Foundation lesson
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.
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
- The displacements −3, −1, +1, +3 have squares 9, 1, 1, 9. Their average is 5, so the RMS is √5, about 2.236.
- Double each displacement: −6, −2, +2, +6. The squares are 36, 4, 4, 36, and their average is 20.
- 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.
Four times the area, twice the side
Each cell is the mean square of the four displacements, 5. The big square is the mean square after every displacement is doubled, 20. Four cells fit in it, and its side is two cell sides.
What it shows, in words
The displacements -3, -1, 1, 3 have squares 9, 1, 1, 9, and their average is 5. That is one cell. Doubling every displacement gives a mean square of 20, which is the whole square, and four cells fill it exactly.
The cell’s side is 2.236 and the whole square’s side is 4.472. Four times the area, twice the side: the side did not grow by four, because a side is a length and the 20 is an area. That is the whole difference between a mean square and a root mean square, and it is why waiting four times as long in the Brownian paper doubles the typical distance rather than quadrupling it.
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.
If you came here from a passage, Back returns you to the exact place you left.