NCERT Solutions for Class 10 Maths Chapter 1

Important NCERT Questions

Real Numbers

NCERT Books for Session 2022-2023

CBSE Board and UP Board Others state Board

EXERCISE 1.3

Page No:11

Questions No:1

# Prove that √5 is irrational.

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It’s a little tricky one🤔,

Let √5 is a rational number.

Therefore, we can find two integers a, b (b ≠ 0) such that √5 = a/b 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.

a = √5b

⇒ a² = 5b²

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

(5k)² = 5b²

⇒ 5k² = b²

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, √5 cannot be expressed as p/q or it can be said that √5 is irrational.

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