TOPICS

Exercise - 1.3

Number Systems

**Question-1 :-** Write the following in decimal form and say what kind of decimal expansion each has :

(i) ^{36}⁄_{100} ,
(ii) ^{1}⁄_{11} ,
(iii) 4^{1}⁄_{8} ,
(iv) ^{3}⁄_{13} ,
(v) ^{2}⁄_{11} ,
(vi) ^{329}⁄_{400}

(i) 36/100 = 0.36 (Terminating) (ii) 1/11 = 0.09090... = 0.09 (Non-Terminating Repeating) (iii)4^{1}⁄_{8}= 33/8 = 4.125 (Terminating) (iv) 3/13 = 0.230969230769... = 0.230969 (Non-Terminating Repeating) (v) 2/11 = 0.181818... = 0.18 (Non-Terminating Repeating) (vi) 329/400 = 0.8225 (Terminating)

**Question-2 :-** You know that 1/7 = 0.142857 Can you predict what the decimal expansions of
^{2}⁄_{7} , ^{3}⁄_{7} , ^{4}⁄_{7} , ^{5}⁄_{7} , ^{6}⁄_{7}
are, without actually doing the long division? If so, how?

(i) 2/7 = 2 x 1/7 = 2 x 0.142857 = 0.285714 (ii) 3/7 = 3 x 1/7 = 3 x 0.142857 = 0.428571 (iii) 4/7 = 4 x 1/7 = 2 x 0.142857 = 0.571428 (iv) 5/7 = 5 x 1/7 = 2 x 0.142857 = 0.714285 (v) 6/7 = 6 x 1/7 = 2 x 0.142857 = 0.857142

**Question-3 :-** Express the following in the form p/q, where p and q are integers and q ≠ 0.

(i) 0.6 ,
(ii) 0.47 ,
(iii) 0.001

(i) Let x = 0.6666... and there is one digit is repeating, so multiply by 10 both sides 10x = 10 X (0.6666...) = 6.666... 6.666... = 6 + x, (Since x = 0.666...) 10x = 6 + x 10x - x = 6 9x = 6 , x = 6/9, i.e., x = 2/3 (ii) Let x = 0.4777... and there is two digit is repeating, so multiply by 100 both sides 10x = 10 X (0.4777...) = 4.777... 10x = 4.3 + 0.4777... 10x = 4.3 + x, (Since x = 0.4777...) 10x - x = 4.3 9x = 4.3 , x = 4.3/90, i.e., x = 43/90 (iii) Let x = 0.6666... and there is one digit is repeating, so multiply by 10 both sides 10x = 10 X (0.6666...) = 6.666... 6.666... = 6 + x, (Since x = 0.666...) 10x = 6 + x 10x - x = 6 9x = 6 , x = 6/9, i.e., x = 2/3

**Question-4 :-** Express 0.99999 .... in the form p/q. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

Let x = 0.9999... and there is one digit is repeating, so multiply by 10 both sides 10x = 1 x 0.9999... = 9.999... 10x = 9 + x (Since x = 0.9999...) 9x = 9 x = 9/9 x = 1

**Question-5 :-** What can the maximum number of digits be in the repeating block of digits in the
decimal expansion of 1/17? Perform the division to check your answer.

```
It can be observed that, 1/17 = 0.0588235294117647
There are 16 digits in the repeating numbers of the decimal expansion of 1/17.
```

**Question-6 :-** Look at several examples of rational numbers in the form p/q (q ≠ 0),
where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions).
Can you guess what property q must satisfy?

Terminating decimal expansion will occur when denominator q of rational number p/q is either of 2, 4, 5, 8, 10, and so on... 9/4 = 2.25 11/8 = 1.375 27/5 = 5.4 It can be observed that terminating decimal may be obtained in the situation where prime factorisation of the denominator of the given fractions has the power of 2 only or 5 only or both.

**Question-7 :-** Write three numbers whose decimal expansions are non-terminating non-recurring.

3 numbers whose decimal expansions are non-terminating non-recurring are as follows. 0.505005000500005000005…, 0.7207200720007200007200000…, 0.080080008000080000080000008…

**Question-8 :-** Find three different irrational numbers between the rational numbers 5/7 and 9/11.

5/7 = 0.714285 9/11 = 0.81 3 irrational numbers are as follows. 0.73073007300073000073… 0.75075007500075000075… 0.79079007900079000079…

**Question-9 :-** Classify the following numbers as rational or irrational :

(i) √23, (ii) √225, (iii) 0.3796, (iv) 7.478478..., (v) 1.101001000100001...

```
(i) √23 = 4.79583152331...
As the decimal expansion of this number is non-terminating non-recurring, therefore, it is an irrational number.
(ii) √225 = 15
It is a rational number as it can be represented in p/q form.
(iii) 0.3796
As the decimal expansion of this number is terminating, therefore, it is a rational number.
(iv) 7.478478... = 7.478
As the decimal expansion of this number is non-terminating recurring, therefore, it is a rational number.
(v) 1.101001000100001...
As the decimal expansion of this number is non-terminating non-repeating, therefore, it is an irrational number.
```

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