Prove that √5 is irrational | How to prove that √5 is irrational

Prove that √5 is irrational

To prove that √5 is irrational, we need to show that it cannot be expressed as the ratio of two integers.Let us assume contrary to the statement that √5 is rational. Then we can write:√5 = a/b

where a and b are co-prime integers.

Squaring both sides,

we get:

5 = a² / b²

Multiplying both sides by b²

we get:

5b² = a²

Since a² is divisible by 5, a must be a divisible by 5.

Let a = 5k, where k is a positive integer.

Fill this value into the last equation,

5b² = (5k)²

Simplifying, we get:

b² = 5k²

This means that b² is also divisible by 5, which implies that b must also divisible with 5.

It means both a and b are divisible by 5 but a and b are coprime.

Now there is a contradiction that a and b have no common factors. So, our assumption that √5 is rational is false. Hence √5 is irrational.

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