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

Go back to  'Ex.1.4'

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 $$5.$$

Learn from the best math teachers and top your exams

• Live one on one classroom and doubt clearing
• Practice worksheets in and after class for conceptual clarity
• Personalized curriculum to keep up with school