The Proof That Shook Ancient Greece: Why √2 Can't Be a Fraction

The Number That Broke a Brotherhood

Around 520 BC, a Greek mathematician named Hippasus was working with the Pythagorean brotherhood — a philosophical and mathematical society that believed, fundamentally, that every number could be expressed as a ratio of two integers.

Then he proved that √2 couldn't.

The legend — historically uncertain but enduring — is that his fellow Pythagoreans threw him overboard into the sea for it. Whether the story is true almost doesn't matter. The mathematical discovery was real, the implications were disruptive, and the proof he or someone from that era constructed is still used today, essentially unchanged.

It's one of the oldest, cleanest arguments in mathematics. And it begins by assuming the opposite of what you want to prove.

What "Irrational" Actually Means

An irrational number is simply a number that cannot be written as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$.

Not "crazy." Not "unpredictable." Just: no ratio of whole numbers equals this.

Rational numbers include all fractions and all terminating or repeating decimals — which is covered in the fraction-to-decimal article. Irrational numbers have decimal expansions that go on forever without any repeating block.

Note that 22/7 is rational — it repeats. π is irrational — it never repeats, even though it's close to 22/7. The decimal resemblance is coincidental.

The Proof by Contradiction

Proof by contradiction works like this: assume the thing you want to disprove is true. Then show that the assumption leads to an impossible conclusion. Therefore the assumption must be false.

Claim: √2 is irrational.

Proof: Suppose for the sake of argument that √2 is rational. Then there exist integers $p$ and $q$, with no common factors (the fraction is in lowest terms), such that:

Square both sides:

So $p^2$ is even (it equals 2 times something). If a perfect square is even, its square root must also be even — because odd numbers have odd squares. So $p$ is even. Write $p = 2k$ for some integer $k$.

Substitute back:

So $q^2$ is also even, which means $q$ is also even.

But now both $p$ and $q$ are even — they share a common factor of 2. That contradicts our starting assumption that the fraction was in lowest terms.

This same argument extends: √3, √5, √7, and the square root of any integer that isn't a perfect square are all irrational. You can prove each one using the same contradiction structure.

Square Roots in Practice — Where They Actually Appear

The diagonal of a unit square is √2. That follows directly from the Pythagorean theorem: $\sqrt{1^2 + 1^2} = \sqrt{2}$. Every 45-45-90 triangle has a hypotenuse of $\text{leg} \times \sqrt{2}$.

Standard deviation uses the square root of variance — because variance squares the deviations (to remove sign), and you need to square-root back to return to the original units. The standard deviation article explains why we don't just take absolute values instead.

The quadratic formula produces square roots when the discriminant isn't a perfect square — which happens for most real-world equations.

Square roots of imperfect squares are irrational. Your calculator shows a decimal approximation. The exact answer — the infinitely precise one — is the radical expression: $\sqrt{2}$, not 1.41421356.

Perfect Squares, Imperfect Squares, and Simplifying What You Can

A perfect square is an integer whose square root is also an integer: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100...

For everything else, simplification means pulling perfect-square factors out of the radical:

The simplified radical form is exact. The decimal approximation is not. This distinction matters in multi-step calculations — rounding early lets errors compound.

The simplifying radicals article covers the systematic method for any radical, including cube roots and higher.

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