Video Math Tutor: Basic Math: Lesson 3 - Operations on Numbers

8/14/2019 Video Math Tutor: Basic Math: Lesson 3 - Operations on Numbers

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BA S I CMA T HASelf-Tutorial

by

LuisAnthonyAstProfessionalMathematics Tutor

LESSON3:

OPERATIONSONNUMBERS

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E-mail may be sent [email protected]


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=Disclaimer: T h i s Se lf-T u t o r i a l is m e a n t f or s t u d e n t s ( 7t h g r a d et h r o u g h c o lle g e ) w h o n e e d a b a s i c r e v ie w o f a r i t h m e t i c on

i n t e g e r s (p o s it i ve a n d n e g a t i ve w h o le n u m b e r s ) a n d O r d e r o f

Op e r a t i o n s . I t i s NOTm e a n t fo r y o u n g c h i ld r e n w h o a r e

le a r n i n g a r i t h m e t i c fo r t h e fi r s t t i m e .

|ABSOLUTE VALUE|FTh e Absolute V a l u e of a nu mber is t he distan ce thatnu mber is fromzero on t he r eal nu mber line.

Mat h S y m b o l:

L The a bsolute value of a nu mber is NEVER n egative. Itseith er positive or zero.

Form al definit ion of Absolut e Value ofx:

This definition m eans th at th e absolute value of a real n umber x is ju st

itself if th e nu mber is zero or positive, and you cha nge t he s ign if th e

nu mber is negat ive. The x does n ot mea n nega tive x, but r at her t he

opposit e ofx. This is discussed in more deta il a litt le lat er on.

EXAMP LE 1: Find .

SOLUTION: This is h ow you can see wha t t he a bsolut e value ofth e number 3 is:

1 unit + 1 unit + 1 un it = 3 units a way from 0

| |||

d||d0 3

1444442444443

The absolute value of 3, denoted by is equal to th e nu mber of un its it is

awa y from zero, since it is th ree un its awa y, th en .


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F The a bsolut e value of 3 is also 3, since 3 is t hr eeun its a way from zero:

1 unit + 1 unit + 1 un it = 3 units a way from 0

| |||

d||d3 0 1444442444443

Together:

3 un its awa y from 0 3 un its awa y from 0

| | |

d||d||d

3 0 3 14444424444431444442444443

EXAMP LE 2: What is ?

SOLUTION: becau se 4 isfourun its a way from zero.

d|||d

0 4 1444442444443

four un its away

EXAMP LE 3: What is ?

SOLUTION: since 0 is well ZERO un its awa y from itself,

it is th e ONLY absolut e value t ha t is equal t o zero. All oth ers ar e positive.

d

0


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OPPOSITENUMBERS

FOpposi te Number s ar e th e same distan ce from zero on t he r ealnu mber line, but th ey are on opposite sides on t he n um ber line.

Mat h S y m b o l: or ( )

EXAMP LE 4:

3 is th e opposite of 3 since th ey ar e both th ree

un its a way from zero, but ar e in opposite directions.

d|||||d

3 0 3 % Opposite Num bers &

L Opposite nu mbers h ave th e sam e absolute value, so

EXAMP LE 5:

Y Wha t is t he opposite of 2? A n s w e r : 2


Y Wha t is t he opposite of? A n s w e r :


A nega tive sign in front of a n um ber, var iable, or gr oup ing symbol is used

to symbolize the opposite. So

x is t he opposite ofx

(8) is the opposit e of 8, which is 8


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( th e nega tion key is used to find th e opposite ofa n um ber of an expression; so it is used to represent both a negat ive

nu mber a nd t he n egation (the opposite) of a n um ber.

Wh a t t o d o: On t h e Ca lcu la t or S creen :

Fin d t he opposite of 2

M

Or M

=Be sur e you use th e negation key an d NOT th e subtr action k ey

when you wan t t o find th e opposite of some expression.

F Opposites a re a lso ca lled Add i t i ve Inverses(more on t his in Lesson #4).

GGGGGGGGGGGGGGGGGG

ARITHMETICOF INTEGERS

A+D+D+I+T+I+O+NFAddi t ion is th e opera tion of combin ing nu mber s to provide anequivalent single value. The n um bers being added ar e called the

summ an ds and th e result is called the sum.

S u m m a n d + S u m m a n d = S u m

Mat h S y m b o l: +

Addition of nu mbers can be visua lized on t he n um ber line. Sta rt at th e

origin , th en move to th e right, if the n um ber is positive, an d left if

th e num ber is negat ive.


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EXAMP LE 6: Add 2 + 3 on a nu mber line.

Step 1: Go 2 un its t o th e right , since 2 is positive:

2 units

||||||||

0

Step 2: Go 3 un its t o th e right , since 3 is positive:

3 units

||||||||

0 2

Since we stopped at 5, tha t is our su m, so 2 + 3 = 5.

Lets t hr ow in a n egative num ber n ow.

EXAMP LE 7: Add 3 + 4 on a nu mber line.

Step 1: Go 3 un its t o th e left, since 3 is n egat ive:

3 units

||||||||

0

Step 2: Fr om 3, go 4 un its t o th e right , since 4 is positive:

4 units

||||||||

3 0

Ther efore, 3 + 4 = 1


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The pr evious example ma y also be done on a single num ber line:

||||||||

3 0 1

3 + 4 = 1

It is cumber some t o always u se th e nu mber line for a ddition, so we

genera lly just add n um bers m ent ally, use our fingers an d toes, or s imply

use a calculat or. We use th e following ru les to help us with addit ion:

RULES OF ADDITION

R u l e 1 : positive + positive = positive

Add t he n um bers, th e sign of the su m is a lso positive.

R u l e 2 : negat ive + negat ive = negat ive

Add th e nu mber s, the sign of th e sum is also negat ive.

Rules 1and2 mea n: if you ha ve like signs, add up th e nu mber s,

an d place th e sign you see for th e final su m.

R u l e 3 : positive + negative or negative + positive

This r ule is a little more complicat ed

What you do is tem pora rily ignore t he signs you see, subtr act t he sma ller

nu mber from t he larger, th en include t he sign of th e nu mber whose

absolut e value was bigger (whichever nu mber is bigger with out looking a t

th e plus/minu s signs, is the s ign you will use for t he fina l resu lt.) Dont

worry, we will do a few examples of th is to mak e it eas ier to lear n.

R u l e 4 : any nu mber + its opposit e = 0

This is called th e Add i t i ve Inverse proper ty of addition. The

opposite is th e additive inverse of a nu mber .

R u l e 5 : 0 + a n um ber or a nu mber + 0 = th e nu mber

This is called th e Add i t i ve Ident i ty proper ty of addition. Zero

is the a dditive ident ity.


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EXAMP LE 8: Her e a re some m ini-examples. Try doing th em by

ha nd first , then use your calculat or.

One of th e keys to su ccessfully using your ca lcula tor is t oPRACTICE, PRACTICE , an d PRACTICE . Yes, th ese are simple problems,but m y recomm enda tion is to always tr y to do problems by han d firs t (if

possible), then use a calculat or . Why? Well, if you only use your

ca lcula tor for th e tough problems, you m ay not ha ve th e experience to do

th e problem. Stu dent s ma y blank when t rying to find th e right

keystrokes t o do a problem. The m ore you pra ctice, the bett er you get.

(Wh a t t o d o: On t h e Ca lcu la t or S creen :

Exam ple ofR u l e 1 :

Find 8 + 5

M


Find t he su m of 8 and 10M

Exam ple ofR u l e 3:

Add 12 an d 6

M

Another example ofR u l e 3 :

What is five plus n egative four ?M


Find t he su m of 9 an d its opposite.M


What is 0 + 7?M


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L When using a calculat or , you dont n eed to enclose the

negat ive num ber within pa ren th eses, but you m ay do so,

for clar ity, when wr iting it out on pa per . This way, you

dont confuse t he n egative sign with th e su btr action

sign.

Some of our previous problems could be writ ten as follows:

5 + 4 can be writ ten as : 5 + (4)

8 + 10 is th e sa me a s: (8) + (10)

And

9 + 9 is equ al t o 9 + (9)

GGGGGGGGGGGGGGGGGG

SUBTRACTIONFSub t r ac t i on is th e inverse operat ion of addition. Withaddition, you add numbers together, while with subtraction, you take

away numbers.

Mat h S y m b o l:

The nu mber t ha t is ta ken a way from th e origina l num ber is called the

S u b t r a h e n d , the original nu mber is called th e Minuend , an d the result

of a su btr action is called th e Dif ference .

Minuend Subtrahend = Difference

RULE FOR SUBTRACTION

To subtr act one nu mber from another, substitut e the subtr ah end

by its opposite, then ADD th e numbers togeth er.

By doing this, all subt ra ctions become a dditions.


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EXAMP LE 9: Per form t he following operat ions :

Y What is 10 3?

S o l u t i o n : 10 3 = 10 + (3) = 7

Y Subt ra ct 5 from 8.

S o l u t i o n : 8 (5) = 8 + (5) = 8 + 5 = 13

( To subt ra ct u sing a calculat or, use th esubt ra ction key: to perform th e opera tion. Do not mix up this key with

th e nega tion key: . If you do, th is is the er ror messa ge you will see:

If you see th is, press to go to th e err or. The screen cha nges back, butth e cur sor will blink over th e err or. J ust corr ect it, an d press . This

should solve th e problem.

Lets do a s imple subtr action:


Find : 11 (6)M

N o t e : Look carefully at th e minu s a nd n egative signs on t he calculat or

display. They ar e slightly differen t. The n egative sign is sma ller a nd

slightly higher on t he calculat or display th an th e minus s ign. In t hese

Notes, I follow stan dar d ma th nota tion a nd u sua lly make th em th e same

(but I m ight cha nge th em a litt le, like using b o ld t y p e for emphasis).


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HOT TIP!

When writing negat ive an d minus signs by ha nd, mak e

th em longer th an you h ave in t he pa st. This way, you a re

clear th at a n um ber is negat ive or a subtr action is being

perform ed. I ha ve foun d over t he years th at stu dents t ha t

write very short signs can confuse t hem with decima l

points, t he = sign, or n ot even kn ow th ey ar e t her e.

E x a m p l e : Write inst ead of , which looks like:

GGGGGGGGGGGGGGGGGG

|DISTANCEBETWEEN|

|TWONUMBERS|FWha t if we would like to figure out how far one n um ber is away froman oth er? How can we do th is? For example, I want to find t he dista nce

between 3 an d 4. This is easy to find u sing a n um ber line:

||||||||

3 0 4

J ust coun t t he nu mber of un its between 3 and 4:

||||||||

3 0 4

7 Units

You can coun t st ar ting a t 3 or from 4. It doesnt m at ter .

At t he beginn ing of th is lesson, we lear ned t ha t a bsolute value is u sed tofind t he dist an ce a nu mber is awa y from zero. Now we can combine t his

with our kn owledge of subtr actions, t o get a special form ula th at will help

us find t his dista nce.

The dista nce between 3 an d 4 is =


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In general:

The distance between n um bers a and b on a

nu mber line is:

F

=Do not mix up th is dista nce form ula for t he one t ha t is used to

measu re t he distan ce between t wo points in a plan e:

This form ula is discussed in an oth er lesson.

GGGGGGGGGGGGGGGGGG

MUL TI PLI CA TIONFMul t ip l icat ion is a sh ort -ha nd n ota tion for r epeat ed addition orsubtraction.

Mat h S y m b o l:

For examp le, 2 +2 +2 +2 +2 = 10 can be re-writt en a s a mu ltiplicat ion:5 2 = 10

The symbol is fine to use wh en you ar e just mu ltiplying n um bers

togeth er, but when algebra is involved, it can be confused with th e

varia ble x.

There ar e severa l ways to represen t m ultiplicat ion. Let a and b

represent arbitrar y numbers. a t imes b can be repr esent ed by an y ofthe following:

N ot a t ion : Com m en ts:

ab Not used very much in a lgebra [See above].

abA raised dot. Not usua lly used with n um bers , since it

ma y be confused with a decima l point . OK to use with

variables.


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(a)(b)

[a][b]

{a}{b}

When th ere is NO symbol between nu mber s an d variables,

or jus t va r iables, th en t his is a wa y of showing implied

mu ltiplicat ion. H ere, expressions ar e placed

within a set of grouping symbols, but th ere is nothin g

between groups of symbols. This is a preferred notat ion.

ab

Also implied mu ltiplicat ion. The var iables a re just next to

each other . This is th e most typical wa y of showing th isopera tion with var iables. Not u sed with nu mber s.

(a)b Im plied mult iplicat ion. Not used very often .

a(b)

Im plied mu ltiplicat ion. Th is is frown u pon for

mu ltiplica tion of var iables, since it can be confused with

anoth er m ath nota tion ca lled function nota tion. It s OK

to use with n um bers, or n um bers (in place ofa) with

var iables (in p lace ofb).

The n um bers/var iables being multiplied togeth er a re called Factors , an dth e resu lt of a mu ltiplicat ion is called th e P r oduc t .

(Factor) (Factor) = Product

RULES OF

MULTIPLICATION

R u l e 1 : positive posit ive = posit ivenegative negative = positive

The pr odu ct of nu mber s with like signs is positive.

R u l e 2 : positive nega t ive = nega t ive or

negative positive = negative

The pr oduct of nu mbers with differen t signs is negat ive.

R u l e 3 : 0 an y nu mber = 0 or

an y nu mber 0 = 0

This is called t he Zero-Factor proper ty of mult iplica tion.


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These Rules can be abbreviat ed as:

(+)(+) = + ()() = +

(+)() = ()(+) =

(0)(#) = 0 (#)(0) = 0 #

=Be car eful an d NOT mix up th e mu ltiplicat ion a nd a ddition r ules.

For example:

(5) + (2) = 7, bu t (5)(2) = 10

0 + 5 = 5, but 0 5 = 0

A nu mber x being mu ltiplied by 1 can be r epresen ted as follows

(excluding oth er n ota tions):

(x)(1) = (1)(x) = 1(x) = (1)x = 1x = (x) = x

N o t e : Notice th e last two on right ha nd side. A negat ive sign t o the left

of a pa ren th esis (or an y grouping symbol) may be seen as mu ltiplicat ion by

1 oras finding th e opposite of a n um ber.

( The key is used to repr esent mu ltiplicat ion. Todifferen tiat e it from the lett er , th e calculat or displays th e mu ltiplicat ion

using an asterisk: . To demonst ra te


What is 13 27?M

The Texas In stru ments gra phing calculat ors u nder sta nd implied

mu ltiplica tion, so a ll of th e following ar e th e same a s 13 27:

13 * 27

13(27)

(13)27

(13)(27)

(13)*(27)


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=A raised dot is OK for mu ltiplicat ion on paper , but NOT on a

ca lcula tor. The key is to ent er a decima l point ONLY.

The sa me can be sa id for oth er grouping symbols: { } and [ ]. These ha ve

differen t mea nings on a calculat or.

So if you need to ent er the following in your calculat or :

3[ (4 + 8) 2 ]

Use an additiona l set of par ent heses in place of th e squar e brackets:

3( (4 + 8) 2 )

GGGGGGGGGGGGGGGGGG

EXPONENTSFJ ust as m ultiplicat ion is an abbreviat ed form of repea ted a ddition,

Exponen ts ar e used to abbreviat e repeat ed multiplicat ion.

Mat h S y m b o l:

Using the nu mber 3 as a n example, th is is what is meant by repeated

multiplication:

3 3 3 3 3 = = 243

5 factors of 3

is rea d a s: th ree to th e fifth power or s imply: th ree to th e fifth .

Using a litt le algebra now:

xxx x =

n factors ofx

is read a s: x to th e n th power or simply: x to the n .


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F The expr ession: 3 3 3 3 3 is sa id to be inExpa n ded Form while is in Expon en t ia l Form ,Exponen t ia l

Notat ion , or Expon en t N ota t ion .

Some ter min ology:

xn This is th e Poweror Exponent

This is th e Base

is rea d: x ra ised to th e firstpower or x to th e first .

N o t e : If a nu mber or var iable is ra ised to th e first power, we can write

it with out th e 1, th at is: .

is rea d: x squar ed, x ra ised to the secondpower or x to th e second .

=Dont sa y: x two or x t o th e two.

is rea d: x cubed, x ra ised to the thirdpower or x to the th ird.

=Dont sa y: x th ree or x to the th ree. Similar war nings for t he r estbelow.

is rea d: x ra ised to the fouth power or x to the four th .

is rea d: x ra ised to the fifth power or x to th e fifth .

Oh is read : x ra ised to the y th power or just x to th ey.

( Use th e power key: to input an exponen t. Ifyou just wan t to squa re a value, use th e key. There is an option to

input a cube, but I find stu dent s dont ever use it , so dont worr y about it.

N o t e : On some calculat ors, t he power or exponen t keys m ay look

like:xy,yx , or a b . Also, looking a t your calculat or , th ere a re oth er

keys/fun ctions th at ha ve powers in th em, like: , , (am ong oth ers),

but we will discuss th ese in other Lessons.


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An o t h e r N o t e : When I a m describing keystrokes to studen ts, I say

power when I refer to this key: . For exam ple, I would sa y is:

th ree power four . It s a persona l choice. You can st ill just sa y: th ree to

th e four th power.

Lets do some calculator drills:


What is ?M

Or :

M

Of cour se, you could also just type:

M

What is negative th ree ra ised to

th e four th power?

=Since we ar e ra ising a negativenu mber t o a power, it MUSTbe

enclosed within par enth eses,

oth erwise, you ma y get th e wrong

an swer (depending on t he power).

So we want : ,NOT:

M

This is th e incorr ect way:M

is really th e opposite of .We get th e wrong an swer.

GGGGGGGGGGGGGGGGGG


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DIVISIONFDivision is th e inverse or r everse opera tion of multiplicat ion. It tells ushow man y times one n umber is cont ained with in an oth er n um ber.

Mat h S y m b o l:

The n um ber being divided is called th e Dividend. The num ber doing the

dividing is called th e Divisor. The r esu lt of a d ivision is called th e

Quot ien t .

Dividend Divisor = Quotient

To divide a nu mber x by a nu mber y, we can use a ny of the following

notations:

N ot a t ion : Com m en ts:

x yNot used very much in algebra . OK to use. Appear s in

opera tions dea ling with fractions. Grea t for n um bers.

x /y

or

OK to use, but ma y be a litt le vague wh en dea ling with

complicat ed expressions . Used typically to writ e fractions

in a m ore compa ct form , such a s in an exponen t or ma tr ix.

This is th e bestway to express division in a lgebra,

especially when t he item s get more complicat ed. The top

par t is called th e Numer a t o r an d the bott om par t is

called th e Denomina tor . Try to use th is nota tion wh en

possible.

or

y x

These a re used when perform ing long division and ar e

the same as:

If a n um ber does not exactly divide into another nu mber , th en t her e is a

R ema i nder .

F Dividend = (Divisor) (Quotient) + Remainder


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E XAMP LE 10: Usin g some nu mber s: 20 3 or is:

RULES OF DIVISION

R u l e 1 : positive positive = positive

nega t ive nega t ive = posit ive

The qu otient of num bers with like signs is positive.

R u l e 2 : posit ive nega t ive = nega t ive ornega t ive posit ive = nega t ive

The quotient of nu mber s with differen t s igns is n egative.

R u l e 3 : 0 an y non-zero nu mber = 0

R u l e 4 : an y non-zero num ber 0 = undefined.

FWe wont cons ider th e case. This is discus sed in

calculus.

These Rules can be abbreviat ed as:

=Dividing by zero is NOTa llowed! Zero cann ot be in t he

denomina tor of a fraction. Dont go there! Do not pa ss Go. Do not

collect $200. Oops! I got a litt le ca r ried a way t her e. Sor ry. L


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FRECIPROCALSF

Y The Reciproca l of a n um ber x is .

Y The reciproca l of a fraction is

N O

Y A nu mber mu ltiplied by its r eciprocal is always equa l to 1. Her e ar e a

couple of exam ples:

and

This is called t he Mul t ip l icat ive Inverse property.

Y The r eciproca l itselfis also called th e mu ltiplicat ive inverse.

Y Zero is the only nu mber t ha t does not h ave a reciprocal.

Y With r eciprocals, all divisions can now be re-writ ten as m ult iplica tions:

a b is th e sam e as: , or

( The key is used to per form divisions . It is

displayed on t he calculat or screen as a forwa rd slant : /. The r ecipr ocal ofa num ber can be foun d by using th e reciprocal key: . In pra ctice, th is

key is very ra rely used. Most stu dents just use: th en th e nu mber.

L One of th e places ma ny student s ma ke mistakes u sing

th e ca lcula tor is in t he a rea of divisions . Be extra careful

an y time you need t o perform th is opera tion or doing

fra ctions on th e calculat or.

HOT TIP!

When in doubt , just enclose th e entire num erator within an

extra set of par ent heses a nd do likewise with t he entire

denominator.

E XAMP LE 11:

To evalua te us ing th e ca lcula tor, visua lize th is as : .


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The express ion: sh ould be seen as:

( The last two expressions would be ent ered a s:


What is ?

M

What is ?

M

FNEGATIVES WITH DIVISIONF

Y The following ar e all equa l:

= = =

Y This equation is always t ru e:

FOTHER EQUATIONS WITH DIVISIONF

E qu a t ion : Com m en ts:

Any n um ber, over itself, is one.

A num ber, divided by 1, is just itself.


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ORDER+ OF OPERATIONSNow tha t we ha ve all these ar ithmetic operat ions, what ha ppens when we

combin e them? How do we evalu at e somet hin g like: 2 + 3 4 ? Do we add

the 2 and t he 3, then mu ltiply by 4, or do we mu ltiply th e 3 an d t he 4 firs t,then add 2 t o th is? Before we can an swer t his, we need to ta lk about th e

order of operat ions . We will st ar t t his discussion with group ing symbols.

FGr oup i ng Symbo l s ar e used t o show certa in ma th emat ical operat ionsshould be done before oth ers in an expression. Her e is a list of th e most

comm on symbols used in gr oup ing:

S ym bols: Com m en ts:

These a re th e most comm only used grouping symbols.E x a m p l e : 1 (2 + 3)

( Only use par ent heses for grouping.

or Squar e

Brackets

Used to enclose items th at already ha ve parent heses.

E x a m p l e : 4[1 (2 + 3)]

( Used for ma tr ices.

or Cu rlyBrackets

Used to group items th at ha ve squar e brackets.

E x a m p l e : 6 + {5 4[1 (2 + 3)]}

( Used t o enclose item s in lists.

or Vinculum

Used to group items th at ha ve braces.

E x a m p l e :

First, evalua te everything in t he nu mera tor.Next, evaluate

everything in t he denominator. Finally, divide th e

nu mera tor by the denominat or.

A detailed exam ple of th is will be shown lat er .

AbsoluteValue

Bars

Do everyth ing within t he bar s first , then ta ke its absolutevalue.

A detailed exam ple of th is will be shown lat er .

Do everyth ing inside th e ra dical fir st , then ta ke th e root.. Exa mples of th is ar e in th e Algebra Lesson: Rad icals.

F th e ra dical sign is rea lly just

Th e vin cu lu m is u sed t o ext en d t he sign :


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Ther e ar e oth er group ing symbols/operat ions , but we wont en coun ter

th em u nt il oth er lessons.

FALTERNATE GROUPING NOTATIONF

Say we ar e given somet hing t ha t looks like th is:

Which is ra th er complicat ed looking. Inst ead of using braces an d bar s, th e

par enth eses and brackets can altern at e, so th e previous problem would

look like:

OK, its st ill a mess, but it is a litt le eas ier on th e eyes.

L We dont have to use par ent heses. It s per fectly

acceptable t o write an y of th e following for grouping:

= = =

But its usu ally best to stick with pa ren th eses.

FORDER OF OPERATION RULESF

Y Eva lua te expressions with in groupin g symbols. So, if you h ave, say,

1 + (8 3), you would d o 8 3 first . If you ha ve items grouped with in

anotherset of grouping symbols, evalu at e first th e inn er set of grouping

symbols.

An exam ple of th is would be: 5 + 4[3 (1 + 2)]. You would eva lua te (1 + 2)

first , since it is th e inner most set of grouping symbols, then ta ke car e of

everyth ing inside of the br ackets.

Y Per form all exponen tial expressions before oth er ar ithm etic opera tions.

Given: , you need to squa re th e 4 first, NOT add t he 3 with th e 4.


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Y Next, per form a ll mu ltiplica tions and /or divisions

, but evaluat e th e expression in order from left-to-right .

To evaluat e 3 2 + 4 2, for examp le, multiply 3 2 firs t, since it is t o th e

left of 4 2.

This gives us: 6 + 4 2. Now, divide th e 4 by 2.

Our expression will now look like t his now: 6 + 2 .

Fina lly, do th e addition a nd t he r esult is 8 .

Y The last ru le is to do all additions an d/or subt ra ctions

but evaluat e th e expression in order from left-to-

right.

Lets eva lua te 8 3 + 5 1. Going from left-to-r ight, do 8 3 fir st . This

gives us 5:

5 + 5 1. Now add t he fives t o get 10:

10 1. Fin ally, do th e subt raction to get 9 as th e end result .

ORDER OF OPERATIONS

1Firs t, evalua te t he expression with in th e inn erm ost set of grouping

symbols.

2Next, evaluat e expressions t ha t ha ve exponen ts.

3Then , perform a ll multiplicat ions an d divisions , going from left-to-r igh t

in th e expression.

4Fin ally, do a ll additions a nd subt ractions, aga in, going from left-to-r ight

in th e expression.


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HOT TIP!

An a cronym t o help you rem ember th e above is:

P E M D A S

P: Pa rent heses

E: Exponen tsM: Multiplication

D: DivisionA: Addit ionS: Subtr action

PEMDAS can be m emorized eas ily if you remem ber t he followingmn emonic device:

Please Excuse My Dear Aunt Sally

FORDER OF PRECEDENCEF

Y The Or der of Precedence, is th e order of import an ce in per form ing

operat ions . The higher th e precedence, th e more import an t it is;

th erefore, n eeds t o be done before item s of a lower precedence. The order

is the sa me a s th e order of opera tions:

1 Groupin g symbols

2 Exponents

3 Multiplications / Divisions

4 Additions / Subtractions

There ar e additiona l opera tions with oth er orders of precedence, but th ese

will do for n ow.

N o t e : Most gra phing calculat ors an d compu ter progra ms follow th eOrder of Precedence ment ioned h ere. Sma ller, non-gra phing calculat ors

ma y evalua te expressions in a differen t way, so be car eful if usin g th em.

Consu lt your calculat ors Opera ting Man ua l for more inform at ion

Now th a t we ha ve all of th ese ru les, lets do severa l deta iled examples.

Fir st , by hand , showing all steps , then we will verify th e an swer by

evaluat ing th e expressions directly using th e calculat or.


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EXAMP LE 12: Simplify: 4 + 32 2 10

SOLUTION:Y There ar e no grouping symbols, so sta rt with t heexponent:

4 + 32 2 10

= 4 + 9 2 10

Y Next, do the mu ltiplicat ion:

4 + 9 x 2 10

= 4 + 18 10

Y Now, since we only ha ve additions a nd su btr actions, evalua te t he

express ion going from left-to-r ight:

4 + 18 10

= 22 10

Y Fina lly, do th e subt ra ction:

22 10

= 12


Simplify: 4 + 32 2 10

M

E XAMP LE 13: Per form th e indicat ed opera tions:

5 + 2[3 4(7 6) + 22]

SOLUTION:Y Evaluat e wha t is in th e inn ermost parent heses first:

5 + 2[3 4(7 6) + 22]

= 5 + 2[3 4(1) + 22]


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Y Do th e Exponent next. The par enth eses around th e 1 ar e used for

implied mu ltiplicat ion, so th e exponen t is eva lua ted before th e

multiplication:

5 + 2[3 4(1) + 22]

= 5 + 2[3 4(1) + 4]

Y Per form t he mu ltiplicat ion within th e bra ckets :

5 + 2[3 4(1) + 4]

= 5 + 2[3 4 + 4]

Y Evalu at e th e subtr action a nd a ddition inside th e brackets (going from

left-to-right):

5 + 2[3 4 + 4]

= 5 + 2[1 + 4]

= 5 + 2[3]

Y The br ackets ar e now used a s implied mu ltiplicat ion, so perform th at

next:

= 5 + 2 [3 ]

= 5 + 6

Y Finally, add th e num bers together :

=5 + 6

= 1

This seems like we ar e doing a hu ge num ber of steps, but I am doing this

st ep-by-step, an d I re-display cert a in st eps for clar ity. Heres how th is

problem would look like if it were done by ha nd for a tes t ques tion:

5 + 2[3 4(7 6) + 22

]= 5 + 2[3 4(1) + 2

2]

= 5 + 2[3 4(1) + 4]

= 5 + 2[3 4 + 4]

= 5 + 2[1 + 4]

= 5 + 2[3]

= 5 + 6


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= 1If your inst ru ctor a llows you to combine st eps, th e above problem can be

done in even fewer st eps, but I only recomm end you do th is after a greatdea l of pr actice .

5 + 2[3 4(7 6) + 22]

= 5 + 2[3 4(1) + 4]

= 5 + 2[3 4 + 4]= 5 + 2[3]

= 5 + 6

= 1


Per form th e indicat ed opera tions:

5 + 2[3 4(7 6) + 22]M

Remember t o use par enth eses only, not br ackets, when

ent ering th e keystrokes into your calculat or.

E XAMP LE 14: Simplify the expression usin g the order of operat ion

rules:

23

| 5 2 8|

SOLUTION:Y The a bsolute valu e bar s serve as grouping symbols, so

evaluat e what is inside them first :

23

|5 2 8|

Y Multiplicat ion ha s h igher precedence:

23

| 5 2 x 8 |

= 23

| 5 16 |


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Y Now do th e subt ra ction with in th e absolute value:

23

| 5 16 |

= 23

| 11 |

Y Next, perform t he absolute value: | 11| = 11

23

|11 |

= 23

11

Y Then, evaluat e th e exponent :

23

11

= 8 11

Y Fina lly, do th e subt ra ction:

8 11

=3


Simplify: 23 | 5 2 8|M

y


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=

Y Evalu at e all subt ra ctions a nd a dditions in th e num era tor, going from

left-to-right:

= =

Y In t he denominat or, evaluat e the subtr action within th e parent heses

first:

=

Y Evaluat e the exponent next:

=

Y Now, do th e subt ra ction:

=

Y Fina lly, divide the n um era tor by th e denominat or (leaving it in

fractional form ):

=

You could h ave a lso just writt en 0.5 as th e final answer.

An a ltern at e way to evalua te t he original pr oblem is t o simplify the

numera tor a nd th e denominator a t t he same time. Star ting with:


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Y Exponent iate th e items in th e num erat or a nd do th e subtra ction in th e

denominator:

Y Do th e subtra ction in th e num erat or, and squa re th e 2 in t hedenominator:

Y Per form th e addition on top, an d th e subt ra ction below:

Y Fin a lly, divide th e top by the bott om (leaving it in fra ctiona l form):

=

( Be sur e to enclose th e entire numerator anddenominat or with in an extr a set of par enth eses.

In oth er words , cha nge: into:


Simplify:

M

Convert to a fra ction:


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LESSON 3 QUIZWhen doing th ese problems, tr y to also do th em u sing your calculat or , (if

possible) to get more pr actice using it.

1Write t he following expressions without absolut e value bar s,simplifying a lso, if possible:

Y

Y

Y

Y

Y

2 Rewrite t he following without absolut e values, leaving t he

an swer in E XACT form:

Y

Y

Y

3Fin d t he opposites of the following:

Y 5. The opposit e is: _____

Y 8. Th e opposit e is: _____


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Y . The opposit e is: _____

Y . The opposit e is: _____

Y x. The opposit e is : _____

4Per form t he following opera tions:

Y 8 + (2) =

Y 8 + 2 =

Y 8 + (2) =

Y Subt ra ct 2 from 8:

Y 8 2 =

Y 8 2 =

Y 8(2) =

Y (8)( 2) =

Y 8 2 =

Y =

Y =

Y Find th e squa re of negat ive five =

Y =

Y =


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5What is the distan ce between 5 and 7?

6What is the dista nce between x and y?

7Which of th e following is cons idered as th eBESTway to input two times t hr ee in a calculat or?

(2)(3) 2(3) (2)3 2 3

(2) (3) 2 (3) (2) 3

8What is th e expanded form of ?

9Write th e Exponen tial Notat ion for 7 7 7:

blWhat is th e base of ?

bmWhat is th e exponen t of ?

bnWhich operation is performed firstfor: ?

boWhich of the following a re not typical grouping symbols used in ma thexpressions?

bpWhat is PEMDAS?

bqTru e or F alse. Subtr action h as a higher order of precedence th anDivision. ________


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brSimplify. Show all steps. Write work on a separ at e sheet of paper.

Y 3 2 + 8 =

Y (2 + 3) 5 =

Y (6 2)(8 + 1) =

Y

Y

Y

Y

Y

Y

=

Y

Y

ANSWERSONNEXTPAGE


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ANSWERS

1Write t he following expressions without absolut e value bar s,simplifying a lso, if possible:

Y 8

Y 7.2

Y 6

Y 10 16 = 6

Y 5

2 Rewrite t he following with out absolut e values, but leaving t he

an swer in E XACT form:

Y f 2

E x p l a n a t i o n : since 2 f is negat ive, you wa nt to get a positive value,since absolut e values a re a lways positive, so th e way a round th is is to just

switch th e places of th e nu mber s.

Y f A7

E x p l a n a t i o n : since is alr ead y positive (t ry it on a calculat or ), you

just need to remove th e absolute value bar s.

Y 3 A7

E x p l a n a t i o n : Similar to first Y in Problem 2. These t ypes of quest ionsare very typical on t ests.


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3Fin d t he opposites of the following:

Y 5. The opposite is: 5

Y 8. The opposite is: 8

Y . The opposite is: A3

Y . The opposite is: 6

Y x. The opposite is:x

4Per form t he following opera tions:

Y 8 + (2) = 6

Y 8 + 2 = 6

Y 8 + (2) = 10

Y Subt ra ct 2 from 8:

This is wr itt en as : 8 (2) = 8 + 2 = 6

Y 8 2 = 8 + (2) = 10

Y 8 2 = 16

Y 8(2) = 16

Y (8)( 2) = 16

Y 8 2 = 4

Y = 4

Y = 4 4 4 = 64 (or just use calculat or to get th is value).


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Y Find th e squa re of negat ive five = 25

N o t e : The an swer isNOT25, since we wan t: .

A typical err or is to writ e wha t wa s a sked for as:

Y = 1

Y = 301

5What is the distan ce between 5 and 7?

Using th e distan ce form ula, we get: 12

F We could have also writt en: 126What is the dista nce between x and y?

Using th e distan ce form ula a gain, we get: y x

F We could have also writt en: x y7Which of th e following is cons idered as t he BESTway to inpu t two

times t hr ee in a calculat or?

(2)(3) 2(3) (2)3 D 2 3(2) (3) 2 (3) (2) 3

F All of th e above ar e acceptable. I wan t you t o useth e one t ha t would involve th e fewestkeyst rokes; however, if you needto add more keystr okes so th at th e expression seem s clearer to you,

th en go ah ead an d add more.

8What is th e expanded form of ?A n s w e r :6 6 6 6


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9Write th e Exponen tial Notat ion for 7 7 7:

A n s w e r :73

blWhat is th e base of ?A n s w e r :The base is 5. The base isNOT5. By order of pr ecedence,

the squar e is justwith th e 5, an d does not include th e negative sign. If

we want ed th e base to be 5, th en t he expression sh ould ha ve been

written: .

bmWhat is th e exponen t of ?A n s w e r :The exponen t is 5.

bnWhich operation is performed firstfor: ?

A n s w e r :The Exponen t or Squar e the Four. Exponen ts ar e done beforeth e oth er operat ions.

boWhich of th e following ar e not t ypical group ing symbols used in m athexpressions?

D

bpWhat is PEMDAS?A n s w e r : It is a n a cronym t o help you r emember th e order of opera tions

of rea l nu mbers . The letter s sta nd for:

P: Pa rent hesesE: Exponen tsM: Multiplication

D: DivisionA: Addit ion

S: Subtraction

bqTru e or F alse. Subtr action h as a higher order of precedence th anDivision.

A n s w e r :FALSE. Subt ra ction (along with addition) ha s t he lowest

order of precedence present ed in t his lesson.


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brSimplify. Steps ar e sh own below each pr oblem.

Y 3 2 + 8 = 2

A n s w e r : 3 x 2 + 8

=6 + 8

= 2

Y (2 + 3) 5 = 25

A n s w e r : (2 + 3) 5

= (5) 5

= 25

Y (6 2)(8 + 1) = 36

A n s w e r : (6 2)(8 + 1)= (4)(9)

= 36

Y 4

A n s w e r : 2(1 3 + 2 2)

= 2(1 3 + 22)

= 2(1 3 + 4 )

= 2(1 3 + 4)

= 2(2 + 4)

= 2(2 + 4)

= 2(2 )

= 4

Y 12

A n s w e r : 10 [2 + (4 23)]

= 10 [2 + (4 23)]

= 10 [2 + (4 8 )]

= 10 [2 + (4 8)]

= 10 [2 + (4)]

= 10 [2 + (4)]


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= 10 [2]

= 10 + 2

= 12

Y 64

A n s w e r :

=

= 64

Y 6

A n s w e r : 12 + 2[4 + (2 3)2

]= 12 + 2[4 + (1)2]

= 12 + 2[4 + (1)2]

= 12 + 2[4 + 1 ]

= 12 + 2[4 + 1]

= 12 + 2[3]

= 12 + 2[3]

= 12 + (6)

= 6

Y 97

A n s w e r : | 3| + (8 + 2)2

= (3) + (10 )2

= (3) + (10)2

= (3) + 100

=3 + 100

= 97


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Y = 20

A n s w e r :

=

=

=

=

=

=

=

=

=

=

=20

Y 4

A n s w e r :

=

=

=


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=

= 4

Y 5

A n s w e r :

=

=

=

=

=

=

= 11 (6)

= 11 + 6

=5

Video Math Tutor: Basic Math: Lesson 3 - Operations on Numbers

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