Real Numbers

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

Solution:

Question 2.
Write down the decimal expansions of those rational numbers in the question 1, which have terminating decimal expansions.

Solution:

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 factors of q ?

(i) 43. 123456789
(ii) 0.120120012000120000…
(iii) 43. 

Solution:
(i) 43.123456789

Here, the denominator is of the form 2^m5ⁿ.
Hence, the number is a rational number, specifically a terminating decimal.

(ii) Since the given decimal number is non ­terminating non-repeating, it is not rational

(iii) Since the given decimal number is non terminating repeating, it is rational, but the denominator is not of the form 2^m5ⁿ.