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

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

## 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 \(5.\)