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| title | tags | summary |
|---|---|---|
| Units, dimensions and conversion | units, dimensions, si, conversion | How physical quantities carry their unit, why a unit is a type rather than a label, and the conventions that make an arithmetic answer meaningful. |
Units, dimensions and conversion
A number without a unit is not a physical quantity. "3" might be three metres, three seconds, or three kilometres per hour, and the three do not describe the same thing. The purpose of a unit system is to make an arithmetic result interpretable, which means the unit is not a note attached to a value but part of the value itself.
Definition
A quantity is a magnitude together with a unit. The dimension is the kind of thing measured, such as length, time, or mass; a unit is a particular choice of magnitude within that dimension.
The base dimensions of the International System of Units, and the units chosen for them, are:
| Dimension | Unit | Defined by |
|---|---|---|
| Length | metre | Speed of light in vacuum |
| Mass | kilogram | Planck constant |
| Time | second | Cesium hyperfine transition |
| Electric current | ampere | Elementary charge |
| Thermodynamic temperature | kelvin | Boltzmann constant |
| Amount of substance | mole | Avogadro constant |
| Luminous intensity | candela | Defined by the other six |
Everything else is derived: newton from kilogram, metre and second; joule from newton and metre; watt from joule and second.
Mathematics
A unit is a conversion factor to a base unit. Length may be expressed in metres, kilometres or feet; each is a number multiplied by a factor to give metres. Dimensionally aware arithmetic keeps track of that factor, so
1\,\text{km} + 500\,\text{m} = 1500\,\text{m}, \quad \text{never } 1 + 500 = 501
The rule is that the conversion factor travels with the value. A quantity of type length carries the factor to metres with it, so adding it to another length applies both factors at once.
A dimension behaves like a type in a type system. Adding a length to a time is a dimension error, in the same way that adding a string to an integer would be one in a compiler that tracked those types.
Scaling laws
Dimension reasoning earns its keep in scaling arguments, where an error shows
up as an impossible exponent. The gravitational perturbation at distance r
from a body of mass M must scale as
a \propto \frac{G M}{r^2}
because G has units of \text{m}^3\,\text{kg}^{-1}\,\text{s}^{-2}; any other combination either does
not have the dimensions of an acceleration or does have them only for a
particular case, which is the signal that the argument is incomplete.
Unit conventions
- Angles. An angle is dimensionless in principle, because a full turn is
one. It is written in radians in most formulae, and in degrees, arcminutes
and arcseconds where it is read by a person. The relationship is
180 / \pidegrees per radian. A rotation described in axis and angle uses the angle in a plane and the axis through the pole; the two conventions differ, which is why a rotation is usually written explicitly as right-hand or left-hand. - Prefixes. SI prefixes are powers of ten: kilo is
10^3, mega10^6, micro10^{-6}. A kilometre is a thousand metres, never a thousand miles. - Binary prefixes. A mebibyte is
2^{20}bytes, not10^6. The distinction matters wherever a size is compared against a capacity. - Units in the field. Orbital work is written in kilometres and seconds, not metres, because the quantities are large and the models are fitted to those units. A propagation model expressed in metres and a catalogue expressed in kilometres will disagree by a factor of a thousand, silently.
Applicability
Unit arithmetic prevents a class of error that is otherwise easy to make and hard to see, because the numbers stay plausible. It does not prevent a different class: the same unit used with two different conventions. A height above the equatorial radius and a height above the ellipsoid differ by tens of kilometres; both are lengths, and neither is wrong.
The measure a quantity is taken against matters as much as its unit. An epoch expressed in seconds from an unspecified origin is not comparable with one expressed in a Modified Julian Date, even though both are times.
Standards
- SI: The International System of Units (SI), BIPM, most recent edition
- Non-SI units accepted for use with SI: the same document, section 4
- Earth reference models: NIMA TR8350.2 for WGS 84
See also
- numerical-methods for what the numbers are computed with
- time-scales for a quantity whose unit hides an entire convention
- geodesy-wgs84 for the two measures a height is taken against