Units & the SI System
A measurement is a number times a unit, and the world agrees on seven base units.
Lesson at a glanceShow the key points
- A measured value = (number) × (unit); the unit is an agreed standard of that quantity.
- The General Conference on Weights and Measures (CGPM) fixes units by international agreement.
- Fundamental (base) quantities are independent; all others are derived from them by multiplying and dividing. SI uses seven: length (m), mass (kg), time (s), current (A), temperature (K), amount of substance (mol), luminous intensity (cd).
- Plane angle (radian, rad) and solid angle (steradian, sr) have their own named units.
- SI prefixes: exa 10¹⁸, peta 10¹⁵, tera 10¹², giga 10⁹, mega 10⁶, kilo 10³, hecto 10², deka 10¹, deci 10⁻¹, centi 10⁻², milli 10⁻³, micro 10⁻⁶, nano 10⁻⁹, pico 10⁻¹², femto 10⁻¹⁵, atto 10⁻¹⁸.
- Good standards are invariable and available. Since 20 May 2019 every SI base unit is defined by fixing the value of a constant of nature (Cs frequency, c, h, e, k, N_A, K_cd).
“The station is 5 away.” Five what? Steps, kilometres, minutes? A number alone says nothing. Physics is quantitative, so every measurement needs an agreed unit.
Number times unit
To measure a quantity we compare it with a standard amount of the same kind, the unit. If a unit of mass is agreed, and an elephant is 2000 units while a goat is 40, the elephant is 50 times heavier. A measured value therefore always has two parts:
, for example .
Who decides the units?
A unit is useless unless everyone uses the same one. By international treaty, the General Conference on Weights and Measures (CGPM, from its French name) decides the standard units and publishes any changes.
Fundamental and derived quantities
Not every quantity needs its own independent unit. Once you have a unit of length, a unit of area follows: the area of a square one unit on each side. A unit of length and a unit of time give a unit of speed. So we choose a small set of fundamental (base) quantities that are
- independent of one another, and
- enough to express every other quantity by multiplying and dividing.
Everything else is a derived quantity, with a derived unit. The choice of base quantities is a convention (one could take speed and time as basic, making length derived), and different systems have made different choices. Seven turn out to be enough for all of physics.
The International System (SI)
Adopted by the CGPM in 1971, the Système International d’Unités (SI) is used across the world. Its seven base quantities and units:
| Quantity | Unit | Symbol |
|---|---|---|
| Length | metre | m |
| Mass | kilogram | kg |
| Time | second | s |
| Electric current | ampere | A |
| Thermodynamic temperature | kelvin | K |
| Amount of substance | mole | mol |
| Luminous intensity | candela | cd |
Two more named units cover angles: the radian (rad) for plane angle and the steradian (sr) for solid angle.
Prefixes
Nature spans huge ranges: the separation of two protons in a nucleus is about 10⁻¹⁵ m, the distance to a quasar about 10²⁶ m. Standard prefixes stand for powers of ten:
| Power | Prefix | Symbol | Power | Prefix | Symbol |
|---|---|---|---|---|---|
| 10¹⁸ | exa | E | 10⁻¹ | deci | d |
| 10¹⁵ | peta | P | 10⁻² | centi | c |
| 10¹² | tera | T | 10⁻³ | milli | m |
| 10⁹ | giga | G | 10⁻⁶ | micro | µ |
| 10⁶ | mega | M | 10⁻⁹ | nano | n |
| 10³ | kilo | k | 10⁻¹² | pico | p |
| 10² | hecto | h | 10⁻¹⁵ | femto | f |
| 10¹ | deka | da | 10⁻¹⁸ | atto | a |
Write (a) 0.000 004 7 m, (b) 3 200 000 000 Hz and (c) 5 × 10⁻¹⁰ s using SI prefixes.
What makes a good standard?
The length from elbow to fingertip (the old cubit) fails the first test: it differs from person to person. For over a century the kilogram was a platinum–iridium cylinder kept near Paris; tiny drifts in its mass compared with its copies exposed the weakness of a physical object.
- 1A measurement = a number × a unit; units are agreed internationally by the CGPM.
- 2Base quantities are independent; derived quantities are built from them. SI has seven base units: m, kg, s, A, K, mol, cd (plus rad and sr for angles).
- 3Prefixes from atto (10⁻¹⁸) to exa (10¹⁸) handle huge and tiny values.
- 4Good standards are invariable and available; since 2019 all SI units are fixed by constants of nature.
Check yourself
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