Square Root of 2 is Irrational/Classic Proof

Theorem

 * $\sqrt 2$ is irrational.

Proof
First we note that, from Parity of Integer equals Parity of its Square, if a number is even, its square root, if an integer, is also even.

Thus it follows that:
 * $(A) \qquad 2 \mathop \backslash p^2 \implies 2 \mathop \backslash p$

where $2 \mathop \backslash p$ indicates that $2$ is a divisor of $p$.

Now, assume that $\sqrt 2$ is rational.

So $\displaystyle \sqrt 2 = \frac p q$ for some $p, q \in \Z$ and $\gcd \left({p, q}\right) = 1$.

Squaring both sides yields:
 * $\displaystyle 2 = \frac {p^2} {q^2} \iff p^2 = 2q^2$

Therefore, $2 \mathop \backslash p^2 \implies 2 \mathop \backslash p$ (see $(A)$ above).

That is, $p$ is an even integer.

So $p = 2k$ for some $k \in \Z$.

Thus:
 * $2 q^2 = p^2 = \left({2 k}\right)^2 = 4 k^2 \implies q^2 = 2k^2$

so by the same reasoning
 * $2 \mathop \backslash q^2 \implies 2 \mathop \backslash q$

This contradicts our assumption that $\gcd \left({p, q}\right) = 1$, since $2 \mathop \backslash p, q$.

Therefore, from Proof by Contradiction, $\sqrt 2$ cannot be rational.

Historical Note
This proof is attributed to Pythagoras of Samos.

The ancient Greeks prior to Pythagoras, following Eudoxus of Cnidus, believed that irrational numbers did not exist in the real world.

However, from the Pythagorean Theorem, a square with sides of unit length has a diagonal of length $\sqrt 2$.