Every factor is a record of a specific derivation chain
This site's Methodology page explains which figures are exact by definition and which are current best measurements. This article goes one level deeper: how you actually derive a conversion factor from first principles, and why doing that derivation explicitly, rather than looking up a pre-computed number, is the only way to know how much precision you are entitled to claim in the result.
Chaining exact definitions
Three definitions anchor almost every customary-to-metric conversion on this site: the international inch, exactly 25.4 mm; the international pound, exactly 0.45359237 kg; and standard gravity, exactly 9.80665 m/s², fixed by the 3rd General Conference on Weights and Measures in 1901 for precisely the purpose of converting between mass and force. Every other customary unit's conversion factor is a multiplication chain built on these three numbers plus simple integer ratios — 12 inches per foot, 3 feet per yard, 5,280 feet per mile, 16 ounces per pound. Because every step in the chain is exact, the final result is exact too, no matter how many steps it takes. A chain of exact multiplications never accumulates rounding error, unlike a chain of measured ones.
Worked derivation: the watt, from a horsepower, from three defined constants
Mechanical horsepower is defined as 550 foot-pounds-force per second, a unit of power built from a unit of force, the pound-force, that itself depends on standard gravity. Deriving its SI equivalent from nothing but the three definitions above:
- 1 pound-force = 1 lb × standard gravity = 0.45359237 kg × 9.80665 m/s² = 4.4482216152605 N (exact — two exact numbers multiplied together)
- 1 foot-pound-force = 1 ft × 1 lbf = 0.3048 m × 4.4482216152605 N = 1.3558179483314 J (exact)
- 1 horsepower = 550 ft·lbf/s = 550 × 1.3558179483314 W = 745.69987158227 W (exact)
That figure, 745.69987158227 W, is not a measurement or an approximation to some "true" horsepower sitting in a museum somewhere. Horsepower was never redefined by an international treaty the way the yard and pound were; it is simply whatever 550 ft·lbf/s comes out to once you fix the length, the mass and the gravity it is built from. Change any one input, use a locally measured gravity instead of the standard value, say, and the "horsepower in watts" figure changes with it — which is exactly why standard gravity had to be fixed by definition in 1901, rather than measured on-site.
Measured factors: the other kind
Not every factor on this site is built this way. The clearest exception is carbon emissions: converting kWh to kg CO2e uses a grid emissions factor, currently 0.445 kg CO2e per kWh, the IEA's 2024 global average, see the Editorial Policy page for how and when that figure gets revisited, which is a measured, time-varying real-world quantity, not a fixed definition. No number of decimal places changes that. The correct response to a measured factor is to state its source and vintage, not to present it with the same certainty as 25.4 millimetres per inch.
The 2019 inversion: what changed, and what it means for a chain
Before 20 May 2019, a mass-based derivation chain like the one above could stop at "kilogram," because a physical platinum-iridium cylinder in Sèvres was the kilogram by definition — anything derived from it, like the pound-force above, inherited that certainty for free. What was not certain was the Planck constant, h, which had to be measured against that physical kilogram and carried laboratory uncertainty in its eighth significant figure.
Since the 2019 redefinition, that relationship inverted. The Planck constant is now fixed by definition at exactly 6.62607015 × 10-34 J·s, and the kilogram is whatever mass makes a Kibble balance, an instrument that balances a mechanical weight against an electromagnetic force derived from the Josephson and quantum Hall effects, both of which depend on h, read exactly consistent with that fixed constant. The kilogram is now realised experimentally, at national metrology institutes, to about one part in 108. Every mass-based conversion factor on this site, including the pound-force derivation above, technically inherits that small, real uncertainty now, whereas before 2019 it inherited none. In practice, one part in 108 is far below any precision this site or its readers need, but it is worth knowing precisely what changed: "exact by definition" migrated from the kilogram to the Planck constant, it did not simply get more precise.
How to check your own chain
Two checks catch almost every derivation error. Units must cancel algebraically at every step; if they do not, a step used the wrong quantity, not just the wrong number. And converting a value forward through the chain and back through it in reverse should return the exact original value, to within ordinary floating-point rounding; if it does not, a step in the chain is not actually the reciprocal of its partner, and the discrepancy tells you which step to re-examine.
Why this matters beyond curiosity
A conversion factor with no visible derivation is a black box: correct until someone asks what precision it is entitled to, at which point there is no way to answer without redoing the work. A factor built by chaining exact definitions can state its precision honestly — unlimited, for a pure unit-definition chain, or bounded by a specific published source, for a measured one. That is the practical reason to derive rather than look up: not academic completeness, but knowing which of the two situations you are actually in before you build something on top of the number.