**EXERCISE 1.4**

**EXERCISE 1.4**

**1. Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion :**

**Sol.** We know that if the denominator of a rational number has no prime factors other than 2 or 5, then it is expressible as a terminating, otherwise it, has non-terminating repeating decimal representation. Thus,- we will have to check the prime factors of the denominators of each of the given rational numbers.

(i) In** **_{}, the denominator is 3125.

We have, 3125 = 5 × 5 × 5 × 5 x 5

Thus, 3125 has 5 as the only

prime factor.

Hence, ** **_{}** **must have a

terminating decimal representation.

(ii) In _{} the denominator is 8

We have, 8=2 x 2 × 2

Thus, 8 has 2 as the only prime factor.

Hence, _{} must have a terminating decimal representation.

### (iii) In ** **_{}** **, denominator is 455.

We have, 455 = 5 × 7 x 13

Clearly, 455 has prime factors

other than 2 and 5. So, it will not have

a terminating decimal representation.

(iv) In _{}** **, the denominator is 1600.

We have, 1600

= 2 x 2 x 2 × 2 x 2 × 2 × 5 × 5

Thus, 1600 has only 2 and 5 as prime factors.

Hence,_{}

must have a terminating decimal representation.

### (v) In ** **_{}** , **the denominator is 343.

We have, 343 = 7 × 7 × 7

Clearly, 343 has prime factors other than 2 and 5.

So, it will not have terminating decimal representation.

### (vi) In ** **_{}

Clearly, the denominator _{} has only 2 and 5 as prime factors.

Hence, ** **_{}** **must have a terminating decimal representation.

(vii) In ** **_{}** **

Clearly, the denominator _{} has prime factors other than 2 and 5.

So, it will not have terminating decimal representation.

(viii) In _{}** ,** we have 15 = 3 x5

Clearly, 15 has prime factors other than 2 and 5. So, it will not have terminating decimal representation.

(ix) In ** **_{}** ,** we have 50 = 2 × 5 x 5

The denominator has only 2 and 5 as prime factors.

Hence, _{}** **must have a terminating decimal representation.

(x) In _{} , the denominator is 210.

We have, 210 = 2 × 3 × 5 x 7

Clearly, 210 has prime factors other than 2 and 5.

So, it will not have terminating decimal representation.

**2. Write down the decimal expansion of those rational numbers in Question 1 above which have terminating decimal expansions.**

(x) Non – terinatinating.

**3. The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form** ** **_{}** what can you say about the prime factors of ***q*?

*q*?

(i) 43.123456789 (ii) 0.120120012000120000…

(iii) 43.123456789

(i) 43.123456789 (ii) 0.120120012000120000…

(iii) 43.123456789

**Sol**. (i) 43.123456789 is terminating.

So, it represents a rational number.

Thus, 43.123456789 = ** **, where *q* =10^{9} ** .**

*q*

(ii) 0.12012001200012000… is non-terminating and non-repeating. So, it is irrational.

(iii) 43.123456789 is non-terminating but repeating So, it is rational.

Thus, 43.123456789 = ** **_{}** ** where *q* = 999999999.

*q*