# Prime (symbol)

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Not to be confused with prime number. Prime (symbol)_sentence_0

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. Prime (symbol)_sentence_1

Although the characters differ little in appearance from those of the apostrophe (or the single or double quotation mark), the appropriate uses of the prime symbol are different. Prime (symbol)_sentence_2

While an apostrophe is often used in place of the prime (and a double quote in place of the double prime) due to the lack of the prime and double prime symbols on everyday keyboards, such substitutions would not normally be considered appropriate in formal materials or in typesetting. Prime (symbol)_sentence_3

## Designation of units Prime (symbol)_section_0

See also: Positional notation § Sexagesimal system Prime (symbol)_sentence_4

The prime symbol ′ is commonly used to represent feet (ft), and the double prime ″ is used to represent inches (in). Prime (symbol)_sentence_5

The triple prime ‴ as used in watchmaking represents a ligne (about 2.26 millimetres or 0.089 inches). Prime (symbol)_sentence_6

Primes are also used for angles. Prime (symbol)_sentence_7

The prime symbol ′ is used for arcminutes (​⁄60 of a degree), and the double prime ″ for arcseconds (​⁄60 of an arcminute). Prime (symbol)_sentence_8

As an angular measurement, 3° 5′ 30″ means 3 degrees, 5 arcminutes and 30 arcseconds. Prime (symbol)_sentence_9

In historical astronomical works, the triple prime was used to denote "thirds" (​⁄60 of an arcsecond) and a quadruple prime ⁗ "fourths" (​⁄60 of a third of arc), but modern usage has replaced this with decimal values of arcseconds. Prime (symbol)_sentence_10

Primes are sometimes used to indicate minutes, and double primes to indicate seconds of time, as in the John Cage composition 4'33", pronounced (and the composition itself lasting) 4 minutes 33 seconds. Prime (symbol)_sentence_11

This notation only applies to duration, and is seldom used for durations longer than 60 minutes. Prime (symbol)_sentence_12

## Use in mathematics, statistics, and science Prime (symbol)_section_1

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. Prime (symbol)_sentence_13

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′). Prime (symbol)_sentence_14

Usually, the meaning of x′ is defined when it is first used, but sometimes, its meaning is assumed to be understood: Prime (symbol)_sentence_15

Prime (symbol)_unordered_list_0

• A derivative or differentiated function: in Lagrange's notation, f ′(x) and f ″(x) are the first and second derivatives of f (x) with respect to x. Likewise are f ‴(x) and f ⁗(x) . Similarly, if y = f (x), then y′ and y″ are the first and second derivatives of y with respect to x. Other notation for derivatives also exists (see Notation for differentiation).Prime (symbol)_item_0_0
• Set complement: A′ is the complement of the set A (other notation also exists).Prime (symbol)_item_0_1
• The negation of an event in probability theory: Pr(A′) = 1 − Pr(A) (other notation also exists).Prime (symbol)_item_0_2
• The result of a transformation: Tx = x′Prime (symbol)_item_0_3
• The transpose of a matrix (other notation also exists)Prime (symbol)_item_0_4

The prime is said to "decorate" the letter to which it applies. Prime (symbol)_sentence_16

The same convention is adopted in functional programming, particularly in Haskell. Prime (symbol)_sentence_17

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). Prime (symbol)_sentence_18

In physics, the prime is used to denote variables after an event. Prime (symbol)_sentence_19

For example, vA′ would indicate the velocity of object A after an event. Prime (symbol)_sentence_20

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′. Prime (symbol)_sentence_21

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 an organic compound. Prime (symbol)_sentence_22

The carbonyl carbon in proteins is denoted as C′, which distinguishes it from the other backbone carbon, the alpha carbon, which is denoted as Cα. Prime (symbol)_sentence_23

In physical chemistry, it is used to distinguish between the lower state and the upper state of a quantum number during a transition. Prime (symbol)_sentence_24

For example, J ′ denotes the upper state of the quantum number J while J ″ denotes the lower state of the quantum number J. Prime (symbol)_sentence_25

In molecular biology, the prime is used to denote the positions of carbon on a ring of deoxyribose or ribose. Prime (symbol)_sentence_26

The prime distinguishes places on these two chemicals, rather than places on other parts of DNA or RNA, like phosphate groups or nucleic acids. Prime (symbol)_sentence_27

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 on the ends of the DNA molecule. Prime (symbol)_sentence_28

The chemistry of this reaction demands that the 3′ OH is extended by DNA synthesis. Prime (symbol)_sentence_29

Prime can also be used to indicate which position a molecule has attached to, such as 5′-monophosphate. Prime (symbol)_sentence_30

## Use in linguistics Prime (symbol)_section_2

The prime can be used in the transliteration of some languages, such as Slavic languages, to denote palatalization. Prime (symbol)_sentence_31

Prime and double prime are used to transliterate Cyrillic yeri (the soft sign, ь) and yer (the hard sign, ъ). Prime (symbol)_sentence_32

However, in ISO 9, the corresponding modifier letters are used instead. Prime (symbol)_sentence_33

Originally, X-bar theory used a bar over syntactic units to indicate bar-levels in syntactic structure, generally rendered as an overbar. Prime (symbol)_sentence_34

While easy to write, the bar notation proved difficult to typeset, leading to the adoption of the prime symbol to indicate a bar. Prime (symbol)_sentence_35

(Despite the lack of bar, the unit would still be read as "X bar", as opposed to "X prime".) Prime (symbol)_sentence_36

With contemporary development of typesetting software such as LaTeX, typesetting bars is considerably simpler; nevertheless, both prime and bar markups are accepted usages. Prime (symbol)_sentence_37

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". Prime (symbol)_sentence_38

## Use in music Prime (symbol)_section_3

The prime symbol is used in combination with lower case letters in the Helmholtz pitch notation system to distinguish notes in different octaves from middle C upwards. Prime (symbol)_sentence_39

Thus c represents the ⟨C⟩ below middle C, c′ represents middle C, c″ represents the ⟨C⟩ in the octave above middle C, and c‴ the ⟨C⟩ in the octave two octaves above middle C. A combination of upper case letters and sub-prime symbols is used to represent notes in lower octaves. Prime (symbol)_sentence_40

Thus C represents the ⟨C⟩ below the bass stave, while C ͵ represents the ⟨C⟩ in the octave below that. Prime (symbol)_sentence_41

In some musical scores, the double prime ″ is used to indicate a length of time in seconds. Prime (symbol)_sentence_42

It is used over a fermata 𝄐 denoting a long note or rest. Prime (symbol)_sentence_43

## History Prime (symbol)_section_4

The name "prime" is something of a metonymy. Prime (symbol)_sentence_44

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"). Prime (symbol)_sentence_45

It was only later, in the 1950s and 1960s, that the term "prime" began to be applied to the apostrophe-like symbol itself. Prime (symbol)_sentence_46

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". Prime (symbol)_sentence_47

## Computer encodings Prime (symbol)_section_5

Unicode and HTML representations of the prime and related symbols are as follows. Prime (symbol)_sentence_48

Prime (symbol)_unordered_list_1

• U+2032 ′ PRIME (HTML ′ · ′) (lower case p)Prime (symbol)_item_1_5
• U+2033 ″ DOUBLE PRIME (HTML ″ · ″) (upper case P)Prime (symbol)_item_1_6
• U+2034 ‴ TRIPLE PRIME (HTML ‴ · &tprime;)Prime (symbol)_item_1_7
• U+2035 ‵ REVERSED PRIME (HTML ‵ · &backprime;, &bprime;)Prime (symbol)_item_1_8
• U+2036 ‶ REVERSED DOUBLE PRIME (HTML ‶)Prime (symbol)_item_1_9
• U+2037 ‷ REVERSED TRIPLE PRIME (HTML ‷)Prime (symbol)_item_1_10
• U+2057 ⁗ QUADRUPLE PRIME (HTML ⁗ · &qprime;)Prime (symbol)_item_1_11
• U+02B9 ʹ MODIFIER LETTER PRIME (HTML ʹ)Prime (symbol)_item_1_12
• U+02BA ʺ MODIFIER LETTER DOUBLE PRIME (HTML ʺ)Prime (symbol)_item_1_13
• U+0022 " QUOTATION MARK (HTML " · ", &QUOT;) (often misused as double prime)Prime (symbol)_item_1_14
• U+0027 ' APOSTROPHE (HTML ' · ') (often misused as a single prime)Prime (symbol)_item_1_15

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. Prime (symbol)_sentence_49

In a context when the character set used does not include the prime or double prime character (e.g., in an online discussion context where only ASCII or ISO 8859-1 [ISO Latin 1] is expected), they are often respectively approximated by ASCII apostrophe (U+0027) or quotation mark (U+0022). Prime (symbol)_sentence_50