Definitions

# Prime (symbol)

The prime symbol ( ′ ), double prime symbol ( ″ ), triple prime symbol ( ‴ ) etc. are used to designate several different units, and for various other purposes in mathematics, the sciences and linguistics. The prime symbol should not be confused with the apostrophe, single quotation mark or acute accent; the double prime should not be confused with the double quotation mark.

## Designation of units

The prime symbol is commonly used to represent feet, arcminutes and sometimes minutes of time. The double prime represents inches, arcseconds and sometimes seconds of time. Thus, 3′ 5″ could mean 3 feet and 5 inches ("three foot five"), or 3 minutes and 5 seconds (of time). As an angular measurement, 3° 5′ 30″ means 3 degrees, 5 arcminutes and 30 arcseconds.

In watchmaking the triple prime represents a ligne.

## Use in mathematics, statistics, and science

In mathematics, the prime is generally used to generate more variable names for things which are similar, without resorting to subscripts—x′ generally means something related to x. For example, if a point is represented by the Cartesian coordinates (xy), then that point rotated, translated or reflected might be represented as (x′, y′). The prime symbol is not related to prime numbers.

Usually, the meaning of x′ is defined when it is first used, but sometimes its meaning is assumed to be understood:

• A derivative or derived function: f′(x) and f″(x) are the first and second derivatives of f(x) with respect to x. Similarly, if y = f(x) then y′ and y″ are the first and second derivatives of y with respect to x. (Other notation exists.)
• Set complement: A′ is the complement of the set A (other notation exists)
• The negation of an event in probability theory: Pr(A′) = 1 − Pr(A) (other notation exists)
• The result of a transformation: Tx = x

In physics, the prime is used to denote variables after an event. For example, vA′ would indicate the velocity of object A after an event. It is also commonly used in relativity: The event at (x, y, z, t) in frame S has coordinates (x′, y′, z′, t′) in frame S′.

In chemistry, it is used to distinguish between different functional groups connected to an atom in a molecule, such as R and R′, representing different alkyl groups in a ketone.

In molecular biology, the prime is used to denote the positions of carbon on a ring of deoxyribose or ribose. The prime distinguishes places on these two chemicals, rather than places on other parts of DNA or RNA, like phosphate groups or nucleic acids. Thus, when indicating the direction of movement of an enzyme along a string of DNA, biologists will say that it moves from the 5′ end to the 3′ end, because these carbons are hanging from the ends of the molecule. Prime can also be used to indicate which position a molecule has attached to, such as “5′-monophosphate”.

## Use in linguistics

The prime can be used in the transliteration of some languages, such as Russian, to denote palatalization.

The prime is also used in X-bar theory instead of a bar to indicate bar-levels in syntactic structures, because the bar is difficult to typeset. This is still read as "X bar", not "X prime".

Some X-bar notations use a double-prime (standing in for a double-bar) to indicate a phrasal level, indicated in most notations by "XP".

## History

The name "prime" is something of a misnomer. Through the early part of the 20th century, the notation x′ was read as "x prime" not because it was an x followed by a "prime symbol", but because it was the first in the series that continued with x″ ("x second") and x‴ ("x third"). It was only later, in the 1950s and 1960s, that the term "prime" began to be applied to the apostrophe-like symbol itself. Although it is now more common to pronounce x″ and x‴ as "x double prime" and "x triple prime", these are still sometimes pronounced in the old manner as "x second" and "x third".

## Representations

Unicode and HTML representations of the prime and related symbols are as follows.

Character Unicode HTML entity
Prime ( ′ ) U+2032 `&prime;`
Double prime ( ″ ) U+2033 `&Prime;`
Triple prime ( ‴ ) U+2034
Quadruple prime ( ⁗ ) U+2057
Modifier letter prime ( ʹ ) U+02B9
Modifier letter double prime ( ʺ ) U+02BA

The "modifier letter prime" and "modifier letter double prime" characters are intended for linguistic purposes, such as the indication of stress or the transliteration of certain Cyrillic characters.

When the character set used does not include the prime or double prime character (e.g., ISO 8859-1 is commonly assumed on IRC), they are often respectively approximated by normal or italic apostrophes and quotation marks. In LaTeX math mode, `f'` (f with an apostrophe) is rendered as $f\text{'},!$. Furthermore, LaTeX provides an oversized prime symbol, `prime` ($prime$) for use in subscripts. For example, `f_prime^prime` appears as $f_prime^prime$.