o f each we g e t 1 1 r < T(r + s ) < s . It is not difficult t o show that r < $r+s) < s,

even if r Is negative or r and s a r e both negative. You

mfght try to prove this yourself using the number line, if you

wish, The number $(r + s) is t h e averap;e of the numbers r and

s. We have observed, then, that the average of two r a t i o n a l

numbers is between these numbers. On t h e number line what p o i n t

do you suppose corresponds t o t he average of two numbers? It I s

the mid-point of the segment determined by the two numbers, If r and s are rational numbers, is $(r + s ) a r a t i o n a l number?

I t What properties o f the rational number system t e l l s us that i t is?

We can summarize what we have observed: The mid-point of t he -- segment j o i n i n g t w o r a t i o n a l points on t h e number l i n e is a - -- --- r a t r i o n a l point col?responding to the averaEe of the two numbers. -- ---

1 The mid-point oS t h e segment jolning the po in t s f o r - and 3 is the point corresponding to the number , since

1 The mid-point of t h e segment j o i n i n g t h e p o i n t s f o r 8 and 1 - 7

iz t h e point corresponding to the number , since

BJ f i n d i n g the average in this manner it I s p o s s i b l e t o f i n d

r a t i o n a l numbers between each p a i r of consecutive numbers repre- s e n t e d In the row of fractions in Problem 7 of mercises 6-2a. If

we i n s e r t these new fractions the row would begin

If you found a l l the new fractions In this row which could be 0

found in t h i s way, there would be 43 fractions between T and This process could be continued indefinitely. You could f i n d X '

p o i n t s between O ana 1 1 between and 8, Z' and so on. You

could f i n d as many r a t l o n a l numbers as you wish between 0 and

1 by t a k i n g averages, averages of averages, and so on i nde f in i t e ly .

The d i s c u s s i o n above suggests an important property of t h e

r a t i o n a l numbers. This is t h e

P r o p e r t y o f density: Between any - two distinct rational %- bcla ; there is a t h i r d r h t i o n a l number, On the number l i n e , this ------ r;ie; ,n~ t h i t t h e n w b e r of r a t i o n a l points on any segment 5 s

l -~~ l l i : ; i t ed ; no matter how many p o i n t s on a very amall segment have

besn n::rned, it 13 possible bo name as many more a s you please .

Are the integers dense? That is, is t h e r e always a third

in teger between any two integers? Illustrate your answer.

Is there a smallest positive integer? a largest?

Is there a smallest negative integer? a largest?

Is there a smallest positive r a t i o n a l number? a largest

negative rational number?

on t h e number line. Think of the p o i n t s f o r 0 and

Name the rational p o i n t P which is halfway between 0 and I . Name t h e po in t halfway between the point P and 0;

between the point P and &. 1 I n the same way, f i n d three r a t i o n a l numbers between

1 I and - 10'

1 Think of t h e segment w i t h end-points 2 and - Show 1000 '

a p l a n you could follow to name as many r a t i o n a l po in t s as

you please on this segment. Use your plan to name at least five points.

6 3 Decimal Representations f o r the R a t l o n a l Numbers -- It is of t en very helpfu l to be able to express r a t i o n a l

numbers as decimals. When it is necessary to compare t w o

rationals t h a t a r e very c lose together, c o n v e r t i n g to decimal

form makes the comparison easier. The decimal fo rm is p a r t i c u -

l a r l y helpful i f there a r e severa l r a t i o n a l numbers t o - b e a r r anged - 13 27 3 and in order. For example, consTder the fractions - -

9 25' .50y 8' and their corresponding decimals 0.22, 0.5'1, 0.375, and

0 It 9s much e a s i e r to order t h e numbers when they a r e written in decimal form.

Some rational numbers a r e easlly written I n decimal form.

We know how to write, by i n s p e c t i o n ,

1 1 1 1 - = 0.5, t= 0 .25 , = 2 , - = 1

0.2, - 2 25 = 0.04, 5

& = 0.008, and a l s o T 17 = 8.5, 4 = 5.75, 176 = 17.5 10

F o r o t h e r rational numbers, a decimal. expression may n o t be

as o b v i o u s b u t we can always o b t a i n it by the usual process of division. For example

The examples we have discussed seem to suggest t h a t t h e

decimal e x p a n s i o n s f o r rational numbers e i t h e r terminate (like 1 1 = 0 . 5 ) or repeat ( l i k e 7 = 0.3333 333 . . . ) . What would be a

r easonab le way t o study such- decimal expansions? Since we have

used the division of numerator by denominator to obtaln a decimal representation, we might study carefully t h e process which we

c a r r y out i n such cases.

7 Consider t h e r a t l o n a l number 8. If we carry out t h e i n d i -

cated division we w r l t e

30 remainder 6

40 - 0 remainder 0

In d iv id ing by 8, t h e on ly remainders which can occur are 0, 1,

2, 3 , , 5, 6 , and 7. The only r e m i n d e r s which - d i d occur

a re 6 at t h e ff r s t stage, then 4 and f i n a l l y 0. We could continue dividing, getting at each new stage a remainder of zero

7 and a quotient of zero. We could write = ,875000 ... but we seldom do. The decimal .875000... is a repeating decimal with

0 repeating over and over again. When the remainder 0 occurs,

the d i v i s i o n I s &exact. We say a division is exact If the process P

of dlvislon produces a zero r ema inde r and thereafter zeros as

quotients. Such a decimal is often spoken of as a terminating

decimal instead of a repeating decimal, and we shall do so a t

times in t h i s chapter .

What about a r a t i o n a l number which does not have a terminat-

ing decimal representation? Suppose we look st a particular 2 example of t h i s kind, say - 13 *

The process of div id ing 2 by

13 proceeds l i k e this:

remainder 7

e tc .

Here t h e possible remainders are 0, 1, 2, 3 , 4, 5, 6 , 7, 8, 9, lo, 11, 12. Not all the remainders do appear, but 7 , 5 , 11, 6 , 8, and 2 occur first in t h i s order. A t

the next stage in the division the remainder 7 re-occurs and

the sequence of remainders 7, 5, 11, 6 , 8, and 2 occurs

again. In fact the process repeats itself again and again. The corresponding sequence of digits In the quotient--153846--wl11 therefore repeat regularly in the decimal expansion f o r A V' repeating decimal is sometimes referred to as a perlodlc decimal.

In order to w r i t e such a periodic decimal concisely and without ambiguity i t i s customary t o write

The bar (vinculum) over the d l g i t sequence 153846 indicates the s e t of d i g i t s which repeats. Similarly, we write 0.3333 ... as 0.7 ... . If it seems more convenlent we can w r i t e 0.3333... as 0.37 ... or 0.337 ..., and 0 as 0.153846-.,.

The method we have discussed is quite a general one and it

can be applled to any r a t i o n a l number k. If the indicated dlvislon Is performed then the only possible remainders which can occur are 0, 1, 2, 3 , ... ( - 1). We look only a t t he

stages which c o n t r i b u t e t o the d i g i t s tha t repeat i n t h e quotient.

These stages usually occur after t he zeros begin t o repeat i n t he

d iv idend . If the remainder 0 occurs, the decimal expansion terminates a t t h l s stage i n t he d i v i s i o n process. Actually, we

may wrfte a terminat ing decimal expansion like 0.25 w i t h a

repeated zero to provide a periodic expansion, as 0.25000 ..., or we may use the bar, as 2 . . Note that a zero remainder may occur prior to th l s stage wlthout terminating the process; f o r example :

remainder 0

If O does n o t occur as a remainder after zeros are annexed to t h e d lvldend, then a f t e r a t most ( b - 1) steps In the division process one of the possible remainders 1, 2, . , . , (b - 1)

w i l l occur agaln and the digit sequence w i l l start repeating. We see from t h i s argument t ha t any r a t i o n a l number has a - --

decimal expansion -- which is p e r i o d i c .

Berc ises a 1, Find decimals f o r these r a t i o n a l numbers. Continue the

division u n t i l the repeating begins, and write your answer

to a t least t e n decimal p l ace s .

2. Which of the following convert to decimals that repeat zero

(terminate) ?

3 . Write in completely f a c t o r e d form the denominators of' those

f r ac t ions t h a t terminated In Problem 2.

' 1 . Carry to six decimal places t h e following fractions.

6-jL. - The R a t i o n a l Number Cor re spond inq to a Per iod ic Decimal

We saw how t o find by divislon the decimal expansion of a

given r a t i o n a l number. We have found t h a t the decimal expanslon is periodic . But suppose we have the opposite situation, that is,

we are given a periodic decimal. Does such a decimal in fac t represejic a rational number? How can we f i n d out?

We can see how t o approach this prob lem by considering an

example. L e t us write the number 0.132132132132 ... and call it n, so t h a t n = 0.132132m . . . . The per iodic b l o c k of d i g i t s

is 132, so we multiply by 1000 whlch shifts the f l r s t block of'

d l g l t s t o t h e l e f t o f t h e decimal point and glves the r e l a t i o n

lOOOn = 132.132132m . . . n = 0.132132m . . .

By subtracting we obtain 999n = 132

so that

or , in simplest form '1 4 n = - 333'

'4 4 We f i n d by t h i s process t h a t 0.132132132m . . = m. The example here illustrates a general procedure which mathe-

maticians have developed to show t h a t every periodic decimal

represents - a r a t i o n a l number, We see, therefore, - that - there - I s - a

one-to-one correspondence between the set of rational numbers, and --- - t h e s e t of periodlc decimals. It would be quite equiva len t , then, 7--

f o r us to define t h e r a t i o n a l numbers as the s e t of numbers repre-

sented by all such periodic decimals.

Before we leave the subject of decimals we want t o d lscuss

one Interesting f a c t about t e rmina t ing decimals. 1 We saw t h a t rationals l i k e 2 = 0.5,

1 - = 0 .2 , $ = 1.875, 5 27.68 are a l l represented by terminating

decimals. How can we determine when t h i s will be the case? If, f o r i n s p i r a t i o n , we look a t the rationals of this type which we

discussed, we see an obvious clue: the denomlnators seem to have

only the prime f a c t o r s 2 or 5 , or both. (See Problem 3 in mercises 6-3. )

Consider a r a t i o n a l niunber in which the denominator is a power 39 of 2, such as T . 2

4 4 By multiplying by %, or 1, we can write p.

5 5

Slnee 2" 54 may be written

we can write:

Similarly if we have a rational number in which the denominator is

a power of 5 we can proceed as in the fo l lowing example,

5 Can you show t h a t 55m25 = 10 ?

Quite generally, if we have any rational number with only powers of 2 and powers of 5 In t h e denominator, we can use the same technique. For example,

and thls gives a terminating decimal representation. Can you 3 prove that is a tenninating decimal by rnultlplylng by 9

2 ' 5 5 In order to establish a general fact of t h l s kind suppose we ask the following question. What rational number ( p and q

assumed to have only 1 as a common factor) can be represented by 5 whhe~e N is an integer?

Suppose t h a t this Is Indeed the case and that

Theref ore k qaN = p a 1 0 . k Thls says that q divides the product of p and 10 . But

we assumed tha t p and q have only 1 as a common factor.

Hence q must divide lok. But the only poss ib le f a c t o r s of 10 k

a r e numbers which are powers of 2 multiplied by powers of 5 . Thus we have proved t h a t a r a t i o n a l number r has a termlna-

ting decimal representatfon if and only If the denominator of r

c o n s i s t s only of the product of a power of 2 and a power of 5. Then r must be of the form

where p l a an in teger .

Class Exeroise - 1 P e r f o m each of the following subtractions.

(a ) 10n - n I d ) loon - 10n (b) 1OO.n - n { e ) 1,000n - n ( c ) 1,000n - 1011 I f ) 10,00017 - lOOn

2. Write t he products as a a i n g l e number.

(a ) 10 x 0,?8oT . . . I f ) 1,000 x Q.613Jrg3q ... ( b ) 100 x 3.12iF .., {g) 100 x 8.03153 ... - ( c ) 1,000 x 0.035035 ... (h) 100 x 312.899g ... ( d ) 10 x 16.a ... (i) LO x 312.89sT . . . ( e ) 10 x O.OO~-I~CF ... (j) 10,000 x 6.01230m ...

3 . Subtract in each of t h e following.

( a ) 3128.993 . . . ( e l 1.233337 ... 312.899 ... 0.123337 . .

( c ) 162.162m ... 0.162iEZ . , .

I!. For each of the following numbers N ffind the smallest number o f the form lok (10, 100, 1000, e tc . ) so that ( ~ Q ~ - N ) - N I s a terminating decimal. Show t h l s to be t r u c .

( a ) 0.555 . . . (b) 0.73n . . * - ( c ) 0.901901 ... (a) 3.02339 ...

( e ) 163.177 . * .

(f) 672.42v ... (g) 0.123456% . . . (h) 3.4100g . . .

a 5 . Express each of t h e following in the form F, where a and

b are counting numbers.

6. If a is replaced by 1 in F;, a by what numbers between 23

and 50 may b be replaced so t h a t a F C E ~ be represented by

a terminating decimal?

a 1. What r a t i o n a l numbers in form 6 have these decimal expres- s l o n s ?

( a ) O . O ~ . . . ( e ) 0.1625 (b) o.~ili ,.. (f) 0.1665 ... ( c > 0.0555 ... (g) 5.125m ... (d) 0.123223 ... (h ) l0.01f5E .,.

2. Write each denomhator of t h e fo l lowing numbers in completely factored form.

3 . Which of the numbers in Problem 2 have decimals whlch repeat

zero?

4. If a is replaced by 1 in the rational number %, by what a numbers between 63 and 101 may b b e replaced so t h a t i;

can be represented by a terminat ing declmal expression?

6-5. Rat iona l P o i n t s on -- t h e Nwnber Line

If we use decimal representations f o r r a t i o n a l numbers, we see immediately how t o locate and how t o order the corresponding

p o i n t s on the number l i n e . - Consider f o r example t h e r a t i o n a l number 2.39blc ... and

its place on t h e number line. The digit 2 in t h e units p l a c e

tells us immediately that t h e corresponding r a t i o n a l point P l i e s between the integers 2 and 3 on the number l i n e . Graphi-

c a l l y t h e n the f i r s t rough p i c tu re is this:

A more prec ise d e s c r i p t L o n is obta ined by looking a t t he first two

d i g i t s 2 . which tell us immediately t h a t P lies between 2.3

and 2 On t he i n t e r v a l f r o m 2 to 3 , then, divided i n t o

tenths (and magnified t e n times f o r easy comparison) we f i n d P

a s shown below

If we continue the process of successively refining the location of P on the number line we have a picture such as the f o l -

Location of point P corresponding to 2 . 3 9 m ...

lowing : P

P (2.3 ...) I I I 1 vl I I I I I I 2 .o - -.o

/' -- 0 --.

/ - P I I 1 I I I I >\

P (2.39 ...) 2.30 --- 3 0

From such a decimal representation f o r a rational. number we eas i ly f i n d how t o locate t h e number to any desired degree o f

accuracy on the number l l n e .

Moreover, given any two distinct rational numbers An t h i s

form it I s a slmple matter t o tell by Inspec t ion whlch is larger and which is smaller, and whlch precedes t h e other on t h e number

line.

If you think of loca t ing the poin t ' careful ly on the

number line would you prefer to use 7 or 0.- ... ? ~f you

wish to compare 2 with another rational, which form is easier 7 to use, 2 or 0.- ... ?

7

/--

-/--

-/-- - d - -

P 4 I IV 1 I I

Bercises 6-5 1. Arrange each group of decimals in t h e order In whlch t h e

po ln t s to which they correspond would occur on t h e number

l l n e . List f i r s t the p o i n t farthest to the l e f t .

Y P (2.396.. .) 23%0 - wN k 2.400

\

P / A - - ~ \ 1 y,/ldv ,- 1 I I 1 I I \

P (2.3961.. .) 2.396@ - -+- ---- 2.3970 /

/ P --- \ -- - I 1 L I V I I I ,

P (2.39614.. .) 2.39610 2.39620

(a) 1.379 1.493 1 rn 385 5.468 1.372

(b) "9.1126 -2.765 -2,761 -5.630 '2.763 (4 -0.1547'5 0.15467 0.15463 0.15475 0.15598

2. In Problem (lc), which p o l n t s lie on the fo l lowing segments:

( a ) The segment with endpoints 1 and 2?

(b) The sement with endpoints 0 and l?

( c ) The segment with endpoints 0.1 and 0.2?

( d ) The segment with endpoints 0.15 and 0.16?

( e ) The segment with endpolnts 0.$54 and 0.155?

3 Draw a 10 centimeter segment; label the endpoints 0 and

1, and d iv ide the segment into tenths. Mark and label t h e

following points:

4. Arrange each group of ra t ional numbers in order of' increasing size by first expressing them in decimal form.

6-6, I r ra t iona l Numbers

We have learned many things about ra t ional numbers. One of

the moat important is the denslty property; between any two distinct r a t i o n a l numbers on the number l i n e there I s a t h i r d

rational number, This te l l s us that there are many r a t i o n a l numbers and r a t i o n a l points--very: many of them. Moreover, they

are spread throughout the number l i n e . Any segment, no matter how small, conta ins Snfinltely many r a t i o n a l po in t s . One might

t h ink that - a l l the points on the number l i n e are rational po in t s .

Let us locate a c e r t a i n p o i n t on the number l i n e by a very simple

compass and s t ra igh t edge construction. Perhaps this point will

have a surprise for us.

[sec. 6-61

( a ) D r a w a number l i n e and call it/ . L e t A be the

point zero and B be the p o i n t one.

(b) A t B, draw a r a y m perpendicular to ,f . ( c ) On m draw a line segment m, one u n i t long.

Id) Draw segment n. (e ) With A as center and radius AC, draw a c i r cu l a r

a r c which i n t e r s e c t s / . Call the point of i n t e r -

s e c t i o n D.

Figure 6-6

Now consider two questions:

( 1 To what n m b e r (if any) does point D correspond?

2 Is this number a rational number?

Cons ide r the flrst question, "TO what number does point D

corre~pond?~' F i r s t f i n d the length of , s lnce and have t h e same l e n g t h . We s h a l l use as unit of measure the unit

distance on the number l i n e . In Figure 6 4 , t r i a n g l e ABC is a

r i g h t t r i a n g l e . The measure of is 1. The measure oP

is 1 We can use the Pythagorean property to f i n d AC.

[ A C ) * = ( E l 2 + ( ~ 8 ) ~

The p o s i t i v e n m b e r whose square is 2 is defined as the square r o o t o f 2 and is w r i t t e n f i ---

Thus,

Therefore, the po in t D corresponds to the number fi, Is f i a r a t i o n a l number? I s i t the quotient of two integers, and can it be represented as a fraction

$9 i n which p and q are integers

and q # O? To answer this questlon, we sha l l fol low a l i n e of reasonlng

which people very often use. It is the t y p e of reasoning which

Johnts mother used one day when John was la te from school. When h i s mother scolded hlm he sald t h a t he had run a l l the way home.

"NO, you dldn tt run a l l t h e way," she sald flrmXy. John was

ashamed, and asked, "HOW d l d you know?" "1f yau had run a l l t h a t

way, you would have been out of b r e a t h , " she said. "YOU a re n o t

out o f breath. Therefore y o u did not run."

John's mother had used Indirect reasoning. She assumed the

opposite of the statement she wished t o prove, and showed that t h i s assumption l ed t o a conclusion which could not posslbly be

true, Therefore her assumption had t o be fa lse , and the original statement had t o be true.

We s h a l l prove tha t f i i s not a r a t i o n a l number. We use

indirect reasonfng. We s h a l l assume t h a t is a rational

number and show that this asstunption leads to a n impossible conclusion.

Assune t h a t fi is a rational number. Then we can write 6 as 2 where p and q are In tegers and q # 0. Take E

q ' 4 in simplest form. This means t h a t p and q have no common --- fac , to r except 1,

2 If = , then

2 2 = %, and so 2q2 = p . Since p and

q

q a r e i n t ege r s , then p2 and q2 are a l s o in tegers . If

P2 = pq2 then p2 must be an even number. ( ~ n i n t ege r is even

ff it 5s equal to 2 times another Integer.) Thus, p . p must be even. An odd number times an odd number is an odd number.

(Do you remember why?) Thus, p must be even, and can be written as 2a, where a I s an Integer.

2 Then p =2$ may be written as (2a12 = 2q 2

and (2a) (2a) = 2q 2

2 and 2*(2a) =2q 2

and 2 2a = q 2

This t e l l s us t h a t q2 is a l s o an even number since it in equal t o 2 times another in teger . So q is also an even number.

Thus our assumption, t h a t f i is a rational number E in 9

simplest form, has led us t o the conclusion that p and q both

have the f a c t o r 2. This is lrngosslble, since the simplest form f o r a fraction I s t he one In which p and q have no common - f a c t o r o the r than 1, So the statement "6 is a r a t i o n a l number" must be fa l se .

S i n c e the measure of segment i n Figure 6-6 is fi, t h e n f i must be the number which corresponds t o po in t D. It has been shown t ha t 4 2 is n o t a rational number. Therefore, there is at least t h i s one point on the hwnber l i n e which corre- aponds t o some number whfch ia not a rational number. In other worda, even though the rational points are denae, the set of points on the number line containa more points than there are r a t l a m 1 numbers.

A number like which is not a rational number, Is

called an irrational number. The prefix " i t changes the mean- i n g of " ra t iona l " t o "not r a t i o n a l . t t

b e r c i s e s 6-6 - 3 . Construct a figure like Figure 6-6, and label po in t D "&?".

Then use your compass to locate the point whlch corresponds to the number -(&)), and label it.

2 . Draw a number l l ne , using a u n l t of the same l eng th as t h e

unit in Problem 1. Use the l e t t e r A f o r t h e p o i n t 0 and

the l e t t e r B f o r the point 2. A t B construct a segment

perpendicular to the number l i n e and 1 u n l t in l ength , and

call it , D r a w . What is the measure of segment m? 7. Use the drawing f o r Problem 2, and locate on t he number l l n e

the points which correspond to 6 and ( Label t h e

points.

9 . Do you t h i n k 6 is a rational number or an irrational number? Why?

*5. Uslng the same method as in Problems 2 and 3, locate the p o i n t

. Can yoy work out a way t o l oca te the poin t f o r fl For f i

6. Locate the p o i n t s which correapond t o these numbers?

(a) 2 6 (b) 3 f i t c > -OK) 7 . Do you t h fnk tha t ( 2 f i ) is a r a t i o n a l nwnber or an i r r a -

tional number?

8. BRAINBUSTER: Prove that f i is an i r r a t i o n a l number. (use

i n d i r e c t reasoning very similar to the line of reasoning whfch

we used to show that f i is i r r a t i o n a l . A t one p o l n t you will have to lolow t h a t if p2 has 5 as a fac to r , then p

also has 5 as a factor. Prove thls simple f a c t . Before you try to prove that f i I s i r ra t iona l , t h ink of t h e unique

fac tor iza t ion property of counting numbers. If the prime number 5 were not a factor of p then how could it be a

2 fac to r of p ? )

In the preceding discussion it was proved tha t f i I s not a

rat ional number. It I s a great surprise to f i n d that we can so

easily construct a line segment whose length is not given by a

rational number. Moreover, it appears that there are many ather numbers, such as 6 and 6 which are not r a t iona l s . If you th ink about the rationals and the irrationals a bit you can see

how to write many, many irrationals. For example, every number of the form g f i , where a I s r a t i o n a l , w i l l be Irrational. 6 Hence the set [E &I can be put I n t o one-to-one correspondence with the s e t of r a t i ona l s [%I. Y e t t he s e t (E&] is obviously

on ly s very small par t of the irrationals!

Indeed, we have suffered a great disfllusionment--the rational

numbers, d e s p i t e being dense on the number l i n e , a c t u a l l y leave empty more positions t han they f i l l !

When we s e t up a one-to-one correspondence between a given s e t and t h e s e t o f counting.numbers (or a subset of the set of

count ing numbers) mathematiclans say we have "enmerated" the se t . t I We can enumerate" the set of all r a t i o n a l numbers, but Georg

Cantor ( 1845-1318) discovered in 1874 that the s e t of i r r a t i o n a l numbers cannot be "enumerated" by any method. There are so many

I r r a t i o n a l numbers that it is impossible to set up a one-to-one correspondence between the s e t of these numbers and t he set of counting numbers. No matter how you try to dfsp lay irrational

numbers some l r r a t i o n a l numbers will alway~ be left out ; more will be l e f t out than have been included, as a ratter of f a c t . This is

what we mean when we say t h a t the r a t l o n a l numbers leave more places empty on t he number l i n e than they fill.

If you are interested in l earn ing more about this important phase of mathematics you might refer to One Two Three ... Infinity --- by George Garnow (pages 14-23). A br ief but interesting hlstory

of Cantor's life can be found in Men of Mathematics by E, T. Bel l -- (chapter 29 ) .

6-7. A Decimal Representation for fi Numbers l i ke and f l correspond t o p o i n t s on the

number l i n e ; they specify lengths of l i n e segments and they

satisfy our n a t u r a l no t i on of what a number is. Perhaps the most

unusual aspect about is the way It was deffned: is

t h e positive number n which when squared y i e l d s 2, so t h a t

This d i f f e r s from our previous way of de f in ing numbers, a lnce up

to now we have d e a l t mainly wlth integers and numbers defined as

ratios of in t ege r s . In o r d e r to he lp us gain a better understanding of f i we

s h a l l look f o r a new way of describing f l in terms of more

familiar notions. IS, for example, we could somehow express f i as a decimal this would he lp us to compape it w i t h the rational

numbers we know. It would also tell us where to place it on t h e

number line.

Let us thf nk about the definition of t h e number fi, namely 2 ( f i ) = 2. If we t h i n k of squaring 1 and 2 we no te irnmedi-

ately t h a t

l2 < ( & I * < 2* and hence 1 <f l< 2 .

This t e l l s us that fi is greater than 1 and less t h a n 2,

but we already know that. We might try a c l o s e r approximation by

t e s t l n g t h e squares o f 1.1, 1.2, 1 1. I t , 1.5. A l i t t l e

arithmetic of this s o r t (try it!) leads us t o the resul t

and therefore we conclude t h a t 1. '1 ( f i< 1.5. The arithmetic involves slightly more work a t the next stage

but we see t h a t

and therefore

If we try to extend the process further we g e t a t t h e next stage

l * ~ t l ~ l < JT < 1*1!15.

You can see that this procese can be continued as long as

our enthusiasm lasts, and gives a better decimal approxlrnatlon at every stage. If we continued to 7 place decimals we would find

This is a very good approximation of , for

and

By the use of the definlng property, ( = 2, then, we

can flnd decimal approximations f o r f i which are as accurate as we wish. We are l e d to write

where t h e three dots indicate that the digits continue without

terminating, as the process above suggests. Geometrically the procedure we have fol lowed can be described

as follows I n the number l i n e . Looking f i r s t at the integers of the number l i n e on the segment from 0 to 10, we saw that 6 would be between 1 and 2.

mlarglng our view of this segment (by a ten-fold magnification) we saw that f i is on the sement with end-points 1.4 and 1.5

and again magnifying t h l a picture, f l lies within the interval (1.41, 1.42)

and so on till t he 8th stage shows us t h a t f i lies between 1.4142135 and 1.4142136.

This process shows us how t o read t h e successive dlglts in the decimal representation f o r fi, A t the same time it gives a way

t o define t h e position of the p o i n t on the real l i n e .

When we write the number as 1 , 1 2 1 , . . it looks suspiciously l i k e many r a t i o n a l numbers we have seen, such as

1 = 0.3333333 . . . and - = 0.111285714 . . . . 7 7

We pause to ask, how are they d i f f e r e n t and how can we tell a

rational from an i r r a t i o n a l number when we see only the decimal representations of t h e numbers?

The one s p e c i a l feature of the decimal representation of a r a t i o n a l number is that it is a per iodic decimal. As we have seen, every perfodic decimal represen ts a rational number. Then

the decimal representation o f f i cannot be periodic, f o r f i is i r r a t i o n a l . We can be sure t h a t as we continue to f i n d new digits in the decimal representation,

no group of digits w i l l ever repeat i n d e f i n i t e l y . We can only be

certain that a decimal names a rational number when the perlod of the decimal is indicated, usual ly with a vinculum (-1.

Exercises 6-7 - 1. Between what two consecutive integers are t h e following i r r a -

t i o n a l numbers? (Write your answer as suggested for ( a ) ) .

Use t h e tableon pages 200-201,

(a) ~ 7 5 LL< J%<L]

(b) JE

(d) (Hint: 4280 I s 42.80 x lo2, so begin eati-

mating by thinking of fim 10)

(4 -5

2. Express (a), (b), and ( c ) as decimals to six places.

(a ) (1.731)*

(b) (1.7321~

(4 ( 1 . 7 3 3 1 ~

( d ) Flnd the difference between your answer f o r (a) and the number 3; f i n d the difference between your answer for ( b ) and the number 3 ; flnd the difference between your answer f o r ( c ) and the number 3 .

( e ) To the nearest thousandth what i s the b e s t decimal expression for fi?

Which of the numbers suggested is the better approxfmation of the fol lowing i r r a t i o n a l numbers?

3 . 6: 1.73 or 1.74

4. : 3.87 or 3.88

5 . f i : 25.2 or 25.3

Flnd, to t h e nearest t en th , the nearest decimal expression for these irrational numbers:

6 . f i 7. 4 x 8.

* 9 , For what p o s i t i v e number n ( t o one decimal p l a c e ) i s

n2 = lo?

*lo. For what positlve number n (to one decimal p l a c e ) is

n2 = 149?

6-8. I r r a t i o n a l Numbers and the Real Number System -7-

We have seen t h a t all r a t i o n a l numbers have per iod ic decimal

representations and t h a t a l l periodic decimals correspond t o

r a t i o n a l numbers. We saw a l s o t h a t is no t r a t i o n a l and t h a t therefore , it is represented by a non-periodic declmal. Hence we

have ca l l ed f i an irrational number. We now use this decimal form to d e f i n e t h e s e t of irrational

numbers. We def ine - an irrational number as an^^ number with a - -- non-periodic declmal representation.

The system composed of all r a t i o n a l and irrational numbers we

c a l l the real number system. From thls we see that real number can be characterized -

a decimal representation. - If the decimal representation I s periodic the number 5 -- - -

r a t i o n a l number, otherwise - the number is an i r r a t i o n a l number. -- With every point P on the real nwnber line we associate one

and on ly one number of this form by a process o f successive loca- tions In decimal i n t e r v a l s of decreasing length. The drawings

on the fol lowing page i l l u s t r a t e the f irst few steps i n f l n d i n g

the decimal corresponding to a po in t P on the number 1Lne. Con- sider point P between 3 and l ~ ,

Note t h a t any two distinct points PI and P2 w l l l cor re - spon5 to d i s t i n c t decimal representatfons, f o r if they occur as

on the number line we need only subdivihe the number l i n e by a nurficiently fine decimal subdivision (tenths, hundredths,

thouzandths, e t c . ) to assure that PI and P2 are separated by a point of subdivision.

Conversely, g iven any decimal, we have found how to locate t h e

corresponding po in t of t h e real number line by considering succes-

s i v e r a t i o n a l decimal approxlmations provided 'by t he number.

(~emember how we started to locate t h e po in t 2 . . In Section

6-5. ) Thus there Is a one-to-one correspondence between t h e set of

r e a l numbers and t h e s e t of points on the number l i n e . The s e t of r e a l numbers contains t h e s e t of rational numbers

as a subset . We have learned that theae rational numbers form a

mathemztical system w i t h opera t ions , addition and multiplication

and t h e i r inverses, subtraction and division. The same is true of the en t i r e s e t of real numbers. We can add real numbers,

rational o r irrational, snd we can multiply real numbers. The

resul t ing number system has all of the proper t i e s of the rational number system. In add l t i on it has one Important property which

the r a t i o n a l number system does not have. This w l l l be discussed below.

Before we l i s t these properties we should pause t o ask what we know about the operations themselves. You should n o t have much

trouble understanding the meaning of addition In the real number system in terms of the number l i n e . Even tho-dgh there is no

sirnpler name for a sum such as n+ 6 than the symbol " f l + J S ' ' i t s e l f , you can think of a method of constructing the

point f i+ 6 on the number l i n e by placing segments of' length

fi and f i end to end. The meaning of multlpllcation is somewhat harder to illustrate.

Given segments of l eng th and it is possible to des-'

c r i b e a geometric construction of a p o i n t which we would naturally

call & 6. However, you w i l l have to study Chapter 9 before

you w l l l be prepared to understand such a construction. The two operations can a l so be given meaning in terms of the decimal representation which we have described, but here, too, difficulties are encountered which you are n o t yet ready to handle. This should

not cause you undue concern. Even a mathematician often has to

accept thlngs which he does not f u l l y understand in order to g e t on with the work which I s of immediate interest to him. But if these

ideas he accepts are Important, he always returns to them as soon

as he can and masters them. You will return to the real number

system again as you study mathematics in the future, and each time

you w i l l understand more of t h e d e f i n i t i o n and the meaning of the

opera t ions . We list first the familiar properties which the real number

system shares with the r a t i ona l number system.

Property - 1. Closure

(a) Closure under Addition. The real nwnber system is closed under the operat ion of additlon, i.e., if a and b are real numbers t h e n a + b is a r e a l number.

(b) Closure under Subtraction. The real number system is closed lander the operat ion of subtraction ( t h e inverse of addition), i.e., if a and b are real numbers then a - b is a r ea l number.

( c ) Closure under Multiplication. The real number system I s

closed under the operation of multiplication, I .e . , if a

and b are r e a l numbers then a . b is a real number.

( d ) Closure under D i v i s i o n . The real number system is closed under the ope ra t i on of dlvlsion (the inverse of multlplica- t i o n ) , I . e . , if a and b are real numbers then a + b

(when b 0 ) Is a r e a l number.

The operat ions of addi t ion , sub t rac t ion , mu l t ip l i ca t ion , and

division on real numbers display the properties which we have

a l ready observed f o r ra t ionals . These may be summarized as follows:

Proper ty 2. Commutativity

(a ) If a and b are reai nwnbers, then 'a + b = b + a .

(b) If a and b are r e a l numbers, then a b = b . a.

Proper ty 2. A s s o c l a t i v i t ; ~

( a ) If a , b, and c are rea l numbers, then a + ( b + c )

= (a + b) + c.

(b) If a , b , and c are real numbers, then ( a . b ) * c = a*(b*c).

Proper ty 2. Jdentl ties

(a ) If a is a r e a l nwnber, then a + 0 = a , l .e . , zero is t h e

identity element f o r the opera t ion of addition.

(b) If a is a r e a l number, then a a l = a, i.e., one is t h e

i d e n t i t y element for t h e opera t ion o f m u l t i p l i c a t i o n .

P roper ty 2, Distributivity

If a, b, tind c are r e a l numbers, then a. (b + c ) = (a.b)+(a4c).

Praper ty 6. inverse^

( a ) If a is a r e a l number, there is a rea l number ( - a ) , c a l l e d

the addit ive i nve r se of a such that a -k ( -a) = 0 .

( b ) If a is a r e a l number and a j 0 there is a r ea l number

b, called the rnu l t ip l fca t ive i n v e r s e of a such t h a t

a . b = 1,

Proper ty 7. Order - - The real number system is ordered, ire., if a and b a r e d i f -

ferent r e a l numbers then e l t h e r a < b or a > b ,

Proper ty - 8. Density

The r e a l m b e r system Is dense , i . e . , between any two d i s t i n c t

real numbera there is always another rszl number. Consequently,

bettreen any two real nwnbers we f i n d a s many more real numbers as

we w i s h . In fact we e a s i l y see that: (1) There is always a

rational number between any two distinct r e a l numbers, no matter

how close. ( 2 ) There is always an i r r a t i o n a l number between any

two d i s t i n c t r ea l nwnbers , no matter how c lose . (You w i l l l e a r n how t o illustrate this later in t h i s s e c t i o n . )

The ninth property of the system of r e a l nwnbers is one which

is not shared by the rationals.

Property 2. Completeness

The r e a l number system Is complete, that is, not on ly does a point

on the number line correspond to each real number, but conversely, a real rxnber corresponds to each po in t on the number l i n e .

The rational numbers d i f f e r from the real numbers in t h i s

respect. A point on t h e number line corresponds to each r a t f o n a l number, but no rational number corresponds to c e r t a i n poin ts on t h e number l i n e . We have seen that in t he aystem of rationals

t h e r e is no number which when squared y ie lds 2. However, in t h e real number system as def ined, such a number is included.

If a and b are posltive r a t i o n a l numbers and b = an we

mite

(read "a is an n t h root of b " ) .

It happens t h a t the n t h root of any positive rational number which is not itself the n t h power of a r a t i o n a l number 2 s an i r r a t i o n a l number. This means tha t such numbers as,

are irrational numbem, whereas,

* are rational numbers. Hence, in t h e system of r a t i ona l numbers we cannot hope to extract nth roots of any r a t i o n a l numbers which are not n t h powers of ra t iona l numbers. However, when we imtroduce the IrrationaLs to form the real number system we can flnd n t h roots of a l l pos i t ive rational numbers, and of a l l positive r ea l numbers as w e l l . Thus a very useful property of the real number system f s:

The real number system contains nth roo t s , "& of a a l l positive rational numbers 6, b f 0; the real

number system conta lns n t h roots of a l l p o s i t i v e real numbers.

This assures us that we can flnd among the real numbers such numbers as

and any other n t h r o o t s of p o s i t l v e rational nwnbers, as well as

numbers which are swns, differences, products, and quotients of

such nwnbers. For example:

are real numbers. In addition t o irrational nwnbers whlch arise from f i n d i n g

roots of r a t i o n a l numbers there are many more i r r a t i o n a l numbers which are ca l l ed transcendental nwnbers. One example of a t r a n -

scendental number is the number whlch you have met i n your study of c i r c l e s . Recall that u is the ratio of the measure of

the circumference of a c i r c l e t o the measure of its diameter. It was surprisingly hard to prove that W is irrational, but it has been done. The decimal representation

22 - 22 is is not repeating. The number " is not , although 7 a f a i r approximation to . (compare t h e 'decimal representation

22 w i t h that of .) Of 7

When you study logarfthms In high school, you will be studying numbers that, with a f e w exceptions, are transcendental numbers. I

N is any positive real number and x I s t h e exponent such t h a t

then we say t h a t x is the logarithm of N t o the base 40; If 2

N is a power of 10, say N 10 , then clearly I# = 10 , so

2 is the logarithm of lo2 to the base 10. In such a ease, the

logarithm is a rational number. But f o r most numbers the logarithm will be a (transcendental) irrational number.

The trigonometric r a t i o s , such as the sine and the tangent of an angle, are o t h e r expressions which usually t u r n out to be

transcendental irrational numbers. These r a t i o s are defined i n

Chapter 9.

mercises 6-8a

1. Whlch of the following numbers do you thlnk are rational and which i r r a t i o n a l ? Make two lists.

( a ) 0.231m .. . ( 9 ) g (b) 0.23123112311123 .., (h) 9 - 0

2 Write each of t h e r a t i o n a l numbers in Problem 1 as a decimal

and as a f r a c t i o n .

3 , For each of t h e i r ra t iona l numbers in Problem 1 wrfte a

decimal correct to the neareat hundredth.

I. ( a ) Write three terminating decimals for rational numbers.

( b ) W r t t e three repeating decimals for rational numbers.

( c ) Write t h r e e decimals f o r irrational numbers.

You4 have learned how to insert other r a t f o n a l numbers between

two given r a t i o n a l s . Now that you have studied decimal representa-

t i o n for r e a l numbers, you can see how t o i n se r t e i t h e r rational o r i r r a t i o n a l numbers between real numbers. Look at these decimals

f o r two numbers a and b.

a = 4.219317317317m . ..

These numbers are quite close together, but any decimal which

begins 4.22 . . . v i i l l be g rea te r than a and less than b. We

can then continue t h e decinlal I n such a way as to make I t rational

or to make it irrational. We can even make the decimal termina- t i n g if we w i s h t o do so. For example, 11.222 and 4.225m ... are r a t i o n a l numbers while 4.2256225662.25666,. is irrational.

A l l of these numbers lie between a and b.

In orde r to be sure that a real number w r i t t e n in the decimal form is an irrational number we must make certaln t h a t it is not

periodic . An easy way t o do thfs is to repeat a number of one or

more digits and t o follow t h i s repeating number by a different

d i g i t once, then twice, then three times, and so on. For example in the number b = 4.2365655655565555 ... the d i g l t 6 is

repeated and 1s followed by a 5, once, twice, and so on. In t h e

number 4.237'823788237888 ... the number 237 is repeated and is

followed by 8 once, then twlce, then three times, and so on.

Because the d i g i t 8 occurs an increasing number of t lmes I n t h e formation of t h i s declmal, the decimal cannot have a d e f l n l t e

period. The nwnber represented must therefore be irrational.

hercises 6-8b - 1. (a) Write a decimal f o r a r a t i ona l nwnber between 2.384m,..

and 2,3693G.. . . (b) Write a decimal for an irrational number between t h e

numbers In (a),

2. Write decimals for (a) a r a t i o n a l number and (b) an irra-

t lona 1 number between 0.7 46019. . . and 0.3 42806. . . 3. Write decimals for (a ) a ra t ional number and (b) an i r r a -

t icqal number between 67.283 ... and 67.28106006 ... 4. Do ycu think that the real number system conta ins square roots

of a l l integers? Support your answer by an example.

5. A n approxlmation which the Babylonians used f a r was the

i n t e r e s t i n g r a t io , HOW good an approximation is t h i s ?

Is it as good as

6-9. Geometric Properties of the Real Number L l n e --- The one-to-one correspondence between the real numbers and

the po in t s of t h e nwnber l i n e gives us f o r the first time a sat is factory geometric representation of' numbers. For t h i s reason it is customary to refer to the number line as -- the real number

l i n e . We h o w t h a t there a re no gaps or missing points in the real

number line. We can speak of t r ac ing the real number line continuously and know that the segment descr ibed at any stage has a

length which is measured by a rea l number. Thus in the number

line Indicated below

we h o w that has a length of measure f i - 1, the length of

has measure 3 - , has measure - 1, I s

measured by n - f i We can think of a p o i n t movlng continuously from 0 to 1.

At every loca t ion we may associate w i t h it a real nwnber.

Because of t h l s continuous property of our r e a l number system,

we sometimes refer to It as the continuum of real numbers. - Rat iona l Approximations & I r r a t i o n a l s

Whenever we give an irrational number in its decimal form,

f o r example, N = 0.019234675 ... , we see that we automatically

define a sequence of rational numbers which give c lose r and closer approximations to the irrational nwnber N. We can read such a

sequence of ra t iona l approximatZons as,

Rationals and Irrationals in the World Around Us - We see many examples of rationals every day--the p r i c e of

groceries, the amount of a bank balance, the rate of pay, the

amount of a weekly salary, the grade on a test paper. Although we have n o t considered the 1rrat;ionals for very

long, it is easy to see many examples which Involve i r r a t i o n a l numbers. For example, consider a circle of radlus one unit. What l a its circumference? Why 2 w units, of course. In fact, any circle whose radius is a rational number has a circumference

which is i r r a t i o n a l . Also, the circular closed region of radius r has an area, the measure of which is an irrational number ( n r2) whenever r is r a t i o n a l and usually whenever r is irra- t i o n a l .

The volume of a c i r cu l a r cyl inder is found by the formula 2 V = r h and it8 lateral surface area A by A = 2 w r h where

h is the altitude of the cyl inder. Here also the volume and area are given by i r r a t i o n a l numbers if the radius r and alti-

tude h are given as r a t iona l s . Also, we l e a r n how to draw some lengths of irrational measure

by the following simple succession of right triangles:

You will notice however tha t this process gives some rational

l eng ths since f i = 2, f i = 3 , and so on.

Draw this succession of r ight angles on your paper. Continue

u n t i l you have drawn a length to represent f i 0 .

b e r c i s e s 6-9 1. Which of the fol lowing numbers are rational and whlch are

irrational?

The nwnber of units in: nit. (a ) the circumference of a c l r c l e whose radius is q

(b) the area of a square whose sides are one uni t long.

( c ) the hypotenuse of a r igh t t r i ang le whose sides are 5 and 12 units long.

( d ) the area of a square whose sides have length f i units. (e ) the volume of a cyl inder whose heigh t is 2 units and

whose base has radius 1 unit.

( f ) the area or a right t r i ang le with hypotenuse of length

2 u n i t s and equal sides.

2 With t h e use or the f a c t s that f i M 1 . 4 1 h and t h a t

f i fil 1.732 show that n* f i w & ~ 2 . 4 4 9 .

3 . \hen we b e g l n to compute w i t h irrational numbers we sometimes

encounter relationships whlch look rather pecu l i a r a t f i rs t

bu t which make p e r f e c t sense on closer inspec t ion .

Here are two examples:

The multiplicative inverse o f I s $ G. The multiplicative i nve r se o f ( f i -t- a) is n- n)*

+(a ) Verify these assert ions approximately by using the

decimal approximations given in Problem 2.

(b) BRAINBUSTER: Verify these assertions exact ly by com-

puting w i t h t h e irrational numbers themselves.

* , Find t h e rad lus of a c i r c l e whose circumference i s 2 . Give

an approximate value f o r the radius. (Use 3 1 f o r T , )

INDEX

The reference is to the page on which the term occurs.

accuracy, 223 addition, 3, 12, 18

property, 76, addit ive inverse, 1 , 19, 38, 47, 237 altitude

8' of cone, 484 of pyramid, 477

angle(s1 complementary, 365 corresponding, 159

Antarctic Circle, 529 antipodal points, 515 antipode, 515 apex of pyramid, 476 approximat ion, 276 arc, 162 Arctic Circle, 529 area, 145

of circle, 458 of regular polygon, 457 of surface of sphere, 535, 536, 538 of trapezoid, 454

associativity, 236, 270 average 243 axes, 24 axis of symmetry, 172 base, 472

of pyramid, 476 bisect, 164 bisector, 164

perpendicular, 191 British k p e r i a l gallon, 153 Cantor, Georg, 262 capacity, 150 centimeter, 139 chance events, 311 circle, 64, 170, 171, 173, 175, 511

Antarctic, 529 Arctic, 529 area of, 64, 458 circumference of, 64 great, 516, 520 i n t e r i o r of, 520 semi-, 517 small, 516, 517

length of, 540

closed number phrase, 53 closure, 236, 270 combinations, 303 commutativity, 236, 270 compass, 157 complementary angles, 365 completeness, 271 compound sentence , 68 concurrent lines, 167 conditf on, 30 cone, 484

al t i tude of 484 height of, 484 right circular, 484 slant height of, 484 vertex of, 484

congruent, 180 triangles, 176

conjecture, 549 Goldbach, 549

constant, 397 of propor t ional i ty 397

construotion(s), 160, l b 3 continuum o f real numbers, 276 coordinates, 21, 23, 24 corresponding angles, 159 cosine, 359 cotangent, 367

number, 3, 235

cubic meter, 148 cylinder

right circular, 469 m t z i g , 572, 573 decimal

expansion, 246 non-periodi c 267 periodic , 246, 250, 2 267, 275 point, 134 repeating, 247 representation 246, 265, 267, , 273 terminating, 267

decimeter, 139 dekometer, 139 density, 240, 244, 271 Descartes, 569 diameter, 514 digi t

significant, 220, 231, 232 In the product, 231

dimensfonal, 421 direct variation, 396 directed s e p e n t , 3, 12

distance between parallel planes, 464 from a palnt to a plane, 464

distributivity, 236, 271 divis ion, 42, 128 edges , 466 Egyptian, 193 empirical probability, 323 enumerate, 262 equat~on(s), 58, 60

equivalent, 75 linear 97

equality, 78, 83 equivalent equations, 75 error

greatest possible, 216, 219, 228 percent of, 223 Q

relattve, 223 ~ ~ c l i d , 162 mtler, 447, 549 Eulerts formula, 447, 569 event ( a )

independent, 335 mutually exclusive, 330, 342

exponent, 121, 126, 129, 149, 273 negative, 121, 126, 129 notation, 134 zem as, 121

factorial , 296, 297 fiducial oint , 3 formula, Franklin, 563 Fulkeraon 572, 573 aalileo, 407 gallon, 153

British Imperial, 153 Goldbach Conjecture, 549 grade, 377 gram, 150 graph, 30 gravitation

Newbonfs law o f , 410 great circle, 516 greatest possible error, 216, 219

of a SUM, 228 Greenwich meridian, 527 half-line, 3, 22 Heawood, 563, 569 hectometer, 139 height

of cone, 484 of pyramid, 477

hexagon, 170

hexagonal pyramid, 477 right prism, 210, 469

Hilbert, David, 577, 578 hyperbola, 405 hypotenuse, 194, 195 identite, 236, 270 inch, 1 3 independent event, 335 indirect reasoning, 259 inequalit , 58, 91 integer(sg, 3, 9, 225 i n t e r i o r

of a c i r c l e , 520 of a sphere, 520

inverse, 271 additive , 237 multiplicative, 237 variation, 401

irrational number, 257, 260, 267, 275 Johnson, 572, 573 kilometer, 139 lat era1

edge of oblique prism, 472 of pyramid, of right pri:;: 466

face(s) of oblique priam, 472 of pyramid, 476 of right prism, 466

l a t i t u d e , 529 parallels of, 517, 526

laws of variation, 394 Lehmer, D. N., 546 length, 138

of small c i r c l e , 540 light ear, 118 line ( s3

concurrent, 167 half -, 3, 22 parallel, 158 perpendicular, 158 real number, 276

linear equation, 47 liter, 150 logarithm, 273 longitude, 528 mas, 150 median, 169 mega-, 141 meridlan

Greenwich, 527 zero, 527

meter, 138 square, 145

metric system, 137, 154 ton, 151

micro-, 141 micron, 141 millimeter, 139 multiplication, 34, 125

property, 83 multiplicative inverse, 237 mutually exclusive events, 330, 342 negative

exponent 121, 126, 129 number, 6 , 9

Newtonts law of gravitation, 410 non-perlodic decimal, 267 notation

exponent, 134 powers-of -ten, 115, 133

nwnber(s) counting, 3, 235 irrational, 257, 260, 267 275 line, 1, 2, 3 5 , 255, 276

real, 276 negative, 8, 9 phrase, 52

closed, 53 open, 53

positive 3 prime, 546 rational, 9, 255 real, 267

continuum of, 276 number l ine, 276 number syat;em, 276

sentence, 51 transcendental, 273

oblique prism, 471, 472 octagon, 170 one -to -one correspondence, 2, 262, 268, 276 open

phraae, 53 sentence, 59

opposite, 9, 13, 19 oMer, 2,37, 271 ordered pair, 25, 33, 94 origin, 3, 21 pantograph, 158 parabola, 103, 407 para l l e l (s)

lines, 158 of latitude, 517, 526 ruler, 158

parallel0 ram, 206 Pascal, 286 Pascal triangle, 286 pentagon, 170 percent of error, 223 periodic decimal, 248, 250, permutatf on, 292

symbol, 295 perpendicular, 190

b i sec to r 191 l ine , 158 t o the plane, 462

phrase, 52 number, 52

closed, 53 open, 53

p i (a), 64, 232, 273, 275, 535 plane(s)

distance between parallel 464 distance from point to, 464 perpendicular to, 462

plottin , 25 point ( sy

antipodal, 515 fiducial, 3

PO lygon area of regular, 457 regular, 170 slmple closed 434

polygonal path 434 polyhedron, 428 positive number, 3 power, 272 powers -of -ten notation, 115, 133 precision, 219, 223 prime number, 546 prism

hexagonal right, 210, 469 l a t e r a l eage of

oblique 472 right, 466

lateral face of oblique 472 r ight , 466

oblique, 471, 472 rectangular right, 208 r i gh t , 208 t r iangular right, 468

probability, 312, 313, 341 empirical , 323 of A and B, 334 of A o r B, 328

property addition, 76, gl multfplication, 83 Pythagorean, 193, 202

prot rac tor , 158 pyramid, 210

altitude of 477 apex of, 476 base of , 476 hefght of, 477 hexagonal, 477 lateral edge of, 477 lateral face of, 476 regular, 478 slant height of, 480 square, 477

Pythagoras, 193 Pythagorean property, 193, 202 quadrants , 27 quadrilateral, 170, 206 radius, 163 Rankin, 559 r a t i o n a l number, 9, 255 real nurnber(s), 267

continuum of, 276 l ine, 276 sy~tern, 276

reasoning indfrect 259

rectangle, 206 rectangular

r i g h t prism, 208 aystem, 24

regular polygon, 170 pyramid, 478

re la t ive error, 223 repeating decimal, 247 rhombus, 452 right

circular cone, 484 circular cylinder, 469 prism, 208 triangle, 192

root, 272 ruler

parallel, 158 scale, 386 scientific notation, 114, 221 segment

directed, 3, 12 selections, 303

symbol, 303, 304

semicircle, 517 sentence

compound, 68 number, 51 open, 59

set of solutions, 60 significant dl g i t , 220, 232

in the product, 231 a i d l a r triangles, 380 simple

closed p o l gon, 434 surface, 4t2

simplex, 422 sin, 366 sine, 359 slant height

of cone, 484 of p ramid, 480

slope, 37 t small circle, 516, 517 solid

spherical, 533 solutfon set, 60 sphere, 511

i n t e r i o r of, 520 surface area of, 535, 536, 538 tangent to, 515 volume of, 533, 535, 542

spherical soap bubbles, 537 s o l i d , 533

square ( a ) meter, 145 pyramid, 477 table of, 200-201

square root, 196 table of, 200-201

straight edge, 157 subtraction, 46 surf ace

of a sphere, 533, 535, 538 simple, 442

symmetry, 172 axis of, 172

table of square roots, 200-201 of squares, 200-201 of tri ommetric ratios, 368

tangent, 35 f to the s here 515

tetrahedron, 918, 477 terminating decimal, 247 tolerance, 217

topological, 564 transcendental number, 273 transversal, 159 trapezoid, 206 454

area of, 454 triangle(s)

congruent 176 Pascal. 286 - r ight , 192 simflar, 380

triangular r i g h t prism, 468 trigonometric ra t ios

table of, 368 Tropic

of Cancer, 529 of Caprfcorn, 529

T-square, 158 twin primes, 552 varf at ion, 392

di rec t , 396 inverse, 401 laws of, 394

varies airec t ly, 397 inversely, 414

vertex of cone, 484 vert ices, 465 vinculum, 248 265 Vinogradov 549 volume. 1d

01- sphere, 533, 535, 542 weight, 152

of water, 152 Williams, 573 zero

as exponent, 121 meridian, 527

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