Foundation lesson
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 micrometre (1 μm) is 10⁻⁶ metres, or 0.000001 m.
- The paper's displacement after one minute, about 6 μm, is 6 × 10⁻⁶ m.
- Squaring it squares the number and the unit: (6 × 10⁻⁶ m)² = 36 × 10⁻¹² m² = 3.6 × 10⁻¹¹ m².
The places, counted, and what squaring does to them
Each step along either ruler is one place after the decimal point, and one factor of ten. Both rulers are drawn at the same scale.
A square micrometre sits at twelve places along the lower ruler. Six is where the micrometre itself sits, one ruler up.
What it shows, in words
The upper ruler counts places after the decimal point in metres. The centimetre is two places along, the millimetre three and the micrometre six, so a micrometre is 10⁻⁶ m, or 0.000001 m. A reader who loses count by one rung has the length wrong by a factor of ten.
The lower ruler counts the same places for areas and is twice as long. The micrometre at six places becomes a square micrometre at twelve, 10⁻¹² m². This is why the Brownian paper's displacement after a minute, about 6 μm or 6 × 10⁻⁶ m, has a square of 3.6 × 10⁻¹¹ m² rather than anything measured in 10⁻⁶ m². Squaring a measurement squares the number and the unit together, the power of ten included.
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.
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