# Squares and square roots

Squaring multiplies a number by itself, and the square root undoes it. Four times a square is the square of twice the number, so four times a mean square is twice a root mean square.

Written for this edition, not translated from Einstein.

## If the mean square grows four times, why does the typical distance only double?

Both +3 and −3 have square 9, so squaring removes the sign, and it weights a large distance more than a small one. A square root returns to the original unit: the square root of a square micrometre is a micrometre.

The root mean square, or RMS, uses both steps: square each displacement, average the squares, then take the square root. The result is a typical distance from the start, in the original unit.

$$
\sqrt{4q}=2\sqrt q\quad(q\ge 0)
$$

The square root of four q is twice the square root of q, for nonnegative q.

## Worked example: Doubling every displacement, and what the RMS does

1. The displacements −3, −1, +1, +3 have squares 9, 1, 1, 9. Their average is 5, so the RMS is √5, about 2.236.
2. Double each displacement: −6, −2, +2, +6. The squares are 36, 4, 4, 36, and their average is 20.
3. The mean square went from 5 to 20, four times as large. The RMS went from about 2.236 to √20, about 4.472: twice as large.

## Where this lesson stops

Four times the mean square is twice the RMS. That is why, in the Brownian paper, waiting four times as long doubles the typical distance.

## This lesson builds on

- [Adding and averaging](/foundations/bridge-sum-average/)