9 math number system

8/14/2019 9 Math Number System

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Finish Line & Beyond

NUMBER SYSTEM

Natural Numbers

Whole Numbers

Integers

Rational Numbers

Irrational Numbers

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Natural Numbers: All numbers starting from 1 and going upto the highest possible

number is called natural numbers. Natural number is denoted by N.

Whole Numbers: If zero is included with natural numbers then the set is called acollection of whole number. Whole numbers are denoted by W.

Integers: If we include negative numbers alongwith whole numbers then the

collection is called integers. Integers are denoted by Z.

Rational Numbers: Any number which is written in the form ofq

p, where p and q

are integers and q 0 , is called a rational number. Rational number is denoted by r.

Question 1: Is zero a rational number?

Answer: Zero can not be written in the formq

p, where p and q are integers and q

0, because when divided by any number 0 will always give infinity as result. Whilerational numbers always give terminating decimal values.

Question 2: Find six rational numbers between 3 and 4.

Answer: Step1:2

43+=

2

7

Step2:

2

2

73+

=

14

13

Step3:

2

42

7 +=

4

15

Step4:

2

2

7

14

13 +=14

31

Step5:

2

2

7

4

15 +=

8

39

Step6:

2

1413

1431+

=7

11

You can notice that by calculating averages between two numbers we get a number

which is exactly between these two numbers. This way you can go on calculatinginfinite numbers of numbers.

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Question3: Find five rational numbers between5

3and

5

4

Answer: Step1:

2

5

4

5

3+

=107

Step2:

2

10

7

5

3 +=

20

13

Step3:

4

3

2

5

4

10

7

=+

Step4:

20

19

210

13

5

3

=+

Step5:

40

31

2

5

4

4

3

=+

Question4: State if following statements are true or false:(a) Every natural number is a whole number.

(b) Every integer is a whole number.(c) Every rational number is a whole number.

Answer: (a) As natural number is all numbers starting from 1 and the whole number

includes zero as well so this statement is true. On the other hand every wholenumber is not natural number as zero is not a natural number.

(b) Only positive integers are whole numbers.(c) Rational numbers are not whole numbers as they are not complete.

Irrational Numbers: If a number cannot be written in the formq

p, where p and q

are integers and q 0 . Example: 2 , 3 , . These numbers result in non-

terminating and non-recurring decimals so they are called irrational numbers.

The collection of all rational and irrational numbers is called real number and isdenoted by R. In other words every point on the number line represents a unique

real number. To understand the number line try to visualize a scale with so many

marks between any two numbers. The following pictures depicts how 2 can be

depicted on the number line.



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In the given picture suppose each side of the given square is measuring

1 unit. Then the diagonal will be 2 . Now if the diagonal is rotated so that it

becomes horizontal and falls on the number line then the point up to which diagonal

will reach will be the exact position of 2 on the number line.

Real Numbers and their decimal expressions:

1. Some rational numbers, when converted to decimal form give terminating

decimal.

2. Some rational numbers, when converted to decimal form give non-terminatingrecurring decimal.

Example: 875.08

7 = , in this case a point comes when we get 0 as remainder. Sothis is a case of terminating decimal.

3

10

=3.33333, in this case we always keep on getting 1 as remainder and quotient

keeps on repeating. This is the case of non-terminating recurring decimal.

142857.07

1 = , this is also a case of non-terminating recurring decimal.

In case of irrational number we get a non-terminating and non-recurring decimal.

Question5: Write the following in decimal form and comment on their kind ofdecimal expression.

(a) 36.010036 = , as we get zero as remainder at last so it is a terminating decimal.

(b) 09090909.011

1 = , we dont get zero as remainder and the quotient keep onrepeating, so this is non-terminating recurring decimal.

(c) 125.48

14 = , terminating decimal.


0 1 2


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Question5: You know that 142857.07

1 = , Without doing the long division fin the

values of7

6,7

5,7

4,7

3,7

2

Answer: Multiplying the numerator with the decimal expression in question you get

very interesting observation:

0.142857 2= 0.285714, So, 285714.07

2 =Similarly, 0.142857 3 = 0.428571

0.142857 4 = 0.5714280.142857 5 = 0.174285

0.142857 6 = 0.857142Question6: Express the following in the form

q

p, where p and q are integers and

0q

(a) 0.6 =3

2

9

6 =

(b) 0.47=99

47

(c) 0.001=900

1

Put 9 for every non-zero digit in the denominator and zero for zero in thedenominator.

Question7: What can the maximum number of digits be in the repeating block of

digits in the decimal expression of17

1?

A fraction in lowest terms with a prime denominator other than 2 or 5 (i.e. coprime

to 10) always produces a repeating decimal. The period of the repeating decimal, 1p,where p is prime, is either p 1 (the first group) or a divisor ofp 1 (the second

group).

Examples of fractions of the first group are:

17 = 0.142857 ; 6 repeating digits







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197 = 0.01030927 83505154 63917525 77319587 62886597 93814432

98969072 16494845 36082474 22680412 37113402 06185567 ; 96

repeating digits

The following multiplications exhibit an interesting property:

27 = 2 0.142857... = 0.285714...

37 = 3 0.142857... = 0.428571...

47 = 4 0.142857... = 0.571428...

57 = 5 0.142857... = 0.714285...

67 = 6 0.142857... = 0.857142...

Question8: What property a rational number must satisfy to have terminating

decimal expression

Answer: If the denominator is either 2 or 5 as its factor then the result will beterminating decimal. As 10 is the product of 2 and 5 so to have terminating decimal

2 or 5 are required. If there is a prime number other than 2 or 5 in the denominatorthen the decimal can or cannot be treminating.

Operations on Real Numbers:

If a and b are positive real numbers then:

(a)baab

=

(b)b

a

b

a =

(c) ( )( ) bababa (d) ( )( ) =baba a-b(e) ( )( ) bdbcadacdcba +(f) ( )ba + = baba +2Question9: Simplify the following expressions:

(a) ( )( )5275 +



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Lets assume that 5=a, 7 =b, 2=c and 5 =d

Then ( )( )dcba + = bdbcadac +Putting values of a, b, c and d we get

35725510 +

Laws of exponents:

(a) am . an = am+n

(b) (am)n = amn

(c) am/an = am-n , m>n

(d) ambm = (ab)m


9 math number system

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