# Prove that square root of 5 is irrational

To prove that square root of 5 is irrational, we will use a proof by contradiction.

What is a proof by contradiction ?

Suppose we want to prove that a math statement is true. Simply put, we assume that the math statement is false and then show that this will lead to a contradiction. If it leads to a contradiction, then the statement must be true

To show that
√5
is an irrational number, we will assume that it is rational.

Then, we need to find a contradiction when we make this assumption.

If we are going to assume that

√5
is rational, then we need to understand what it means for a number to be

rational.
Basically, if square root of 5 is rational, it can be written as the ratio of two numbers as shown below:

Square both sides of the equation above

Multiply both sides by y

^{2}
5 × y

^{2} =

x^{2}
/
y^{2}

× y

^{2}
We get 5 × y

^{2} = x

^{2}
## In order to prove that square root of 5 is irrational, you need to understand also this important concept.

{
Another important

**concept** before we finish our proof: Prime factorization

**Key question**: is the number of prime factors for a number raised to the second power an even or odd number?

For example, 6

^{2}, 12

^{2}, and 15

^{2}
6

^{2} = 6 × 6 = 2 × 3 × 2 × 3 (4 prime factors, so even number)

12

^{2} = 12 × 12 = 4 × 3 × 4 × 3 = 2 × 2 × 3 × 2 × 2 × 3 (6 prime factors, so even number)

15

^{2} = 15 × 15 = 3 × 5 × 3 × 5 = (4 prime factors, so even number)

There is a solid pattern here to conclude that any number squared will have an even number of prime factors.

In order words, x

^{2} has an even number of prime factors.

}
Let's finish the proof then!

5 × y

^{2} = x

^{2}
Since 5 × y

^{2} is equal to x

^{2}, 5 × y

^{2} and x

^{2} must have the

__same number__ of prime factors.

We just showed that

x

^{2} has an even number of prime factors.

y

^{2} has also an even number of prime factors.

5 × y

^{2} will then have an odd number of prime factors.

The number 5 counts as 1 prime factor, so 1 + an even number of prime factors is an odd number of prime factors.

5 × y

^{2} is the same number as x

^{2}. However, 5 × y

^{2} gives an odd number of prime factor while x

^{2} gives an even number of prime factors.

This is a contradiction since a number cannot have an odd number of prime factors and an even number of prime factors at the same time

The assumption that square root of 5 is rational is wrong. Therefore, square of 5 is irrational