Prove that root 5 is irrational

Question – Prove that root 5 is irrational.

Solution –

Proof: Let us assume, the contrary that √5 is not  an irrational number.

This means that √5 is a rational number

Then, there exist two integers a and b, where (b ≠ 0) such that √5 = a/b

Let a and b have a common factor other than 1, then we can divide by the common factor and assume that a and b are coprime

√5 = a/b

a = √5 b

Squaring both side,

(a)²= 5 (b)² …..(1).

Therefore (a)² is divisible by 5 and hence a is divisible by 5. [∴ Theorem : Let p be a prime number , if p divides a² , then p divides a, where a is a positive integer]

So we can write a = 5k, for some integer k. substituting this in (1) we get

25k²= 5 (b)²

(b)² = 5k²

(b)² is also divisible by 5 and hence b is divisible by 5.

Therefore, a and b have the least common factor 5, which is a contradiction to the fact that a and b are coprime. This contradiction has arisen because of our incorrect assumption that √5 is rational.

So, we concluded that √5 is  an irrational rational.

Related Questions –

• Prove that 3+ 2√5 is irrational
• Prove that the following are irrational (i) 1/√2 (ii) 7√5 (iii) 6+√2