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 p/q, what can you say about the prime factors of q?
(1) 43.123456789
(i) 0.120120012000120000...
(iii) 43.123456789 ( bar from 1 to 9)
Answers
(i) and (iii) are rational numbers while (ii) is an irrational number.
Explanation:
(i) 43.123456789 contains a certain number of digits that can be represented in the form of p/q.
∵ it is a rational number.
Since it has a certain number of digits and the number which includes certain number of digits is characterized always as a terminating number and for a terminating number, the denominator only contains the prime factor of 2 and 5.
(ii) 0.120120012000120000. . .
The denominator's prime factor will not have a value = 2 or 5.
∵ It is an irrational number because it is non-repeating and non-terminating as well.
(iii) 43. 123456789 with bar
∵ It is a rational number as the bar symbolizes that the decimals repeat over and again.
So, it is non- terminating and a rational number.
Learn more: rational numbers
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Explanation:
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