Produced by Charles Wells Revised 2017-01-19 Introduction to this website website TOC website index blog

Delimiters are pairs of symbols used in the symbolic language either for enclosing expressions or as operators.

- This list includes only the major meanings of the most common delimiters, according to my judgment. See Remarks about usage.
- The Wikipedia article on brackets in math includes other delimiters and many additional usages about delimiters.
- "Bracket" has other meanings. See the Glossary entry.

- In American usage, the symbols
"$($" and "$)$" are called
**parentheses**. - “Parenthesis” is singular, “parentheses” is plural.
- In British and Australian usage, the usual name for these symbols is "brackets", but they may be called
**round brackets**to make it clear that they are not square or curly brackets. - In the USA, parentheses may be called
**round parentheses**to distinguish them from other delimiters. - The input to a function is typically enclosed in parentheses. For example, if we define $f(x)={{x}^{3}}-2$, then $f(3)=25$. However, there are some standard exceptions to this, explained in the abmath section on prefix notation.
- The symbol $n\choose k$ denotes the binomial coefficient.
- Other uses of round parentheses are given in the sections Bare delimiters, The notation $(a,b)$, and Matrices.

- The delimiters “$[$“ and ”$]$” are called
**square brackets**. - Square brackets are occasionally used as bare delimiters.
- Square brackets may be used instead of parentheses to enclose matrices.
- Square brackets may be used instead of parentheses to enclose the argument to a function in an expression of its value, as in "$f[x]$" instead of "$f(x)$". The square brackets are required in Mathematica for function arguments.
- Square
brackets are used as outfix notation with special meanings:
- For real numbers $r$ and $s$, $[r,s]$ is the closed interval $\{x|r\leq x\leq s\}$.
- For integers $m$ and $n$, $[m,n]$ is the least common multiple of $m$ and $n$.
- $[g,h]$ denotes the commutator of two elements of a group.

- the symbols "$\langle$"
and "$\rangle$" are called
**angle brackets**. - In printed material they are usually noticeably distinct from the greater-than and less-than symbols "$\lt$" and "$\gt$", but they may not be distinguished in handwriting.
- Angle brackets are used as outfix notation to denote various constructions, most notably an inner product as in “$\left\langle v,w \right\rangle$”.
- $n$-tuples are sometimes written "$\langle a,b,c,\ldots\rangle$" instead of "$(a,b,c\ldots)$".
- In my research for the Handbook I could not find a citation for the use of angle brackets as bare delimiters, but I betcha someone somewhere has used them that way.
- Angle brackets are also called
**chevrons**or**pointy brackets**, the latter mostly in speech. I have never heard a mathematician call them "chevrons".

- The symbols "$\{$" and "$\}$ are called
**braces**,**curly braces**or**curly brackets**. - Braces are used as bare delimiters when there are nested parentheses, in much the same way as square brackets.
- Braces are used in the list notation for sets
- They are also used in setbuilder notation..
- Braces are used as outfix notation for functions. In particular, the fractional part of a real number $r$ may be denoted by "$\{r\}$". For example, $\{3/2\} = 0.5$.
- A left brace may be used by itself in a definition by cases.

- A pair of delimiters may or may not have significance beyond
grouping; if they do not they are
**bare delimiters**. - The three types of character used as bare delimiters in mathematics are parentheses, square brackets and braces.
- Typically, parentheses are the standard delimiters in symbolic expressions.
- Combinations of different delimiters are used most often with nested parentheses to aid the reader to match the correct pairs. For example, \[{{\left[ {{\left( {{x}^{2}}\sin x+{{(2x-1)}^{-4}} \right)}^{3}}-{{\left( 1+\cos x \right)}^{2}} \right]}^{5}}\] Note that some round parentheses have been made bigger to aid in the matching. This is a common technique facilitated by LaTeX.

The notation "$(a,b)$" may denote any of these functions:

- "$(a,b)$" may denote the ordered pair with first coordinate $a$ and second coordinate $b$. This usage extends to $n$-tuples, for example the ordered triple $(3,1,2)$.
- For $a$ and $b$ real numbers, "$(a,b)$" may denote the
**open real interval**\[\left\{ x\,|\,a\lt x \text{ and } x\lt b\right\}\] - "$(a,b)$" may denote the
**greatest common divisor**of the integers $a$ and $b$. - These varying usages may occur in the same document. See the Handbook, Citation 139, for an example.
- Ordered pairs are often written as "$\langle a,b\rangle$" instead of "$(a,b)$".

Matrices may be enclosed by parentheses; for example: \[\left( \begin{matrix} {{a}_{11}} & {{a}_{12}}& {{a}_{13}}\\ {{a}_{21}} & {{a}_{22}} & {{a}_{23}}\\ {{a}_{31}} & {{a}_{32}} & {{a}_{33}} \\ \end{matrix} \right)\]

This matrix is one single **math object** with $9$ parameters. It is not in any sense a set of $9$ numbers. It is *one matrix.*

Square brackets may be used for this as well, as in \[\left[ \begin{matrix} {{a}_{11}} & {{a}_{12}}& {{a}_{13}}\\ {{a}_{21}} & {{a}_{22}} & {{a}_{23}}\\ {{a}_{31}} & {{a}_{32}} & {{a}_{33}} \\ \end{matrix} \right]\]

However, vertical bars, as in
\[\left|
\begin{matrix}
{{a}_{11}} & {{a}_{12}}& {{a}_{13}}\\
{{a}_{21}} & {{a}_{22}} & {{a}_{23}}\\
{{a}_{31}} & {{a}_{32}} & {{a}_{33}} \\
\end{matrix} \right|\]
is by definition the determinant of the matrix, which is a *single number.* It is not a matrix at all.

- Brackets and Brackets (mathematics) in Wikipedia.
- Brackets vs. parentheses in English language and usage Stack Exchange.
- Parentheses and brackets in Separated by a Common Language.

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