# 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/q, what can you say about the prime factor of q?

(i) 43.123456789 (ii) 0.120120012000120000....

(iii) 43.123456789

**Solution:**

Let x be a rational number whose decimal expansion terminates.

Then x can be expressed in the form p/q, where p and q are coprime, and the prime factorization of q is of form 2^{m} × 5^{n}, where n and m are non-negative integers.

(i) 43.123456789

Since this number has a terminating decimal expansion, it is a rational number of the form p/q and q is of form 2^{m} × 5^{n}

This means the prime factors of q will be either 2 or 5 or both.

(ii) 0.120120012000120000...

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

(iii) 43.123456789

Since the decimal expansion is non-terminating recurring, the given number is a rational number of the form p/q and q is not of form 2^{m} × 5^{n}. This means the prime factors of q will also have a factor other than 2 or 5.

**Video Solution:**

## 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/q, what can you say about the prime factor of q? (i) 43.123456789 (ii) 0.120120012000120000.. (iii)43.123456789

### NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.4 Question 3 - Chapter 1 Exercise 1.4 Question 3:

**Summary:**

43.123456789 and 43.123456789 are rational numbers of the form p/q and q is of the form 2^{m} × 5^{n} and the prime factors of q will be either 2 or 5 or both whereas 0.120120012000120000... is an irrational number