Classify the following numbers as rational or irrational:

(i) \( \sqrt{23}\)

(ii) \( \sqrt{225}\)

(iii) 0.3796

(iv) 7.478478...

(v) 1.101001000100001...

Asked by Vishal kumar | 2 years ago |  301

1 Answer

Solution :-

We know

(i) \( \sqrt{23}\)

\( \sqrt{23}\) = 4.795831…..

It is an irrational number

(ii) \( \sqrt{225}\) = 15

Therefore, \( \sqrt{225}\) is a rational number.

(iii) 0.3796

It is terminating decimal. Therefore, it is rational number

(iv) 7.478478….

The given number 7.478478…. is a non-terminating recurring decimal, which can be converted into \( \frac{p}{q}\) form.

While, converting 7.478478…. into \( \frac{p}{q}\) form, we get

x = 7.478478.... … (a)

1000x = 7478.478478.... … (b)

While, subtracting (a) from (b), we get

\( \cfrac{\begin{matrix}1000\mathrm x = 7478.478478...\\-\mathrm x = 7.478478....\end{matrix}}{999\mathrm x = 7471}\)

x = \( \frac{7471}{999}\)

Therefore, 7.478478…. is a rational number.

(v) 1.101001000100001....

We can observe that the number 1.101001000100001.... is a non-terminating nonrecurring decimal. Thus, non-terminating and non-recurring decimals cannot be converted into \( \frac{p}{q}\) form.

Therefore, we conclude that 1.101001000100001.... is an irrational number.

Answered by Shivani Kumari | 2 years ago

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