number system

NUMBER SYSTEM

BY :IKA KOMALA SARI

NILA PATMALAR. ULFAHTUL HASANAH

VIERA VIRLIANI

B.INGGRIS MATEMATIKA

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NUMBER SYSTEMHuman beings have trying to have a count

of their belonging, goods, ornaments, jewels, animals, trees, goats, etc. by using techniques.

1. putting scratches on the ground2. by storing stones-one for each commodity kept taken out

This was the way of having a count of their belongings without knowledge of counting

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NUMBER SYSTEM

The questions of the type:

HOW MUCH? HOW MANY?

Need accounting knowledge

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The functions of learning number system

Are 11 functions, that to:

Illustrate the extension of system of number from natural number to real (rational and irrational) numbers

Identify different types of numbersExpress an integers as a rational numberExpress a rational number as a

terminating or non-terminating repeating decimal and vice-versa

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Find rational numbers between any two rationals

Represent a rational number on the number line

Cites example of irrational numbersRepresent 2, 3, 4 on the number lineFind irrational numbers between any

two given numbers

2 3 4

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Round off rational and irrational numbers to given number of decimal places

Perform the four fundamental operation of addition, subtraction, multiplication, and division on real numbers

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1.1 EXPECTED BACKGROUND KNOWLEDGE

It is about the accounting numbers in use on the day to day life

Accounting numbers

Day life

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1.2 Recall of Natural Numbers, Whole Numbers, and Integers

Natural Numbers1, 2, 3, …

There is no greatest natural number, for if 1 added to any natural numbers. we get the next higher natural number, call its successor.

Example :4+2=6

12:2=6

22-6=16

12×3=36

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Addition and multiplication of natural numbers again yield a natural numbers

But the subtraction and division of two natural number may or may not yield a natural numbersExample:

Number line of natural numbers

1 2 3 4 5 6 7 8 9 …

2-6 = -4 6 : 4 = 3/2

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Whole Numbers The natural number were extended by

zero (0)0, 1, 2, 3, …

There is no greatest whole numbersThe number 0 has the following properties:a+0 = a = 0+a a-0 = a but 0-a is not defined in whole

numbersa×0 = 0 = 0×a Division by 0 is not defined

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The whole number in four fundamental operation is same

The line number of whole number

0 1 2 3 4 5 6 7 …

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Integers Another extension of numbers

which allow such subtractions. It is begin from negative numbers until the whole number.

The number line of integers

… -3 -2 -1 0 1 2 3 4 …

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Representing Integers on number line A B C D … -4 -3 -2 -1 0 1 2 3 4 5 …

Then A = -3 C = 2 B = -1 D = 3

A < B, D > C, B < C, C > A The rule:1. A > B, if A is to the right of B2. A < B, if A is to the left of B

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1.3 Rational NumberRational Numbers

Consider the situation, when an integer a is divided by another non-zero integer b. The following case arise:

1. When A multiple of BA = MB, where M is natural number or integer. Then, A/B =M

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1.3 Rational Number

2. Rational number is when A is not A multiple B. A/B is not an integer. Thus, a number which can be put in the form p/q, where p and q are integers and q ≠ 0.

Example:

All Rational Numbers

-2 5 6 11 3 -8 2 7

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1.3 Rational NumberPositive and Negative Rational Number

1. p/q is said positive numbers if p and q are both positive or both negative integers2. p/q is said negative if p and q are of different sign. Example:

+ -34

-1-5

-74

6-5

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1.3 Rational NumberStandard Form of a Rational Number

We can see that-p/q = -(p/q)-p/-q = -(-p)/-(-q)= p/qp/-q= (-p)/q

-p p -p p q -q -q q

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1.3 Rational Number

Notes: A rational number is standard

form is also referred to as “a rational lowest form” . There are two terms interchangeably

Example:18/27 can be written 2/3 in

standard form (lowest form)

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1.3 Rational Number

Some important result:

1. Every natural number is a rational number but vice-versa is not always true

2. Every whole number and integer is a rational number but vice-versa is not always true

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1.7 FOUR FUNDAMENTAL OPERATIONS ON RATIONAL NUMBERS

Addition of Rational Numbers1. Consider the addition of rational

numbers , + =

for example : + = =

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2. Consider the two rational numbers p/q and r/s

p/q + r/s = ps/qs + rq/sq =

for example :

¾ + 2/3 = = =

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from the above two cases, we generalise the following rule:

(a)The addition of two rational numbers with common denominator is the rational number with common denominator and numerator as the sum of the numerators of the two rational numbers.

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b)The sum of two rational numbers with different denominator is a rational number with the denominator equal to the product of the denominators of two rational numbers and the numerator equal to sum of product of the numerator of first rational with the denominator of second and the product of numerator of second rational number and the denominator of the first rational number.

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Examples:Add the following rational numbers :

(i) 2/7 and 6/7 (ii) 4/17 and -3/17

Solution:(i) 2/7 + 6/7 = 8/7(ii) 4/17 + (-3)/17 = 1/17

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Add each of the following rational numbers, examples:(i) 3/4 and 1/7

Solution :(i) we have 3/4 + 1/7

= 3x7/4x7 + 1x4/7x4 = 21/28 + 4/28 = 25/28

3/4 + 1/7 = 25/28 or 3x7+4x1 / 4x7= 21+4/728 = 25/28

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Subtraction of Rational Numbers(a) p/q – r/q = p-r/q

Example :

7/4 – ¼ = …7/4 –1/4 = 7 – 1 4 = 6/4 = 2x3 = 3/2

2x2

3/5 – 2/15 = …3x12/5x12 – 2x5/12x5= 36/60 – 10/60 = 26/60 = 13x2/30x2 = 13/30

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Multiplication and Division of Rational Numbers

(i) Multiplication of two rational number (p/q) and (r/s) , q0, s0 is the rational number pr/ps where qs0

= product of numerators/product of denominators

(ii) Division of two rational numbers p/q and r/s , such that q0, s0, is the rational number ps/qr, where qr0

In the other words (p/q) (r/s) = p/r x (s/r)Or (First rational number) x (Reciprocal of the second rational

number)Let us consider some examples

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Examples :(i) 3/7 and 2/9 (ii) 5/6 and (-2/19)

Solution :(ii) 3/7 x 2/9 = 3x2/7x9 = 3x2/7x3x3 = 2/21

(3/7))x(2/9) = 2/21

(ii) 5/6 x (-2/19) = 5x(-2)/6x19 = - 2x5/2x3x19 = -5/57 5/6 x (-2/19) = -5/57

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(i) (3/4) (7/12)

Solution:

(i) (3/4) (7/12)

= (3/4) x (12/7) [Reciplocal of 7/12 is 12/7]

= 3x12/4x7 = 3x3x4/7x4 = 9/7

(3/4) (7/12) = 9/7

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1.8 DECIMAL REPRESENTATION OF A RATIONAL NUMBER

You are familiar with the division of an integer by another integer and expressing the result as a decimal number. The process of expressing rational number into decimal from is to carryout the process of long division using decimal notation. Example: Represent each one the following into a decimal number (i) (ii) :

5

12

25

27

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Solution: Using long division, we get (i)

Hence , = 2,4

(ii) (-1, 08) hence, = -1, 08

4,2

0,2

0,210

,12

5

x

5

12

x200

20025

27

25

25

27

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From the above example, it can be seen that the division process stops after a finite number of steps, when the remainder becomes zero and the resulting decimal number has a finite number of decimal places. Such decimals are known as terminating decimals.Note that in the above division, the denominators of the rational numbers had only 2 or 5 or both as the only prime factor

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Alternatively, we could have written as = = 2,4

Other examples: Here the remainder 1 repeats.

The decimal is not a terminating decimal

= 2,333… or 2,3

5

12

25

212

x

x

10

24

33,2

00,19

0,16

00,7

3

3

7

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` = 0,2

85714 Note: A bar over a digit or a group of digits implies that digit or group of digits starts

repeating itself indefinitely.

28571428,0

456

6014

2028

307

1049

5035

4056

6014

000.2

77

2

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Expressing decimal expansion of rational number in p/q form

Examples:Express in form p/q !

Express in in form p/q !

0,48

=100

48

25

12

Let x = 0,666 (A) 10 x = 6,666 (B)(B)-(A) gives 9x = 6 or

x = 2/3

0,666

The example above illustrates that:

A terminating decimal or a

non-terminating

recurring decimal

represents a rational number

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Note :The non-terminating recurring decimals like 0,374374374… are written as 0,374.The bar on the group of digits 374 indicate that group of digits repeats again and again.

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1.9 RATIONAL NUMBERS BETWEEN TWO RATIONAL NUMBERS

Is it possible to find a rational number between two given rational numbers. To explore this, consider the following example.Example : Find rational number between and

Let us try to find the number ( + ) ( ) = now, = = And = =

4

35

6

2

1

4

3

5

6

2

1

20

241540

39

4

3104

103

x

x

40

30

5

6

85

86

x

x

40

48abviously, < <40

30

40

39

40

48

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1.9 RATIONAL NUMBERS BETWEEN TWO RATIONAL NUMBERS

is a rational number between the rational numbers and

Note : = 0,75. = 0, 975 and = 1,2Than: 0,75 < 0, 975 < 1,2This can be done by either way :

(i) reducing each of the given rational number with a common base and then taking their average(ii) by finding the decimal expansions of the two given rational numbers and then taking their average

4

340

39

5

6

4

3

40

39

5

6

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1.4 Equivalent Forms of a Rational Number

A rational number can be written in an equivalent form by multiplying or dividing the numerator and denominator of the given rational number by the same numberExample :2/3 = 2x2 = 4/6 and 2/3 = 2x4 = 8/12

3x2 3x8It’s mean 4/6 and 8/12 are equivalent

form of the rational number 2/3

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1.5 Rational Numbers on the Number Line

We know how to represent intergers on the number line. Let us try to represent ½ on the number line. The rational number ½ is positive and will be represented to the right of zero. As 0<½<1, ½ lies between 0 and 1. divide the distance OA in two equal parts. This can be done by bisecting OA at P

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1.5 Rational Numbers on the Number Line

Let P represet ½. Similarly R, the mid-point of OA’, represents the rational number -½.

A R 0 P A … -2 -1 0 1 2 3 …

Similarly , can be represented on the number line as below: B’ A’ O A P B C

… -2 -1 0 1 2 3 … As 1 < 4/3 < 2 therefore, 4/3 between 1 and 2

3

4

-1/2 1/2

4/3

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1.6 COMPARISON OF RATION NUMBER

In order to compare to rational number, we follow any of the following methods:(i)If two rational numbers, to be compare

have the same denominator compare their numerators. The number having the greater numerator is the greater rational number. Thus for the two rational numbers and , with the same positive denominator.

as 9>5. so,

17

5

17

9

17

5

17

9,17

17

5

17

9

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(ii) If two rational number are having different denominator, make ther denominator equal by taking their equivalent form and then compare the numerator of the resulting rational numbers. The number having a greater numerator is greater rational number.

For example, to compare two rational numbers and , we first make their

denominator same in the following manner:7

311

6

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(iii) By plotting two given rational numbers on the number line we see that rational number to the righ of the other rational number is greater.

77

33

117

113

x

x77

42

711

79

x

x As 42>33, or 77

33

77

42

7

3

11

6

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For example, take and , we plot these number on the number line as below:

-2 -1 0 1 2 3

3

2

4

3


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0<⅔<1 and 0< ¾<1. it means ⅔ and ¾ both lie between 0 and 1. by the method of diving a line Into equal number of parts, A represent ⅔ and B represent ¾

As B is to the right of A, ¾>⅔ or ⅔<¾So, out of ⅔ and ¾, ¾ is greter number.

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Thank’s for your attention

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