Why Is √2 Irrational?

We said that √2 cannot be written as a fraction of two integers.

But how do we know?

Let's prove it.

Suppose, for a moment, that √2 is rational.

Then we should be able to write it as:

√2 = p/q

where p and q are integers, q ≠ 0, and the fraction is in its simplest form.

Square both sides:

2 = p²/q²

Multiply both sides by q²:

p² = 2q²

The right side is divisible by 2.

Therefore, p² is even.

If the square of an integer is even, the integer itself must be even.

p = 2k

for some integer k.

Substitute p = 2k into p² = 2q²:

(2k)² = 2q²

4k² = 2q²

q² = 2k²

So q² is also even.

Therefore, q must also be even.

q = 2m

for some integer m.

But this creates a contradiction.

We started with p/q in its simplest form.

Yet we have shown that both p and q are even.

p/q can be reduced

So p/q was not actually in its simplest form.

THEREFORE

Our original assumption must be false.

√2 ≠ p/q

Therefore, √2 is irrational.