04 / 10 · Square roots

Roots that never end

The spiral of Theodorus

Draw a square with sides of length 1. Its diagonal cuts it into two right triangles, and by Pythagoras the diagonal is √(1² + 1²) = √2. So √2 is a real length you can draw with a ruler.

Yet you can never write it down exactly. √2 = 1.41421356… and the digits go on forever without repeating. The Greeks proved it cannot be any fraction a/b. Numbers like this are called irrational.

hypotenuse² = a² + b²
(√n)² + 1² = n + 1  →  next diagonal = √(n + 1)
√2 = 1.41421356237…

Try it · Add triangles to grow the spiral. The newest diagonal is highlighted with its length. Hover any spoke to read its root.

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Theodorus of Cyrene found a lovely way to draw many roots at once. Stand a new triangle with a side of 1 on the last diagonal, again and again. Each new diagonal is the next root: √2, √3, √4 = 2, √5… The triangles curl into a spiral.

Only perfect squares give whole-number roots. Every other whole number has an irrational root.

Key ideas

Words worth remembering

Pythagoras' theorem: In a right triangle, the square on the long side equals the sum of the squares on the other two.
Irrational number: A number that cannot be written as a fraction; its decimals never end or repeat.

Think about it

Which spokes of the spiral have whole-number lengths? What do those numbers have in common?