Foundation · Explanatory preview
Squares and square roots
Squaring multiplies a number by itself. The nonnegative square root reverses that operation.
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Why does four times a mean square mean twice the RMS?
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.
The square root of four q is twice the square root of q, for nonnegative q.
One 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.
A stopping point: Use the nonnegative square root when reporting a magnitude.
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