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The vertical bar (  | ) is a computer character and with various uses in mathematics, computing, and typography. It has many names, often related to particular meanings: (in logic), verti-bar, vbar, stick, vertical line, vertical slash, bar, glidus, obelisk, or pipe, and several variants on these names. It is occasionally considered an allograph of broken bar (See below).


Usage

Mathematics
The vertical bar is used as a mathematical symbol in numerous ways:
  • : |x|, read "the absolute value of x"
  • set-builder notation: \{x|x<2\}, read "the set of x such that x is two". Often a colon ':' is used instead of a vertical bar
  • : |S|, read "the cardinality of the set S"
  • conditional probability: P(X|Y), read "the of X given Y"
  • : a | b, read " a divides b" or " a is a factor of b", though Unicode also provides special ‘divides’ and ‘does not divide’ symbols (U+2223 and U+2224: ∣, ∤)
  • the in : a|b, read " a nand b"
  • : distance P|ab notes the shortest distance between point P to line ab, so P|ab is perpendicular to line ab
  • evaluate (subscript notation): f(x)|_{x=4}, read " f of x, evaluated at x equals 4" (see at Wikibooks)
  • length: | s|, read "the length of the string s"
  • restriction: f|_{A}: A \to F denotes a restriction of function f where it is defined over a domain that is a superset of A
  • Sometimes a vertical bar following a function, with sub- and super-script limits a and b is used when evaluating definite integrals to mean " f( x) from a to b" or " f( b) – f( a)"
  • A vertical bar can be used to separate variables from fixed parameters in a function, for example f(x|\mu,\sigma)
  • The order of a group G is written |G|
  • The of a matrix A is sometimes written |A|, and when the matrix entries are written out, the determinant is denoted by surrounding the matrix entries by vertical bars instead of the brackets or parentheses of the matrix, for instance
  • :\begin{vmatrix} a & b & c\\d & e & f\\g & h & i \end{vmatrix}
  • Occasionally to denote norms of greater-than-one-dimensional vectors (note that absolute value is a one-dimensional norm), although a double vertical bar (see below) is more often used to avoid ambiguity.

The double vertical bar, is also employed in mathematics.

  • norm: \|\mathbf{x}\|, read "the norm of the x"
  • parallelism in : AB \parallel CD indicates that the line AB is parallel to the line CD
  • propositional truncation (a former which truncates a type down to a mere proposition in homotopy type theory): for any a : A (read "term a of type A") we have |a| : \left\| A \right\| (here |a| reads " image of a : A in \left\| A \right\|" and |a| : \left\| A \right\| reads " propositional truncation of A")


Physics
The vertical bar is used in bra–ket notation in . Examples:

  • |\psi\rangle: the quantum physical state \psi
  • \langle\psi|: the corresponding to the state above
  • \langle\psi|\rho\rangle: the of states \psi and \rho
  • supergroups in physics are denoted G( N| M) which reads " G, M vertical bar N"; here G denotes any supergroup, M denotes the bosonic dimensions, and N denotes the Grassmann dimensionsLarus Thorlacius, Thordur Jonsson (eds.), M-Theory and Quantum Geometry, Springer, 2012, p. 263.


Computing

Pipe
A pipe is an inter-process communication mechanism originating in , which allows the output (standard out and, optionally, standard error) of one process to be used as input (standard in) to another. In this way, a series of commands can be "piped" together, giving users the ability to quickly perform complex multi-stage processing from the command line or as part of a Unix shell script ("bash file"). In most (command interpreters), this is represented by the vertical bar character. For example:

[[grep]] -i 'blair' filename.log | more
     

where the output from the "grep" process is piped to the "more" process.

The same "pipe" feature is also found in later versions of and Microsoft Windows.

This usage has led to the character itself being called "pipe".


Disjunction
In many programming languages, the vertical bar is used to designate the logic operation or, either bitwise or or or.

Specifically, in C and other languages following C syntax conventions, such as C++, , Java and C#, a | b denotes a bitwise or; whereas a double vertical bar a || b denotes a (short-circuited) logical or. Since the character was originally not available in all and keyboard layouts, can transcribe it in form of the ??!, which, outside string literals, is equivalent to the | character.

In regular expression syntax, the vertical bar again indicates logical or (alternation). For example: the Unix command -E 'fu|bar' matches lines containing 'fu' or 'bar'.


Concatenation
The double vertical bar operator "||" denotes string in PL/I, standard ANSI , and theoretical computer science (particularly ).


Delimiter
Although not as common as commas or tabs, the vertical bar can be used as a in a . Examples of a standard data format are 1998B and HL7. It is frequently used because vertical bars are typically uncommon in the data itself.

Similarly, the vertical bar may see use as a delimiter for regular expression operations (e.g. in ). This is useful when the regular expression contains instances of the more common forward slash (/) delimiter; using a vertical bar eliminates the need to escape all instances of the forward slash. However, this makes the bar unusable as the regular expression "alternative" operator.


Backus–Naur form
In Backus–Naur form, an expression consists of sequences of symbols and/or sequences separated by '|', indicating a choice, the whole being a possible substitution for the symbol on the left.


Concurrency operator
In calculi of communicating processes (like ), the vertical bar is used to indicate that processes execute in parallel.


APL
The pipe in APL is the modulo or residue function between two operands and the absolute value function next to one operand.


List comprehensions
The vertical bar is used for list comprehensions in some functional languages, e.g. Haskell and Erlang. Compare set-builder notation.


Phonetics and orthography
In the Khoisan languages and the International Phonetic Alphabet, the vertical bar is used to write the (). A double vertical bar is used to write the alveolar lateral click (). Since these are technically letters, they have their own code points in the Latin Extended-B range: U+01C0 for the single bar and U+01C1 for the double bar.

Longer single and double vertical bars are used to mark prosodic boundaries in the IPA.


Literature

Punctuation
In medieval European manuscripts, a single vertical bar was a common variant of the virgula used as a , ,. and caesura mark..

In and other Indian languages, text blocks used to be written in stanzas. Two bars || represent the equivalent of a .


Poetry
A double vertical bar or is the standard caesura mark in literary criticism and analysis. It marks the strong break or common to many forms of , particularly Old English verse.


Notation
In the and early printings of the King James Version, a double vertical bar is used to mark that contain an alternative translation from the original text. These margin notes always begin with the conjunction "Or". In later printings of the King James Version, the double vertical bar is irregularly used to mark any comment in the margins.


Encoding
The vertical bar is encoded in Unicode at .


Solid vertical bar vs broken bar
The broken bar ( ¦) in computing was historically an of the vertical bar, and was perceived as such before the broad implementation of character sets (namely, ISO/IEC 8859 series), which made a distinction between the two forms. Since the 1990s, it has been a separate character (in Unicode and not a part of ASCII; it is termed the "parted rule" in Unicode documentation. However, in some fonts. the glyph used for the vertical bar is identical to the glyph used for a broken bar.

The broken bar is encoded in Unicode at .

Due to historical confusion between the two, computer keyboards and displays may not clearly or consistently differentiate them:

  • The typical used in the United Kingdom features separate keys for vertical bar and broken bar; however, in many fonts the vertical bar key produces a broken-bar symbol. Windows keyboard drivers have the vertical bar on , while the broken bar is on the grave accent (`) key, and is typed with .
  • The ANSI QWERTY keyboard has only one key, which used to be labeled with a broken bar but now more commonly carries a vertical bar – since it always produces a vertical bar character.
  • On many , the “> < |” key in the lower left is labelled “> < ¦” but always produces a vertical bar character.
  • On French AZERTY keyboards, the vertical bar can be produced by pressing on Mac computers or on Windows computers.

The broken bar has hardly any practical application and does not appear to have any clearly identified uses distinct from the vertical bar. In non-computing use — for example in mathematics, physics and general typography — the broken bar is not an acceptable substitute for the vertical bar. Aforementioned usages in computing rely on the abstract character with 124 (0x7C) in ASCII (or ASCII-compatible code pages) and do not depend on visual rendering, which actually may be a broken bar in some environments.

Some variants of the family of code pages such as EBCDIC 500 distinguished the broken bar from the solid vertical bar.


In common character maps
,
CP437, CP667, CP720, CP737, CP790, CP819, CP852, CP855, CP860, CP861, CP862, CP865, CP866, CP867, CP869, CP872, CP895, CP932, CP991
124 (7C)N/ABroken bar is no longer a part of ASCII, since the early 1990s
CP775124 (7Ch)167 (A7h)
CP850, CP857, CP858124 (7Ch)221 (DDh)
CP863124 (7Ch)160 (A0h)
CP864124 (7Ch)219 (DBh)
ISO/IEC 8859-1, -7, -8, -9, -13,
CP1250, CP1251, CP1252, CP1253, CP1254, CP1255, CP1256, CP1257, CP1258
124 (7Ch)166 (A6h)
ISO/IEC 8859-2, -3, -4, -5, -6, -10, -11, -14, -15, -16124 (7Ch)N/A
U+007CU+00A6
(CCSID 500 variant)187 (BBh)166 (A6h)
Men-Ku-Ten1-01-35
&#124;&brvbar; or &#166;

Additional related Unicode characters:

  • Double vertical line ( ‖ ): U+2016 used in pairs to indicate norm
  • Parallel to ( ∥ ): U+2225
  • Latin letter dental click (⟨ǀ⟩): U+01C0
  • Latin letter lateral click (⟨ǁ⟩): U+01C1
  • Symbol 'divides' (⟨∣⟩): U+2223
  • Various Box-drawing characters at U+2500 to U+257F


In text processing
In , the vertical bar can be used as delimiter in mathematical mode. The sequence \| creates a double vertical line (a | b \| c is set as a | b \| c). This has different spacing from \mid and \parallel, which are relational operators: a \mid b \parallel c is set as a \mid b \parallel c. In LaTeX text mode, the vertical bar produces an (—), or one can use the \textbar command instead.

The vertical bar is also used as special character in other lightweight markup languages, notably Wikipedia's own .


See also

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