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Prove that √5 is irrational

Solution:

Let us prove that √5 is an irrational number.

This question can be proved with the help of the contradiction method. Let's assume that √5 is a rational number. If √5 is rational, that means it can be written in the form of a/b, where a and b integers that have no common factor other than 1 and b ≠ 0. i.e., a and b are coprime numbers.

√5/1 = a/b

√5b = a

Squaring both sides,

5b² = a² ... (1)

This means 5 divides a².

From this, 5 also divides a.

Then a = 5c, for some integer 'c'.

On squaring, we get

a² = 25c²

Put the value of a² in equation (1).

5b² = 25c²

b² = 5c²

This means b² is divisible by 5 and so b is also divisible by 5. Therefore, a and b have 5 as common factor. But this contradicts the fact that a and b are coprime. This contradiction has arisen because of our incorrect assumption that √5 is a rational number. So, we conclude that √5 is irrational.

☛ Check: NCERT Solutions Class 10 Maths Chapter 1

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Video Solution:

Prove that √5 is irrational

NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.3 Question 1

Summary:

√5 is irrational by the method of contradiction.

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