Prove that √5 is an irrational number

Proof

We prove this by contradiction.

Assume that √5 is a rational number.

Then it can be expressed in the form:

where a and b are integers having no common factor other than 1, and b ≠ 0.

Step 1: Square both sides

Step 2: Analyze the equation

From a² = 5b², we see that a² is divisible by 5. This implies that a is divisible by 5.

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

Step 3: Substitute the value of a

Substitute into the equation:

This shows that b² is also divisible by 5, and hence b is divisible by 5.

Step 4: Reach a contradiction

Now both a and b are divisible by 5. This contradicts the assumption that a and b have no common factor other than 1.

Conclusion