Bracket (mathematics)

In mathematics, various typographical forms of brackets are frequently used in mathematical notation such as parentheses ( ), square brackets [ ], braces { }, and angle brackets $\langle\,\,\rangle$ . Generally such bracketing denotes some form of grouping: in evaluating an expression containing a bracketed sub-expression, the operators in the sub-expression take precedence over those surrounding it. Additionally, there are several uses and meanings for the various brackets.

Historically, other notations, such as the vinculum, were similarly used for grouping; in present-day use, these notations all have specific meanings. The earliest use of brackets to indicate aggregation (i.e. grouping) was suggested in 1608 by Christopher Clavius and in 1629 by Albert Girard.^[1]

In the Z formal specification language, braces denote a set and angle brackets denote a sequence.

Symbols for representing angle brackets

A variety of different symbols are used to represent angle brackets. In e-mail and other ASCII text it is common to use the less-than (<) and greater-than (>) signs to represent angle brackets, because ASCII does not include angle brackets.^[2] Unicode has three pairs of dedicated characters:

U+2329 (〈) and U+232A (〉) (left/right-pointing angle bracket) which are deprecated^[3]
U+27E8 (⟨) and U+27E9 (⟩) (mathematical left/right angle bracket)
U+3008 (〈) and U+3009 (〉) (left/right angle bracket in Chinese punctuation)

In LaTeX the markup is \langle and \rangle: $\langle \,\, \rangle\,$ .

Algebra

In elementary algebra parentheses, ( ), are used to specify the order of operations. Terms inside the bracket are evaluated first; hence 2×(3 + 4) is 14 and 10 ÷ 5(1 + 0) is 2 and (2×3) + 4 is 10. This notation is extended to cover more general algebra involving variables: for example $(x+y)\times(x-y)$ . Square brackets are also often used in place of a second set of parentheses when they are nested, to provide a visual distinction.

Also in mathematical expressions in general, parentheses are used to indicate grouping (that is, which parts belong together) when necessary to avoid ambiguities, or for the sake of clarity. For example, in the formula (εη)_X = ε_Xη_X, used in the definition of composition of two natural transformations, the parentheses around εη serve to indicate that the indexing by X is applied to the composition εη, and not just its last component η.

Functions

The arguments to a function are frequently surrounded by brackets: $f(x)$ . It is common to omit the parentheses around the argument when there is little chance of ambiguity, thus: $\sin x$ .

Coordinates and vectors

In the cartesian coordinate system brackets are used to specify the coordinates of a point: (2,3) denotes the point with x-coordinate 2 and y-coordinate 3.

The inner product of two vectors is commonly written as $\langle a, b\rangle$ , but the notation (a, b) is also used.

Intervals

Main article: Interval (mathematics)

Both parentheses, ( ), and square brackets, [ ], can also be used to denote an interval. The notation $[a, c)$ is used to indicate an interval from a to c that is inclusive of $a$ but exclusive of $c$ . That is, $[5, 12)$ would be the set of all real numbers between 5 and 12, including 5 but not 12. The numbers may come as close as they like to 12, including 11.999 and so forth (with any finite number of 9s), but 12.0 is not included. In some European countries, the notation $[5,12[$ is also used for this.

The endpoint adjoining the square bracket is known as closed, while the endpoint adjoining the parenthesis is known as open. If both types of brackets are the same, the entire interval may be referred to as closed or open as appropriate. Whenever infinity or negative infinity is used as an endpoint in the case of intervals on the real number line, it is always considered open and adjoined to a parenthesis. The endpoint can be closed when considering intervals on the extended real number line.

Sets and groups

Braces { } are used to identify the elements of a set: {a,b,c} denotes a set of three elements.

Angle brackets are used in group theory to write group presentations, and to denote the subgroup generated by a collection of elements.

Matrices

An explicitly given matrix is commonly written between large round or square brackets:

{\begin{pmatrix}1&-1\\2&3\end{pmatrix}}\quad \quad {\begin{bmatrix}c&d\end{bmatrix}}

Derivatives

The notation

f^{(n)}(x)\,

stands for the n-th derivative of function f, applied to argument x. So, for example, if $f(x) = \exp(\lambda x)$ , then $f^{(n)}(x) = \lambda^n\exp(\lambda x)$ . This is to be contrasted with $f^n(x) = f(f(\ldots(f(x))\ldots))$ , the n-fold application of f to argument x.

Falling and rising factorial

The notation (x)_n is used to denote the falling factorial, an n-th degree polynomial defined by

(x)_n=x(x-1)(x-2)\cdots(x-n+1)=\frac{x!}{(x-n)!}.

Confusingly, the same notation may be encountered as representing the rising factorial, also called "Pochhammer symbol". Another notation for the same is x⁽ⁿ⁾. It can be defined by

x^{(n)}=x(x+1)(x+2)\cdots(x+n-1)=\frac{(x+n-1)!}{(x-1)!}.

Quantum mechanics

In quantum mechanics, angle brackets are also used as part of Dirac's formalism, bra–ket notation, to note vectors from the dual spaces of the bra $\left\langle A\right|$ and the ket $\left|B\right\rangle$ .

In statistical mechanics, angle brackets denote ensemble or time average.

Polynomial rings

Square brackets are used to denote the variable in polynomial rings. For example, $\mathbb{R}[x]$ is the polynomial ring with the $x$ variable and real number coefficients.^[4]

Lie bracket and commutator

In group theory and ring theory, square brackets are used to denote the commutator. In group theory, the commutator [g,h] is commonly defined as g⁻¹h⁻¹gh. In ring theory, the commutator [a,b] is defined as ab − ba. Furthermore, in theory, braces are used to denote the anticommutator where {a,b} is defined as ab + ba.

The Lie bracket of a Lie algebra is a binary operation denoted by $[\cdot,\cdot]:\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}$ . By using the commutator as a Lie bracket, every associative algebra can be turned into a Lie algebra. There are many different forms of Lie bracket, in particular the Lie derivative and the Jacobi–Lie bracket.

Floor/ceiling functions and fractional part

Square brackets, as in [π] = 3, are sometimes used to denote the floor function, which rounds a real number down to the next integer. However the floor and ceiling functions are usually typeset with left and right square brackets where only the lower (for floor function) or upper (for ceiling function) horizontal bars are displayed, as in $\lfloor\pi\rfloor=3$ or $\lceil\pi\rceil=4$ .

Braces, as in {π} < ¹/₇, may denote the fractional part of a real number.

For example a highest factor of 10 would be 5. The numbers 1, 2, and 5 all go into 10 but the biggest is the number 5.

Notes

↑ Cajori, Florian 1980. A history of mathematics. New York: Chelsea Publishing, p. 158
↑ Raymond, Eric S. (1996), The New Hacker's Dictionary, MIT Press, p. 41, ISBN 9780262680929 .
↑ "Miscellaneous Technical" (PDF). unicode.org.
↑ Stewart, Ian (1995). Concepts of Modern Mathematics. Dover Publications. p. 90.

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