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)

The denominator is of the form 5^m.
Hence, the decimal expansion of

 is terminating.


(ii)

The denominator is of the form 2^m.

Hence, the decimal expansion of

 is terminating.
(iii)

455 = 5 × 7 × 13
Since the denominator is not in the form 2^m × 5ⁿ, and it also contains 7 and 13 as its factors, its decimal expansion will be non-terminating repeating.
(iv)

1600 = 2⁶ × 5²
The denominator is of the form 2^m × 5ⁿ.
Hence, the decimal expansion of

  is terminating.
(v)

Since the denominator is not in the form 2^m × 5ⁿ, and it has 7 as its factor, the decimal expansion of is non-terminating repeating.
(vi)

The denominator is of the form 2^m × 5ⁿ.
Hence, the decimal expansion of

is terminating.
(vii)

Since the denominator is not of the form 2^m × 5ⁿ, and it also has 7 as its factor, the decimal expansion of

 is non-terminating repeating.
(viii)

The denominator is of the form 5ⁿ.
Hence, the decimal expansion of

is terminating.
(ix)

The denominator is of the form 2^m × 5ⁿ.
Hence, the decimal expansion of

 is terminating.
(x)

Since the denominator is not of the form 2^m × 5ⁿ, and it also has 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.

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 q?

(i) 43.123456789 (ii) 0.120120012000120000… (iii)  

(i) 43.123456789
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) 0.120120012000120000 …
The decimal expansion is neither terminating nor recurring. Therefore, the given number is an irrational number.
(iii)

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.