In mathematics, the prime is generally used to generate more variable names for similar things without resorting to subscripts, with generally meaning something related to (or derived from) . For example, if a point is represented by the
Cartesian coordinates , then that point rotated, translated or reflected might be represented as . Usually, the meaning of is defined when it is first used, but sometimes, its meaning is assumed to be understood: • A
derivative or differentiated function: in
Lagrange's notation, and are the first and second derivatives of the function with respect to . The pattern may be continued, such as in and , with each additional prime denoting the next higher derivative. Similarly, if , then and denote the first and second derivatives of with respect to . Other notations for derivatives also exist (see
Notation for differentiation). •
Set complement: is the complement of the set (other notations also exist). • The negation of an
event in probability theory: (other notations also exist). • The result of a
transformation: • The
transpose of a matrix (other notations also exist) • The
dual of a
vector space The prime is said to "decorate" the letter to which it applies. The same convention is adopted in
functional programming, particularly in
Haskell. In
geometry,
geography and
astronomy, prime and double prime are used as abbreviations for
minute and second of arc (and thus
latitude,
longitude,
elevation and
right ascension). In
physics, the prime is used to denote variables after an event. For example, may indicate the velocity of object A after an event. It is also commonly used in relativity: the event at in
frame , has coordinates in frame . In
chemistry, it is used to distinguish between different functional groups connected to an atom in a molecule, such as and , representing different
alkyl groups in an
organic compound. The
carbonyl carbon in proteins is denoted as , which distinguishes it from the other
backbone carbon, the
alpha carbon, which is denoted as . In
physical chemistry, it is used to distinguish between the lower state and the upper state of a
quantum number during a transition. For example, denotes the upper state of the quantum number while denotes the lower state of the quantum number . 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 end to the end, because these carbons are on the ends of the DNA molecule. The chemistry of this reaction demands that the be extended by DNA synthesis. Prime can also be used to indicate which position a molecule has attached to, such as ==Use in linguistics==