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Legend says that in the 5th century BCE, the Greek mathematician Hippasus discovered that the diagonal of a unit square cannot be written as a simple fraction β€” a discovery so shocking to the Pythagoreans that he was allegedly drowned at sea.
Today, we know this diagonal length is 2\sqrt{2}. A rational number can be written as an integer ratio ab\frac{a}{b} (where b≠0b \neq 0), while an irrational number cannot.
To prove that 2\sqrt{2} is irrational, mathematicians use a powerful technique called proof by contradiction (or reductio ad absurdum).
Before building the full proof, we need one essential fact about how prime numbers divide into squares.

The Prime Divisibility Lemma

A prime number is an integer greater than 1 with no positive divisors other than 1 and itself. Euclid's Lemma states that if a prime pp divides a product abab, then pp must divide aa or pp must divide bb.
Specifically, if a prime pp divides a2a^2, then pp must divide aa. This is because squaring an integer simply doubles the exponent of every prime factor in its prime factorization.
With this lemma in hand, how do we construct the classic proof for 2\sqrt{2} step by step?

Proving 2\sqrt{2} Is Irrational

We begin by assuming the opposite: suppose 2\sqrt{2} is rational. That means 2=ab\sqrt{2} = \frac{a}{b} for positive integers aa and bb, written in irreducible form (where aa and bb share no common factors other than 1, meaning gcd⁑(a,b)=1\gcd(a, b) = 1).
Squaring both sides gives 2=a2b22 = \frac{a^2}{b^2}, which rearranges to a2=2b2a^2 = 2b^2. Because a2a^2 is 2 times an integer, a2a^2 is even; by our divisibility lemma, this means aa must be even.