The following real numbers have decimal expansions as given below. In each case, state whether they are rational or not. If they are rational and expressed in the form p/q, where p, q are integers, q≠ 0, and p, q are co-prime, then what can you say about the prime factors of q?
(i) 37.09158
(iii) 8.9010010001…
(iv) 2.3476817681…
Solution:
(i) 37.09158
We know that
It has terminating decimal
Here
It is a rational number and factors of q will be 2 or 5 or both.
We know that
It has non-terminating recurring decimals
Here
It is a rational number.
(iii) 8.9010010001…
We know that
It has non-terminating, non-recurring decimal.
Here
It is not a rational number.
(iv) 2.3476817681…
We know that
It has non-terminating, recurring decimal.
Here
It is a rational number and the factors of q are prime factors other than 2 and 5.
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