# Ex.1.4 Q3 Real Numbers Solution - NCERT Maths Class 10

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## 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 \begin{align}\frac{p}{q},\end{align} what can you say about the prime factor of $$q$$ ?

(i) $$\,43.123456789$$

(ii) $$\,0.120120012000120000 \dots \dots$$

(iii) $$\,43.\mathop {123456789}\limits^{\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$$

Video Solution
Real Numbers
Ex 1.4 | Question 3

## Text Solution

Reasoning:

Let $$x$$ be a rational number whose decimal expansion terminates.

Then $$x$$ can be expressed in the form \begin{align}\frac{p}{q},\end{align} where $$p$$ and $$q$$ are coprime, and the prime factorisation of $$q$$ is of the form $${2^n}\, \times \,{5^m}$$, where $$n, \;m$$ are non-negative integers.

Steps:

(i) $$\,43.123456789$$

Since this number has a terminating decimal expansion, it is a rational number of the form \begin{align} \frac{p}{q} \end{align} and $$q$$ is of the form $$2^{m} \times \,5^{n}$$

i.e., the prime factors of  $$q$$  will be either $$2$$ or $$5$$ or both.

(ii) $$\,0.120120012000120000 \dots \dots$$

The decimal expansion is neither terminating nor recurring. Therefore, the given number is an irrational number.

(iii) $$\,43.\mathop {123456789}\limits^{\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$$

Since the decimal expansion is non-terminating recurring, the given number is a rational number of the form \begin{align} \frac{p}{q} \end{align} and $$q$$ is not of the form $$2^{m}\times 5^{n}$$

i.e., the prime factors of $$q$$ will also have a factor other than $$2$$ or $$5.$$

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