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Produced by Charles Wells     Revised 2017-01-19

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SYMBOLS: DELIMITERS

Introduction

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.

"$\mathbf{\langle}$ $\rangle$"

• 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.

Bare delimiters

• 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 $(a,b)$

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

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.

References

• 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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