Irrationality of √2 — proof by contradiction

What just happened?

$\sqrt2$ is the length of the diagonal of a square with side 1 (Pythagoras' theorem: $1^2+1^2=2$) — a concrete, measurable length. And yet it cannot be written as an exact fraction $p/q$. We prove this by contradiction: we assume it can, and arrive at a contradiction.

Proof by contradiction, step by step:

Suppose the opposite — that $\sqrt2$ is rational, so it can be written as an irreducible fraction (with no common divisors):

We square both sides and multiply by $q^2$:

Since $p^2=2q^2$, the number $p^2$ is even — and the square of an odd number is odd, so $p$ is even: $p=2k$. We substitute this back:

Now $q^2$ is even, so $q$ is also even. But then $p$ and $q$ are both even — they share the divisor 2 — which contradicts the assumption that the fraction was irreducible.

A contradiction means the assumption was false. Hence $\sqrt2$ is not any fraction $p/q$ — it is an irrational number.

Why was this a shock?

The discovery was made by the ancient Pythagoreans (c. 500 BCE, attributed to Hippasus of Metapontum). They believed that “all is number” — that is, a fraction — so the existence of a length that cannot be written as a fraction shattered their worldview. Legend has it that Hippasus revealed this secret and was punished for it. The discovery forced mathematicians to extend the notion of number to include irrational numbers (such as $\sqrt2,\ \pi,\ e$) and to build solid foundations for analysis.

Why this matters for your exams:

The proof of the irrationality of $\sqrt2$ is a classic example of a proof by contradiction: you assume the negation of the claim and derive a contradiction, which proves the claim. Proofs by contradiction are a required element of advanced mathematics — this scheme (assumption → derivation → contradiction) carries over to many other problems.

Remember: $\sqrt2=\dfrac{p}{q}\Rightarrow 2q^2=p^2$ forces $p$ and $q$ to be even — a contradiction with irreducibility. That is why $\sqrt2$ is irrational, and the “by contradiction” method is one of the most powerful proof tools.