Foundation
Mathematics Foundation
Properties of Real Number
Irrational Number
Rational Numbers
Question
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, you say about the prime factors of q?
(i) 43.123456789 (ii) 0.1201200112000120000... (iii) 43.123456789...
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Solution
(i) 43.123456789
It has certain number of digits, so they can be represented in form of .
Hence they are rational number.
As they have certain number of digits and the number which has certain
number of digits is always terminating number and for terminating number
denominator has prime factor 2 and 5 only.
(ii) 0.120120012000120000...
Here the prime factor of denominator Q will has a value which is not equal to
2 or 5.
So, it is an irrational number as it is non-terminating and non-repeating.
(iii) 43.123456789
(iii) 43.123456789
Here the prime factor of denominator Q will has a value which is apart from
2 or 5, some other factor also.
So, it is an rational number,0.123456789 repeating again and again. It is non- terminating.