# Exercise 1.3 - Chapter 1 Real Numbers class 10 ncert solutions Maths - Free PDF Download 2022 - 2023

Question 1:

Prove that  is irrational.

Let  is a rational number.

Therefore, we can find two integers ab (b ≠ 0) such that

Let a and b have a common factor other than 1. Then we can divide them by the common factor, and assume that a and b are co-prime.

Therefore, a2 is divisible by 5 and it can be said that a is divisible by 5.

Let a = 5k, where k is an integer

This means that b2 is divisible by 5 and hence, b is divisible by 5.

This implies that a and b have 5 as a common factor.

And this is a contradiction to the fact that a and b are co-prime.

Hence,cannot be expressed as or it can be said that is irrational.

Question 2:

Prove that  is irrational.

Let is rational.

Therefore, we can find two integers ab (b ≠ 0) such that

Since a and b are integers, will also be rational and therefore,is rational.

This contradicts the fact that is irrational. Hence, our assumption that is rational is false. Therefore,  is irrational.

Question 3:

Prove that the following are irrationals:

Let is rational.

Therefore, we can find two integers ab (b ≠ 0) such that

is rational as a and b are integers.

Therefore, is rational which contradicts to the fact that  is irrational.

Hence, our assumption is false and is irrational.

Let  is rational.

Therefore, we can find two integers a, b (b ≠ 0) such that

for some integers a and b

is rational as a and b are integers.

Therefore,  should be rational.

This contradicts the fact thatis irrational. Therefore, our assumption that is rational is false. Hence,  is irrational.

Let  be rational.

Therefore, we can find two integers ab (b ≠ 0) such that

Since a and b are integers,  is also rational and hence, should be rational. This contradicts the fact that is irrational. Therefore, our assumption is false and hence,  is irrational.