# Squares and square roots

Authored explanation; editorial review pending.

Both +3 and −3 have square 9. Squaring therefore removes the sign while giving greater weight to a larger magnitude. A square root returns the result to the original kind of unit: the square root of a square micrometre is a micrometre.

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

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

## Worked example

The mean square of −3, −1, +1, +3 is 5. Doubling each displacement gives −6, −2, +2, +6, whose mean square is 20. The RMS changes from approximately 2.236 to 4.472: twice as large, not four times as large.

Use the nonnegative square root when reporting a magnitude.