The vertical bar ( | ) is a computer character and glyph with various uses in mathematics, computing, and typography. It has many names, often related to particular meanings: Sheffer stroke (in logic), verti-bar, vbar, stick, vertical line, vertical slash, bar, obelisk, pike, or pipe, and several variants on these names. It is occasionally considered an allograph of broken bar (see below).
The vertical bar is used as a mathematical symbol in numerous ways:
absolute value: , read "the absolute value of x"
set-builder notation: , read "the set of x such that x is less than two". Often a colon ':' is used instead of a vertical bar
cardinality: , read "the cardinality of the set S"
conditional probability: , read "the probability of X given Y"
divisibility: , 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 Sheffer stroke in logic: , read " a nand b"
distance: distance notes the shortest distance between point to line , so is perpendicular to line
evaluate (subscript notation): , read " f of x, evaluated at x equals 4" (see at Wikibooks)
length: | s|, read "the length of the string s"
restriction: denotes a restriction of function where it is defined over a domain that is a superset of
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
The order of a group G is written
The determinant of a matrix A is sometimes written , 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
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: , read "the norm of the Vector space x"
parallelism in geometry: indicates that the line is parallel to the line
propositional truncation (a Type theory former which truncates a type down to a mere proposition in homotopy type theory): for any (read "term of type ") we have
(here reads " image of in " and reads " propositional truncation of ")
The vertical bar is used in bra–ket notation in quantum physics
: the quantum physical state
: the dual space corresponding to the state above
: the inner product of states and
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 dimensions
[Larus Thorlacius, Thordur Jonsson (eds.), M-Theory and Quantum Geometry, Springer, 2012, p. 263.]
A pipe is an inter-process communication mechanism originating in Unix
, 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 DOS and Microsoft Windows.
This usage has led to the character itself being called "pipe".
In many programming languages, the vertical bar is used to designate the logic operation or
, either bitwise or
or Boolean datatype or
Specifically, in C and other languages following C syntax conventions, such as C++, Perl, 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, ANSI C can transcribe it in form of the C trigraph ??!, 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 grep -E 'fu|bar' matches lines containing 'fu' or 'bar'.
The double vertical bar operator "||" denotes string concatenation
in PL/I, standard ANSI SQL
, and theoretical computer science (particularly cryptography
Although not as common as commas or tabs, the vertical bar can be used as a delimiter
in a flat file
. Examples of a pipe delimited
standard data format are LEDES
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 sed). 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.
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.
In calculi of communicating processes (like pi-calculus
), the vertical bar is used to indicate that processes execute in parallel.
The pipe in APL is the modulo or residue
function between two operands and the absolute value function next to one operand.
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 dental click
(). A double vertical bar is used to write the alveolar lateral click (). Since these are technically letters, they have their own Unicode
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.
In medieval European manuscripts, a single vertical bar was a common variant of the virgula used as a full stop
, scratch comma
and caesura mark.
In Sanskrit and other Indian languages, text blocks used to be written in stanzas. Two bars || represent the equivalent of a pilcrow.
A double vertical bar or is the standard caesura mark
in English language
literary criticism and analysis. It marks the strong break or caesura
common to many forms of poetry
, particularly Old English verse.
In the Geneva Bible
and early printings of the King James Version, a double vertical bar is used to mark marginalia
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.
The vertical bar is encoded in Unicode at .
Solid vertical bar vs broken bar
The broken bar
) in computing was historically an allograph
of the vertical bar, and was perceived as such before the broad implementation of extended ASCII
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 keyboard layout 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 code point 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 EBCDIC 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 (7Chexadecimal)||N/A|[Broken bar is no longer a part of ASCII, since the early 1990s]
|CP775||124 (7Ch)||167 (A7h)|
|CP850, CP857, CP858||124 (7Ch)||221 (DDh)|
|CP863||124 (7Ch)||160 (A0h)|
|CP864||124 (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, -16||124 (7Ch)||N/A|
|EBCDIC (CCSID 500 variant)||187 (BBh)||166 (A6h)|
|Shift-JIS Men-Ku-Ten||1-01-35|| |
|HTML|||||¦ or ¦|
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
, the vertical bar can be used as delimiter in mathematical mode. The sequence \| creates a double vertical line (a | b \| c is set as
). This has different spacing from \mid and \parallel, which are relational operators: a \mid b \parallel c is set as
. In LaTeX text mode, the vertical bar produces an em dash
(—). The \textbar command can be used to produce a vertical bar.
The vertical bar is also used as special character in other lightweight markup languages, notably MediaWiki's Wikitext.