# Exercise 1.4

Question 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:

(i)

Sol :

The denominator is in the form of

Hence, the decimal expansion of is terminating.

(ii)

Sol :

The denominator is in the form of

Hence, the decimal expansion of  is terminating.

(iii)

Sol :

Since,the denominator is not in the form of and it also contains 7 and 13 as its factors , its decimal expansion will be non-terminating repeating .

(iv)

Sol :

The denominator is of the form .

Hence , the decimal expansion of is terminating .

(v)

Sol :

Since,the denominator is not in the form of and it also contains 7 as its factors , its decimal expansion will be non-terminating repeating .

(vi)

Sol :

The denominator is of the form .

Hence , the decimal expansion of is terminating .

(vii)

Sol :

The denominator is of the form but it also has 75 as its factor, the decimal expansion of is non-terminating repeating .

(viii)

Sol :

The denominator is of the form 5n

Hence ,the decimal expansion of is terminating .

(ix)

Sol :

The denominator is of the form

Hence , the decimal expansion of is terminating .

(x)

Sol :

The denominator is in the form of but it contains 3 as its factors , the decimal expansion of is non-terminating repeating .

Question 2

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

Sol :

(i)

(ii)  (working)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(x)

Question 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 factor of ?

(i)

Sol :

Since this number has a terminating decimal expansion, it is a rational number of the form  and q is of the form .

i.e., the prime factors of q will be either 2 or 5 or both .

(ii)

Sol :

The decimal expansion is neither terminating nor recurring. Therefore, the given number is an irrational number.

(iii)

Sol :

Since, the decimal expansion is non-terminating recurring, the given number is a rational number of the form and q is not of the form

i.e., the prime factors of q will also have a factor other than 2 or 5 .