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) 0.1201120012000120000…. (iii)
Solution:
(i) 43.123456789
It is rational because decimal expansion is terminating. Therefore, it can be expressed in
form where factors of q are of the form
where n and m are non-negative integers.
(ii) 0.1201120012000120000…
It is irrational because decimal expansion is neither terminating nor non-terminating repeating.
(iii)
It is rational because decimal expansion is non-terminating repeating. Therefore, it can be expressed in
form where factors of q are not of the form
where n and m are non-negative integers.
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