Metre ($\mathrm{m}$)

The metre, symbol $\mathrm{m}$, is the SI base unit of measurement for length. It is defined by fixing the speed of light in vacuum to $299792458\mathrm{m}{\mathrm{s}}^{-1}$, with the second defined independently. Thus light in vacuum travels one metre in $1/299792458$ of a second.

The exact defining constant does not make every measured length exact. Realizing a length standard can involve optical interferometry, frequency references, alignment and corrections for propagation through air; their imperfections contribute to measurement uncertainty. The definition specifies the reference, whereas an experiment supplies an imperfect realization.^[ref]

A micrometre, symbol $\mu \mathrm{m}$, is a unit of measurement for length equal to one millionth of a metre:The SI prefix micro supplies the exact factor $10^{-6}$.^[ref]

For an invented measurement example, a metal foil of thickness $25\mu \mathrm{m}$ has thickness $0.025\mathrm{mm}$. Forty such layers have combined thickness $1000\mu \mathrm{m}=1\mathrm{mm}$ if they lie flat without gaps. The calculation makes a small length tangible without assuming a particular measuring instrument.

Unit conversion does not remove measurement uncertainty. A reported thickness of $(25\pm 1)\mu \mathrm{m}$ becomes $(0.025\pm 0.001)\mathrm{mm}$: both the value and its uncertainty receive the same exact scale factor.

A decimetre, symbol $\mathrm{dm}$, is exactly one tenth of a metre:The SI prefix deci fixes the factor $10^{-1}$. This exact relation concerns units; the uncertainty of a measured length still depends on the measurement.^[ref]

The decimetre connects familiar lengths to volume particularly neatly. A cube with edges $1\mathrm{dm}$ has volumeThus a litre is exactly a cubic decimetre, while a cubic metre contains 1,000 litres. The exponent acts on the conversion factor as well as the unit: a cubic decimetre is not one tenth of a cubic metre.^[ref]

For example, a rectangular container measuring $2\mathrm{dm}$ by $3\mathrm{dm}$ by $0.5\mathrm{dm}$ has volume $3\mathrm{d}{\mathrm{m}}^3=3\mathrm{L}$, assuming the stated dimensions describe its interior. A length conversion becomes a cubic conversion when used to compute volume.

A millimetre, symbol $\mathrm{mm}$, is exactly one thousandth of a metre: $1\mathrm{mm}=10^{-3}\mathrm{m}$. The SI prefix milli specifies that exact scale factor; it does not state the uncertainty of a measurement expressed in the unit.^[ref]

For example, a thickness of $0.25\mathrm{mm}$ is $250\mu \mathrm{m}$ or $2.5\times 10^{-4}\mathrm{m}$. Powers of a unit require powers of the conversion factor: $1{\mathrm{mm}}^2=10^{-6}{\mathrm{m}}^2$ and $1{\mathrm{mm}}^3=10^{-9}{\mathrm{m}}^3$. Thus a cube with 10 millimetre edges has volume $1000{\mathrm{mm}}^3=1{\mathrm{cm}}^3$, not $1000{\mathrm{cm}}^3$. This distinction prevents large errors when moving between microscopic lengths and areas or volumes.

A nanometre is $10^{-9}$ metres. The prefix nano denotes the exact decimal factor $10^{-9}$. A length of $500\mathrm{nm}$ is $5\times 10^{-7}\mathrm{m}$, a typical visible-light wavelength.^[ref]

The unit is convenient for comparing molecules, thin films and optical wavelengths. A smaller number of nanometres does not by itself identify a different physical regime: the relevant comparison may be with an atomic spacing, a mean free path or a wavelength, depending on the experiment.

A femtometre is $10^{-15}$ metre, written $\mathrm{fm}$. It is a convenient unit of measurement for the dimensions of an atomic nucleus, rather than the much larger dimensions of a chemical atom. For example, a length of $5\mathrm{fm}$ is $5\times 10^{-15}\mathrm{m}$.

Scattering probes such small lengths through momentum transfer rather than a miniature mechanical ruler. The relation between wavelength and momentum explains why increasingly short-distance structure requires increasingly large momentum transfer.