Proof that √5 is an irrational number

Let us suppose that 5 is irrational. This means that it may be written as a fraction p/q, where p and q are integers with no common factors (other than 1) and q is greater than zero.

So, we have:

√5 = p/q

Squaring both sides of the equation:

5 = (p/q)²

5 = p²/q²

Multiplying both sides by q²:

5q² = p²

q² = (p²)/5

Now, we observe that (p²) is divisible by 5.So (p) also be divisible by 5.

Now let's p can be written as 5k, where k is an integer.

Substituting 5k for p in the equation:

5q² = (5k)²

5q² = 25k²

q² = 5k²

This implies that q² is also divisible by 5. Therefore, q must also be divisible by 5.

However, this contradicts our initial assumption that p and q have no common factors other than 1. Therefore, our assumption that √5 is rational must be wrong, and we conclude that √5 is irrational.