Vertical bar

Author: Leandro Alegsa Creation date: November 13, 2022

A vertical bar is the glyph "|". It has several other names besides vertical bar, such as "pipe", "vertical slash", and "bar". In mathematics, it may be put on both sides of a real or complex number to mean its absolute value. When applied on both sides of a matrix, it means its determinant. When placed in between two numbers, it means the divisibility relation (for example. $a\mid b$ means "a divides b").

In mathematics, double vertical bars are also used to refer to various mathematical concepts. These include the parallel relation (as in $\ell _{1}\parallel \ell _{2}$ ) and the norm of a vector (as in $\|\mathbf {v} \|$ ).

Usage

Mathematics

The single vertical line stands for

the relation "divides" between integers
the conditional probability in probability theory
the restriction of the domain of definition of a given function: if are $A\subseteq B$ two sets and a function with domain of definition , then denotes the function which on coincides with and is otherwise undefined.
the function evaluation at one point. $f(x)\vert_{x=1} = f(1)$
the difference of two function values at two points, for example in integral calculus: $F(x) \Big\vert_{a}^{b} = F(b) - F(a)$
"with the property:" or "to which applies:" in the intensional notation of sets:

$\{2z\mid z\in \mathbb {Z} \}$ is the set of even integers,

$\{(x,\,y)\in \mathbb {R} ^{2}\mid x^{2}+y^{2}=1\}$ is the unit circle

the Sheffer's line in logic $A | B := \neg (A \wedge B) \equiv \overline{A \wedge B}$

A pair of simple vertical strokes (like parentheses) denotes

the absolute amount of a number
the determinant of a matrix
the power of a set

The double vertical line denotes parallelism: $g \parallel h$

A pair of double vertical bars (like brackets) denotes a standard

Physics

In Dirac notation in physics, ⟨is $\langle\phi|\mathcal{O}|\psi\rangle$ a matrix element.

Data processing and programming languages

In shells, a vertical line indicates a pipe. Pipes can be used to chain the input and output streams of two programs. (Ex.: ls | more)
In some markup languages, the vertical bar represents the border of a table cell
In various databases (e.g. Informix) the vertical bar is used as a default separator of data fields for exported tables.
In some search engines, such as Google, the vertical bar - placed without spaces between search terms - can replace the OR function (the operator "OR")
In many programming languages (e.g. C, Perl, PHP, and Java), two vertical bars in a row denote the logical OR, and a single vertical bar denotes the bitwise OR.
Two vertical bars one after the other denote the concatenation of two strings in various programming languages (e.g. Rexx)

Other uses

In lexicons and dictionaries, such as the Duden, the vertical line indicates possible separators of a word; alternatively, a midpoint is sometimes used.
In textual sciences, the vertical stroke is often used to mark a page turn.
On the menus of gourmet restaurants, the vertical line is used when only the three main ingredients are indicated by the dishes to separate them.
In the International Phonetic Alphabet (IPA) and in Khoisan languages, a similar click sound letter indicates the dental click.

Since the vertical stroke is not used in normal text, it is not considered a punctuation mark.

Coding

The vertical bars are defined and coded as follows:

International character encoding standard Unicode and encoding in HTML web document format
Characters	Unicode		German designation	HTML
Characters	Position	Name	German designation	hexadecimal	decimal	named
\|	U+007C	Vertical line	Vertical stroke	\|	\|	(not available)
¦	U+00A6	Broken bar	Interrupted stroke	¦	¦	& brvbar;
∣	U+2223	Divides	Divide-Stroke	∣	∣
⎮	U+23AE	Integral extension	Integral line	⎮	⎮
│	U+2502	Box drawings light vertical	Element thin top-bottom	│	│
┃	U+2503	Box drawings heavy vertical	Element thick top-bottom	┃	┃
┆	U+2506	Box drawings light triple dash vertical	Element thin dashed top-bottom	┆	┄
┇	U+2507	Box drawings heavy triple dash vertical	Element thick dashed top-bottom	┇	┇
┊	U+250A	Box drawings light quadruple dash vertical	Element thin dotted top-bottom	┊	┊
┋	U+250B	Box drawings heavy quadruple dash vertical	Element thick dotted top-bottom	┋	┋
╎	U+254E	Box drawings light double dash vertical	Element thin large dashed top-bottom	╎	╎
╏	U+254F	Box drawings heavy double dash vertical	Element thick large dashed top-bottom	╏	╏
ǀ	U+01C0	Latin letter dental click	Latin letter Dental click	ǀ	ǀ

The current version of the ASCII character set contains the vertical stroke, but not the interrupted stroke. This occurs, however, in equally old character sets of the company IBM. However, the ASCII extensions ISO 6937 and ISO 8859-1 (also known as Latin-1) already contain both stroke variants in 1986. Almost all modern computers with Latin fonts use an ISO 8859 variant or the Unicode standard and can therefore process and display both characters without any problems.

There was originally a reason for the two variants of the vertical stroke (continuous and interrupted): In the older fonts, there were those which represented the vertical stroke confusingly in the same way as the lowercase letter l (small L) and the uppercase letter I (capital i). In more sophisticated documents, this resulted in three characters with an (almost) identical representation.