To prove irrational square root two,
Suppose there are integers p and q
with p over q entirely reduced
and when their ratio is taken, root two is produced.

squaring both sides of this faulty equation
yields p squared over q squared is two, a relation
so p squared is even, a fact that is true
if and only if it is divisible by two.

But since two is prime, evenness of p squared
says that p is even, (a fact that Euclid declared).
We can now deduce a little bit more;
the number two q squared is divisible by 4.

But then q squared is divisible by two
And we know that this fact, is simply not true
Because then q is even, that just cannot be
because p,q are both even, contradiction, QED.