Field Book: The Remainder

The remainder is this site's name for the part of a measurement convention that does not match the thing it was meant to fit. A calendar is meant to track the seasons, a unit is meant to equal some reference, and a flat map is meant to stand for a curved surface. Where the fit is not complete, what is left over is the remainder.

The term is a figure of speech, not the remainder of a division in arithmetic, and not a claim that a convention has to leave a gap. It points at one particular mismatch and says what kind it is.

Four kinds

When an article names a remainder, it says which of four kinds it is, and never blends two.

Periodic adjustment: a calendar year and the year of the seasons do not divide evenly into each other, so leap days or leap months make up the difference; leap seconds have done the same job for the gap between UTC and UT1, the time scale set by Earth's rotation.

Rounding: a definition or conversion keeps some digits and drops the rest, as when the inch was fixed in 1959 at 25.4 mm [exact by definition].

Measurement error: a historical measurement differs from a later, more accurate one. When that kind of error ends up written into a definition, this site calls it the frozen error.

Projection trade-off: no flat map can keep area, shape, distance and direction all at once, so every projection keeps some of these and gives up others. A map that keeps area gives up shape; it does not throw area away.

An example

The leap-year rule shows the first kind. A calendar of 365 whole days leaves a fraction of a day over each year, and the Gregorian rule handles it with February 29 in years divisible by 4, skipped in century years not divisible by 400. Even so, the calendar's average year runs about 11 seconds long against the vernal equinox year [estimate, range 10.8–11.7 s] and about 27 seconds long against the mean tropical year [estimate, range 26.4–27.2 s]. The 400-year cycle is fixed, so that gap builds up rather than being corrected. The full working is in the leap-year article.

When there is none

Not every topic leaves a remainder, and an article does not invent one. Each article ends with a line headed “Carry the remainder:”. When there is a remainder, that line names its kind and where it sits. When there is none, the line says instead what the convention chose and what it gave up.

The remainder is one of three terms in this site's field book, alongside the unstated per and the frozen error.