Prove that square root of 5 is irrational 

To prove that square root of 5 is irrational, we will use a proof by contradiction.
Then, we need to find a contradiction when we make this assumption.


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 y²


We get 5 × y² = x²

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

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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, 6², 12², and 15²

6² = 6 × 6 = 2 × 3 × 2 × 3 (4 prime factors, so even number)

12² = 12 × 12 = 4 × 3 × 4 × 3 = 2 × 2 × 3 × 2 × 2 × 3 (6 prime factors, so even number)

15² = 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, x² has an even number of prime factors.

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Let's finish the proof then!

5 × y² = x²

Since 5 × y² is equal to x², 5 × y² and x² must have the same number of prime factors.

We just showed that

x² has an even number of prime factors.

y² has also an even number of prime factors.

5 × y² 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 × y² is the same number as x². However, 5 × y² gives an odd number of prime factor while x² 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