MathsSeeing how numbers work3 / 10▾

  1. Square roots

    01What is a square root?
  2. 02Perfect squares
  3. 03Heron's method
  4. 04Roots that never end

  5. Probability

    05What is probability?
  6. 06The law of large numbers
  7. 07Two dice
  8. 08Bayes' theorem
  9. 09The bell curve
  10. 10Three meanings of probability

03 / 10 · Square roots

Heron's method

Squeezing a rectangle into a square

How does a calculator find √20? One of the oldest methods, used in Babylon and written down by Heron of Alexandria around 60 AD, is still close to what computers do.

Start with any guess, say 2. Build a rectangle with area 20 and one side 2. The other side must be 20 ÷ 2 = 10. A 2 × 10 rectangle is far from square: the true side is somewhere between 2 and 10.

guess x   →   other side = N ÷ x
next guess = (x + N ÷ x) ÷ 2

N = 20:  2 → 6 → 4.667 → 4.476 → 4.4721

Try it · Pick a number and a first guess, then press Next step. Watch the rectangle turn into a square and the digits settle in the table.

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So take the average of the two sides: (2 + 10) ÷ 2 = 6. Now build a 6 × 3.33 rectangle. Still not square, but closer. Average again: 4.667. Then 4.476. Then 4.4721…

Each step roughly doubles the number of correct digits. After four or five steps the rectangle is a square to within the width of a hair, and its side is √20.

Key ideas

Words worth remembering

Iteration
Repeating the same step, each time starting from the last answer.
Convergence
Getting closer and closer to a fixed answer as the steps go on.

Think about it

Why does averaging the two sides always land between the guess and the true root?