## NCERT Solutions for Class 10 Maths Exercise 1.4 Chapter 1 Real Numbers – FREE PDF Download

NCERT Class 10 Maths Ch 1 is one of the most important ones in the NCERT syllabus. Duly following NCERT Solutions for Class 10 Maths Chapter 1 ensures you that there will be no hindrance when you opt for more advanced branches of Maths. This is where CoolGyan comes in. Our free Class 10 Real Numbers solutions will help you understand the chapter thoroughly.

# NCERT Solutions for Class 10 Maths Chapter 1 – Real Numbers

1. Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating decimal expansion.

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(x)

**Ans.**According to Theorem, any given rational number of the form where

*p*and

*q*are co-prime, has a terminating decimal expansion if q is of the form , where m and n are non-negative integers.

(i)

*q *= 3125 = = 2^{0} x 5^{5}

Here, denominator is of the form , where m = 5 and n = 0.

It means rational number has a terminating decimal expansion.

(ii)

*q *= 8 = = 2^{3} x 5^{0}

Here, denominator is of the form , where m = 0 and n = 3.

It means rational number has a terminating decimal expansion.

(iii)

*q *= 455 =

Here, denominator is not of the, where m and n are non-negative integers.

It means rational number has a non-terminating repeating decimal expansion.

(iv)

*q *= 320 = =

Here, denominator is of the form , where m = 1 and n = 6.

It means rational number has a terminating decimal expansion.

(v)

*q *= 343 = = 7^{3}

Here, denominator is not of the form , where m and n are non-negative integers.

It means rational number has non-terminating repeating decimal expansion.

(vi)

*q *=

Here, denominator is of the form , where m = 2 and n = 3 are non-negative integers.

It means rational number has terminating decimal expansion.

(vii)

*q *=

Here, denominator is not of the form , where m and n are non-negative integers.

It means rational number has non-terminating repeating decimal expansion.

(viii)

= 2^{0} x 5^{1}

Here, denominator is of the form , where m = 1 and n = 0.

It means rational number has terminating decimal expansion.

(ix)

*q *= 10 =

Here, denominator is of the form , where m = 1 and n = 1.

It means rational number has terminating decimal expansion.

(x)

Here, denominator is not of the form , where m and n are non-negative integers.

It means rational number has non-terminating repeating decimal expansion.

2. Write down the decimal expansions of those rational numbers in Question 1 which have terminating decimal expansions.

**Ans.**(i) 133125=1355=13×2555×25=416105=0.00416133125=1355=13×2555×25=416105=0.00416

(ii)

(iv)

(vi)

(viii)

(ix)

3. 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 , what can you say about the prime factors of q?

(i) 43.123456789

(ii) 0.1201120012000120000…

(iii)

**Ans.**(i) 43.123456789

It is rational because decimal expansion is terminating. Therefore, it can be expressed in form where q = 10^{9} and factors of *q* are of the form where *n* and *m* are non-negative integers

(ii) 0.1201120012000120000…

It is irrational because decimal expansion is neither terminating nor non-terminating repeating.

(iii)

It is rational because decimal expansion is non-terminating repeating. Therefore, it can be expressed in form where factors of *q* are not of the form where *n* and *m* are non-negative integers.

Thus, 43.123456789=pq,43.123456789=pq, where q = 999999999