Prove that √5 is irrational.

By BYJU'S Exam Prep

Updated on: October 17th, 2023

We have to prove that √5 is an irrational number

It can be proved using the contradiction method

Assuming √5 as a rational number, i.e., can be written in the form a/b where a and b are integers with no common factors other than 1 and b is not equal to zero.

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1. Prove that √5 is irrational.

√5/1 = a/b

√5b = a

By squaring on both sides

5b² = a²

b² = a²/5 …. (1)

It means that 5 divides a².

It means that it also divides a

a/5 = c

a = 5c

By squaring on both sides

a² = 25c²

Substituting the value of a² in equation (1)

5b² = 25c²

b² = 5c²

b²/5 = c²

As b² is divisible by 5, b is also divisible by 5

a and b have a common factor as 5

It contradicts the fact that a and b are coprime

This has arisen due to the incorrect assumption as √5 is a rational number.

Therefore, √5 is irrational.

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