Prime (symbol)

The prime symbol , double prime symbol , triple prime symbol , and quadruple prime symbol are used to designate units and for other purposes in mathematics, science, linguistics and music. Although the characters differ little in appearance from those of the apostrophe and single and double quotation marks, the uses of the prime symbol are quite different. While an apostrophe is now often used in place of the prime, and a double quote in place of the double prime (due to the lack of prime symbols on everyday writing keyboards), such substitutions are not considered appropriate in formal materials or in typesetting. Positional notation#Sexagesimal systemThe prime symbol is commonly used to represent feet (ft), and the double prime is used to represent inches (in). The triple prime as used in watchmaking represents a ligne ( of a French inch or pouce, about ). Primes are also used for angles. The prime symbol is used for arcminutes ( of a degree), and the double prime for arcseconds ( of an arcminute). As an angular measurement, means 3 degrees, 5 arcminutes and 30 arcseconds. In historical astronomical works, the triple prime was used to denote "thirds" ( of an arcsecond) and a quadruple prime "fourths" ( of a third of arc), but modern usage has replaced this with decimal fractions of an arcsecond. Primes are sometimes used to indicate minutes, and double primes to indicate seconds of time, as in the John Cage composition 4′33′′ (spoken as "four thirty-three"), a composition that lasts exactly 4 minutes 33 seconds. This notation only applies to duration, and is seldom used for durations longer than 60 minutes. In mathematics, the prime is generally used to generate more variable names for similar things without resorting to subscripts, with x′ generally meaning something related to (or derived from) x. For example, if a point is represented by the Cartesian coordinates (x, y), then that point rotated, translated or reflected might be represented as (x′, y′).