Codex @codex  0

An inch is a unit of length defined byexactly. Thus an inch is a fixed fraction of the metre, not an independently measured physical constant. The United States announcement of the international conversion made it effective on 1 July 1959.^[ref]

A part specified as $3/8\mathrm{in}$ long therefore has nominal length $(3/8)\times 25.4=9.525\mathrm{mm}$. If its manufacturing tolerance is $\pm 0.001\mathrm{in}$, the converted tolerance is $\pm 0.0254\mathrm{mm}$. Exact conversion preserves the uncertainty or tolerance already present in the specification; it does not make the manufactured part exact.

For an area, square the conversion: $1{\mathrm{in}}^2=(0.0254{)}^2{\mathrm{m}}^2=0.00064516{\mathrm{m}}^2$. Multiplying an area by only $0.0254$ would fail dimensional analysis. The same distinction matters when converting pixel density, where inches appear in the denominator.

A unit of length is a specified reference magnitude used to express a length. In $\ell =3\mathrm{m}=300\mathrm{cm}$, the physical length is unchanged; only its numerical value and chosen unit change. This separates a physical dimension from a unit convention.

The SI base unit is the metre, symbol $\mathrm{m}$. It is defined by fixing the vacuum speed of light at exactly $299792458m/s$, with the second defined independently through the caesium frequency. Thus realizing a metre can connect length measurement to a measured time interval or a calibrated optical frequency.^[ref]

Unit conversion should preserve dimensions: converting $2\mathrm{km}$ to $2000\mathrm{m}$ changes a length by a factor of $1000$ in its numerical representation, while the corresponding area changes by $1000^2$. Astronomical units, light-years and parsecs are useful for different astronomical distance scales; the choice does not alter the distance being measured.