The Irrationality of √2

One of the earliest and most profound results in mathematics is the discovery that no fraction can ever square to give exactly 2. The ancient Greeks — traditionally the Pythagoreans — proved that there is no rational number whose square is precisely 2:

$\text{There is no } \frac{p}{q} \in \mathbb{Q} \text{ such that } \left(\frac{p}{q}\right)^{\!2} = 2$

This result shattered the Pythagorean belief that all of reality could be expressed through whole-number ratios, and it opens the door to the real number continuum — a theme Penrose develops in Chapter 3 of The Road to Reality.

The Number Line

Use the slider below to increase the maximum denominator and watch rational fractions crowd ever closer to $\sqrt{2}$ — without ever reaching it.

The Integer Lattice

The lattice view shows the geometric picture: the curve $p = q\sqrt{2}$ threads endlessly between integer grid points, never passing through one.

The Proof by Contradiction

Suppose for contradiction that $\sqrt{2} = p/q$ where $p$ and $q$ are positive integers with no common factor. Squaring both sides gives:

$p^2 = 2q^2$

Since $p^2$ is even, $p$ itself must be even — say $p = 2k$. Substituting:

$4k^2 = 2q^2 \quad\Longrightarrow\quad q^2 = 2k^2$

So $q$ is also even. But then $p$ and $q$ share the factor 2, contradicting our assumption that the fraction was in lowest terms. Therefore no such fraction exists. $\blacksquare$

Mathematical Insight

What makes this result so striking — and so important for Penrose's narrative — is that it reveals the existence of “gaps” in the rational number line. No matter how finely you subdivide fractions, $\sqrt{2}$ lives in a hole between them. Filling these holes leads to the construction of the real numbers $\mathbb{R}$, which form the backbone of calculus, geometry, and modern physics.

The lattice view makes this vivid: the line $p = q\sqrt{2}$ has irrational slope, so it can never pass through a point with two integer coordinates. Near-misses like $7/5$, $99/70$, and $577/408$ are governed by the theory of continued fractions — the “best rational approximations” to $\sqrt{2}$.