# 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 factors of q?

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

(iii) 43.123456789

**Solution:**

Let x be a rational number that has a terminating decimal expansion.

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

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

(ii) 0.120120012000120000...

The decimal expansion is non-terminating non-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 the form 2^{m} × 5^{n}. This means the prime factors of q will also have a factor other than 2 or 5.

**☛ Check: **NCERT Solutions Class 10 Maths Chapter 1

**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

**Summary:**

For the following real numbers, the decimal expansions are given. In each case, let's decide whether they are rational or not. 43.123456789 is a rational number 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, 43.123456789 is a rational number of the form p/q and q is not of the form 2^{m} × 5^{n} and the prime factors of q will have factors other than 2 and 5 whereas, 0.120120012000120000... is an irrational number.

**☛ Related Questions:**

- Prove that 3 + 2√5 is irrational.
- Prove that the following are irrationals: (i) 1/√2 (ii) 7√5 (iii) 6 + √2
- Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion: (i) 13/3125 (ii) 17/8 (iii) 64/455 (iv) 15/1600 (v) 29/343 (vi) 23/2352 (vii) 129/225775 (viii) 6/15 (ix) 35/50 (x) 77/210
- Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions. (i) 13/3125 (ii) 17/8 = 2.125 (iii) 64/455 (iv) 15/1600 (v) 29/343 (vi) 23/ (23 x 52) (vii) 129/(22 x 57 x 75) (viii) 6/15 (ix) 35/50 (x) 77/210

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