Delimiters (brackets)

A *delimiter* is one of the bracket characters that enclose an expression — parentheses ( ), square brackets [ ], braces \{ \}, and the like. This page is a lookup for how to type each character. Adjusting their size — stretching with \left/\right to fit the contents, or picking from the four \big … steps — lives on a separate page (the math page “Delimiters (\left \right)”), so here we stay with the characters themselves, gathered as parentheses, bars, floor and ceiling, arrows, and a few special brackets.

Delimiters vs. sizing

One distinction first. *Which* bracket you want (the kind of character) and *how big* it should be (its size) are decided separately in LaTeX. This page covers the former: typing the delimiter character itself, such as (, \langle, or \lfloor. The sizing — stretching automatically to the contents with \left( … \right), or stepping up with \bigl( / \Bigl( — is left to the dedicated math page.

Delimiters are almost all symbols used inside math mode. Unless noted otherwise, the commands in the tables below work in standard LaTeX with no extra package (only a few absolute-value/norm commands need amsmath, and those are marked). And **every character in these tables can follow \left/\right to auto-size**, as in \left\langle … \right\rangle or \left\lfloor … \right\rfloor (see the related page for the sizing details).

% 文字の種類はこのページ、サイズ調整は別ページ
\[
  ( a + b )
  \qquad
  \left( \frac{a}{b} \right)
  \qquad
  \left\langle \frac{x}{2} \right\rangle
\]

Parentheses, brackets, braces, angles

The four most basic kinds. Parentheses ( ) and square brackets [ ] are typed straight from the keyboard. Braces need care: { and } are reserved characters that group arguments in LaTeX, so typing them bare produces no visible bracket. To get the brace characters, prefix them with a backslash — **\{ and \}**. Angle brackets are not the inequality signs < >; use \langle (⟨) and \rangle (⟩), as in inner products and bra–ket notation.

The commands \lbrack, \rbrack, \lbrace, and \rbrace are aliases that produce the same characters as [ ] \{ \}. The named forms are sometimes preferred where a bare [ or { would be ambiguous, such as inside a macro argument.

Vertical bars (absolute value, norm)

There are two bars: the single bar for absolute value and the double bar for norm. The single bar is | (also \vert), and the double bar is \| (also \Vert). But writing |x| with the same character for both the opener and the closer leaves LaTeX unable to tell whether each | opens or closes, which can throw off the surrounding spacing.

So the amsmath package supplies semantically correct commands that distinguish opener from closer: for absolute value, the opener \lvert and closer \rvert; for norm, the opener \lVert and closer \rVert. These are spaced correctly as open/close delimiters. To avoid writing them out each time, you can also define dedicated commands like \abs{…} and \norm{…} with mathtools’ \DeclarePairedDelimiter (see the related page).

\usepackage{amsmath}   % \lvert \rvert \lVert \rVert に必要
% ...
\[
  |x| \;=\; \lvert x \rvert,
  \qquad
  \lVert v \rVert \;=\; \sqrt{\langle v, v \rangle}
\]

Floor and ceiling brackets

The floor (round down) and ceiling (round up) brackets. The floor ⌊x⌋ uses the opener \lfloor (⌊) and closer \rfloor (⌋); the ceiling ⌈x⌉ uses the opener \lceil (⌈) and closer \rceil (⌉). All are standard LaTeX, and the l/r in each name marks left (open) versus right (close). The corner at only the bottom (floor) or only the top (ceiling) is how you tell them apart.

\[
  \lfloor x \rfloor \le x \le \lceil x \rceil,
  \qquad
  \left\lfloor \frac{n}{2} \right\rfloor
\]

Arrows and slashes as delimiters

Brackets are not the only delimiters. Vertical arrows can serve as stretchable delimiters: the single arrows \uparrow (↑), \downarrow (↓), and two-way \updownarrow (↕), plus the double arrows \Uparrow (⇑), \Downarrow (⇓), and \Updownarrow (⇕). Combined with \left/\right, they stretch to the height of the contents — useful for vertical maps in commutative diagrams or for marking a range beside a matrix.

Slashes work as delimiters too. The forward slash / is typed directly; the backslash \backslash (\) is used for quotient sets such as G\backslash H. You write \backslash because a bare \ is the line-break command. Placed on one side of a \left/\right pair, these give a diagonally stretching delimiter.

Special brackets and the “invisible” one

Finally, some delimiters that are seldom needed but handy to know. \lgroup (⟮) and \rgroup (⟯) look like heavier parentheses with sharper corners, attractive in large displays. \lmoustache (⎰) and \rmoustache (⎱) are the top and bottom halves of a large brace, used as special stretchable dividers. Both are standard-LaTeX extensible delimiters built from the pieces of a brace.

And one more: the **period . is the “delimiter that draws nothing” — the null delimiter**. Since \left and \right must always be paired, you put a . on the side you want blank. In an evaluation such as \left. \frac{dy}{dx} \right|_{x=0}, the \left. on the left draws nothing while only the \right| stretches as a bar. Because . itself shows nothing, it is less a delimiter than an “empty side that satisfies the pairing rule.”

\[
  \left\lgroup \frac{a}{b} \right\rgroup
  \qquad
  \left. \frac{dy}{dx} \right|_{x=0}
\]

Every delimiter listed here will auto-size to its contents when placed after \left/\right, and \bigl / \Bigl … let you step through fixed sizes by hand. For how to choose between them, the inner growing divider \middle, and defining absolute value/norm in one place with \DeclarePairedDelimiter, see the math page “Delimiters (\left \right).”