# Powers of ten and physical units

A power of ten moves the decimal point. A measurement is a number together with its unit, and the unit goes through every step of a calculation.

Written for this edition, not translated from Einstein.

## How do you write a very small length without counting zeros?

The Brownian paper works its example, in §5, for particles 0.001 mm across. In metres that is 0.000001 m: easy to miscount by one zero, which is a factor of ten. Written as 10⁻⁶ m, the −6 says the decimal point has moved six places to the left, and that length has a name of its own, the micrometre (μm).

The unit goes through the arithmetic with the number. A micrometre is 10⁻⁶ m, so a square micrometre is 10⁻¹² m², not 10⁻⁶ m²: squaring a length squares its unit, power of ten included.

## Worked example: Squaring six micrometres

1. 1 micrometre (1 μm) is 10⁻⁶ metres, or 0.000001 m.
2. The paper's displacement after one minute, about 6 μm, is 6 × 10⁻⁶ m.
3. Squaring it squares the number and the unit: (6 × 10⁻⁶ m)² = 36 × 10⁻¹² m² = 3.6 × 10⁻¹¹ m².

## Where this lesson stops

A measurement is a number and a unit together. Squaring it squares both: (6 × 10⁻⁶ m)² is 3.6 × 10⁻¹¹ m².

## This lesson builds on

- [Fractions and ratios](/foundations/bridge-fractions-ratios/)