Vertical bar (|): symbol, mathematical and typographic uses

Mathematical meanings

In mathematics the vertical bar has several standard interpretations. When a single bar is placed on each side of a real or complex number it denotes absolute value, written as |x| for a real number and extended to complex numbers to indicate magnitude; the notion is often discussed under the heading absolute value. When the same single bars surround a square array they are frequently used to indicate the determinant of a matrix rather than the matrix itself; the determinant is a scalar summarizing certain linear properties of the matrix and is sometimes written with bars instead of brackets (determinant).

Placed between two integers the symbol denotes the divisibility relation: for integers a and b the notation a | b means "a divides b" — that is, b is a multiple of a. This usage is familiar in elementary number theory and is sometimes explained with simple examples such as 3 | 12.

Double bars and related constructions

Double vertical bars (‖ ... ‖) are used for different but related concepts. One common use is to denote a vector norm, a generalization of length that assigns a non‑negative size to a vector; this is often written as ‖v‖ for a vector v. Double bars can also appear in geometry to indicate that two lines or segments are parallel, written as ℓ1 ∥ ℓ2 in many textbooks.

Computing, programming and piping

Outside mathematics the vertical bar has a prominent role in computing. In command‑line environments and many programming languages it is called the pipe and is used to pass the output of one process as input to another. In logic and set notation it can serve as a separator or condition marker (for example, {x | property(x)}), where the bar reads "such that". In typesetting and markup it sometimes functions as a delimiter because its thin, vertical shape is visually unobtrusive.

Typography, history and distinctions

Typographically the vertical bar is distinct from similar characters such as the broken bar (¦), the lowercase letter l (l), and the numeral one (1); careful fonts and spacing help avoid confusion. Historically the exact origin of the modern keyboard vertical bar is tied to typewriter and early computer character sets that adopted a small set of printable symbols and assigned vertical bar its current code positions. Variants include single and double bars, each carrying different conventional meanings depending on discipline.

Summary and examples

• Absolute value: |z| for a complex or real z.
• Determinant: |A| for a square matrix A, sometimes written to mean det(A).
• Divisibility: a | b reads "a divides b".
• Norm and parallelism: ‖v‖ and ℓ1 ∥ ℓ2.

The vertical bar is a compact, versatile symbol whose interpretation depends strongly on surrounding notation. For quick reference see introductory texts on algebra, linear algebra, or command‑line programming for examples of how the bar is used in practice; more detailed discussions are found in subject‑specific sources and style guides.

Related topics: the pipe operator in shells and programming, typographic differences between bar and similar glyphs, and norms versus absolute values are common points of comparison for readers who wish to explore further.

For further reading consult general references on symbols and notation in mathematics and computing; entries on the glyph and its variants appear in many symbol tables and typography guides (glyph, notation).