Units and MeasurementThe International System of Units
Until quite recently, scientists in different countries used different systems of units. Three were in wide use, and they differed in their base units of length, mass and time:
| System | Length | Mass | Time |
|---|---|---|---|
| CGS | centimetre | gram | second |
| FPS (British) | foot | pound | second |
| MKS | metre | kilogram | second |
Today the whole world uses one system: the Système International d'Unités (French for International System of Units), or SI for short. Its scheme of symbols, units and abbreviations was set up by the International Bureau of Weights and Measures (BIPM) in 1971, and revised by the General Conference on Weights and Measures in November 2018. It is used in science, technology, industry and trade everywhere. SI is a decimal system, so converting inside it is just a matter of moving the decimal point: 1 km = 1000 m, 1 mg = 0.001 g. This chapter, and the rest of the book, uses SI units.
Seven Base Units
For 130 years the kilogram was an object: a small cylinder of platinum–iridium kept in a vault near Paris. Every balance in the world was traced back to it. But objects can wear away or pick up dust, so in 2019 the kilogram, the ampere, the kelvin and the mole were redefined using fixed constants of nature. Now any well-equipped laboratory can realise them.
SI has seven base units (Table 1.1). Each one is now fixed by giving an exact value to one of seven constants of nature, and the base units are built from those constants.
Tap the units to see what fixes each one.
Table 1.1 — SI base quantities and units
| Base 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 |
Note
The exact numbers in the definitions need not be learnt and are not asked in tests: they only show how precisely the units are pinned down. When you use the mole, always say what is being counted: atoms, molecules, ions, electrons or some other particles.
All other units are derived from these seven (Appendix A 6 of your textbook). Some derived units have special names, such as the newton (N), joule (J), watt (W) and pascal (Pa). The prefixes for multiples and sub-multiples (kilo, milli, micro …) are in Appendix A 2, and the rules for writing symbols are in Appendices A 7 and A 8.
Plane Angle and Solid Angle
Besides the seven base units, SI defines units for two kinds of angle (Fig. 1.1 in your textbook). A plane angle dθ is the length of an arc ds divided by the radius r of the circle:
Its unit is the radian (symbol rad).
Drag the slider and watch the amber arc and the two radii.
Arc length ds =
A whole circle has an arc of 2πr, so it makes 2πr ÷ r = 2π rad: 360° = 2π rad and 1° = π/180 rad. Since the radian is a length divided by a length, it has no dimensions.
A solid angle dΩ is the same idea in three dimensions. A cone from the centre O of a sphere of radius r cuts a patch of area dA out of its surface, and dΩ = dA ÷ r². Its unit is the steradian (sr). Like the radian, it is dimensionless.
The whole surface of a sphere, 4πr², makes 4π sr, so a hemisphere subtends 2π ≈
Units Outside SI
Some units that are not part of SI are still used every day, in markets, farms, hospitals and laboratories. You should be able to convert them to SI.
Table 1.2 — some units retained for general use (outside SI)
| Name | Symbol | Value in SI unit |
|---|---|---|
| minute | min | 60 s |
| hour | h | 60 min = 3600 s |
| day | d | 24 h = 86 400 s |
| year | y | 365.25 d = 3.156 × 10⁷ s |
| degree | ° | 1° = (π/180) rad |
| litre | L | 1 dm³ = 10⁻³ m³ |
| tonne | t | 10³ kg |
| carat | c | 200 mg |
| bar | bar | 0.1 MPa = 10⁵ Pa |
| curie | Ci | 3.7 × 10¹⁰ s⁻¹ |
| roentgen | R | 2.58 × 10⁻⁴ C/kg |
| quintal | q | 100 kg |
| barn | b | 100 fm² = 10⁻²⁸ m² |
| are | a | 1 dam² = 10² m² |
| hectare | ha | 1 hm² = 10⁴ m² |
| standard atmospheric pressure | atm | 101 325 Pa = 1.013 × 10⁵ Pa |
The SI prefixes turn one unit into a whole ladder of units, each a power of ten apart:
| Prefix | Symbol | Power |
|---|---|---|
| exa | E | 10¹⁸ |
| peta | P | 10¹⁵ |
| tera | T | 10¹² |
| giga | G | 10⁹ |
| mega | M | 10⁶ |
| kilo | k | 10³ |
| hecto | h | 10² |
| deca | da | 10¹ |
| Prefix | Symbol | Power |
|---|---|---|
| deci | d | 10⁻¹ |
| centi | c | 10⁻² |
| milli | m | 10⁻³ |
| micro | µ | 10⁻⁶ |
| nano | n | 10⁻⁹ |
| pico | p | 10⁻¹² |
| femto | f | 10⁻¹⁵ |
| atto | a | 10⁻¹⁸ |
Tip
A prefix belongs to the unit, so a power applies to the prefix too: 1 cm² = (10⁻² m)² = 10⁻⁴ m², not 10⁻² m². In the same way 1 hectare = 1 hm² = (100 m)² = 10⁴ m².
Speed Round: Prefixes and Units
Prefixes are powers of ten; Table 1.2 converts the everyday units. Get five right to move on.