Delimiters (\left \right)

When the contents of a parenthesis or bar grow into a tall fraction or a matrix, an ordinary-sized delimiter looks puny against them. LaTeX gives you two tools: **\left … \right, which stretches a delimiter automatically to the height of its contents, and \big \Big \bigg \Bigg**, four fixed sizes you pick by hand. This page sorts out when to use each, the “invisible delimiter” that blanks out one side, the growing middle divider \middle, and the list of delimiter symbols themselves.

Auto-sizing with `\left` … `\right`

The easiest approach is to put \left before an opening delimiter and \right before the closing one. LaTeX measures the height of the contents and stretches the delimiters just tall enough to cover them. The character written **immediately after \left or \right is the actual delimiter**: ( ) for parentheses, [ ] for brackets, \{ \} for braces (see below), \langle \rangle for angle brackets, and so on.

\[
  \left( \frac{a^2 + b^2}{c^2} \right)
  \qquad
  \left[ \sum_{k=1}^{n} \frac{1}{k} \right]
\]

The first stretches the parentheses up and down to wrap the tall fraction, matching its height exactly. Written plainly as ( \frac{a^2+b^2}{c^2} ), the parentheses would be only one character tall, leaving small brackets stranded beside a large fraction. The second likewise grows the square brackets to fit a body containing a summation sign.

Two rules matter. First, **\left and \right must come in pairs — forgetting one is a compile error. Second, the paired delimiters need not be the same kind**: you can open with a parenthesis and close with a bracket, as in \left( … \right] for a half-open interval. Note that a \left … \right span forms a single group, and its output cannot contain a line break. To wrap a long formula, you must complete the pair within each line of an alignment environment.

Convenient as it is, \left … \right tends to pick a delimiter slightly larger than you might want, and because each delimiter is treated as an inner atom, the fine spacing you would expect around an operator or relation can be thrown off. Used everywhere, it can make brackets look oversized or leave their sizes inconsistent across lines — which is where the manual sizes below come in.

The invisible delimiter (`.` for a blank side)

Although \left and \right must always be paired, sometimes you want a delimiter on only one side. For that, put a **period .** immediately after the \left or \right on the side you want blank. The . is treated as a “delimiter that draws nothing” (a null delimiter): it satisfies the pairing rule while displaying nothing on that side.

\[
  \left. \frac{dy}{dx} \right|_{x=0}
  \qquad
  \left. \frac{x^3}{3} \right|_{0}^{1}
\]

This is the classic way to build an evaluation bar (“evaluate … at x=0”). The \left. on the left draws nothing, while only the \right| becomes a vertical line stretched to the height of the contents, with x=0 at its lower right. In the second example the bar grows to the height of the fraction x^3/3, with 0 at the bottom and 1 at the top — the familiar shape for evaluating a definite integral. You can also write the bar with a manual size, as in f(x) \big|_{0}^{1} (next section), which needs no matching \left.

The growing divider `\middle`

In set-builder notation { x | condition }, you often want a divider — a vertical bar — inside the outer delimiters that grows to the same height as them. Between \left and \right you may place **\middle followed by any delimiter**, as many times as you like. Writing \middle| inserts a vertical line in the middle, stretched to the same height as the outer braces.

\[
  \left\{\, x \in \mathbb{R} \;\middle|\; x^2 < \frac{1}{2} \,\right\}
\]

Here the outer braces grow to fit the contents (which include a tall fraction), and the \middle| bar grows to the same height, cleanly separating the condition on either side. A plain | would stay one character tall and look conspicuously small against the stretched braces. Note that \middle works only between \left and \right (never on its own).

Manual sizes: `\big` `\Big` `\bigg` `\Bigg`

\big \Big \bigg \Bigg let you choose the size yourself from four steps, increasing in that order, with the delimiter written right after, as in \big(. \big is only a little larger than an ordinary delimiter and can be used inline without spreading the lines apart. \Big is roughly 1.5 times as tall, and \bigg / \Bigg are larger still — better suited to displayed equations.

Why pick manual sizing over automatic \left … \right? Three reasons mainly. First, consistent size across lines: \left … \right sizes each line to its own contents, so brackets in a stack of equations tend to vary in height, whereas manual sizing pins every line to the same step. Second, avoiding oversized brackets: where contents only slightly taller than usual would push \left … \right up a size, you can hold it to \big. Third, control over spacing — and for that the l/r forms below are the key.

The bare forms like \big set the delimiter as an “ordinary atom,” adding stray space around it. To set it correctly as an opener or closer, use **\bigl for the opening and \bigr for the closing** (l for left, r for right; each size has \Bigl/\Bigr, \biggl/\biggr, \Biggl/\Biggr). The l/r forms are treated as open/close delimiters and insert no spurious space between the delimiter and a preceding atom. And to place a divider as a relation in the middle — the bar of set-builder notation, say — use the m (middle) form \bigm|, which adds the proper relational space on either side.

\[
  \bigl( a \bigr)
  \quad
  \Bigl( b \Bigr)
  \quad
  \biggl( c \biggr)
  \quad
  \Biggl( d \Biggr)
\]
\[
  \biggl\{\, x \bigm| x > 0 \,\biggr\}
\]

The first line shows the same parenthesis growing step by step from left to right (set to the chosen step’s height regardless of contents). The second is set-builder notation with braces fixed in size by \biggl\{ … \biggr\} and a \bigm| bar placed as a relation. Manual sizing also suits nested delimiters such as \bigl( \Bigl[ \,\cdots\, \Bigr] \bigr), where making the inner pair one size larger keeps the nesting easy to read.

The delimiter symbols

The table lists the main delimiters you can place after \left/\right or the \big family. Parentheses, brackets, and the slash are typed directly from the keyboard, but **because { } are reserved characters in LaTeX, you must write \{ \} to get braces as delimiters**, even in math. The single vertical bar is | (also \vert) and the double bar is \| (also \Vert).

Absolute value, norm & `\DeclarePairedDelimiter`

For absolute value and norm, writing plainly |x| or \|x\| can leave LaTeX unsure whether each | is an opener or a closer, throwing off the spacing. amsmath provides commands that distinguish the sides: for absolute value the **opener \lvert / closer \rvert, and for norm the opener \lVert / closer \rVert**, so they are set with the correct open/close spacing.

Even so, typing \left\lvert … \right\rvert every time is tedious. With \DeclarePairedDelimiter from the mathtools package you can fold the opener, closer, automatic sizing, and manual sizing into a single command. Define it in the preamble and you can write \abs{x} and \norm{v} in the body; the **starred forms \abs*{…} \norm*{…} call \left … \right internally to stretch to the contents**. The unstarred form is normal size, and an optional argument such as [\big] selects a manual size.

\usepackage{mathtools}
\DeclarePairedDelimiter{\abs}{\lvert}{\rvert}
\DeclarePairedDelimiter{\norm}{\lVert}{\rVert}

% 本文では：
\[
  \abs{x} \le \abs*{\frac{a}{b}}, \qquad \norm*{\frac{v}{2}}
\]

Here \abs{x} is an absolute value at normal size, while \abs*{\frac{a}{b}} wraps the fraction in bars stretched to its height. \norm*{…} likewise grows its double bars to the contents. If you use the same delimiter repeatedly, defining it once like this is the robust choice — spacing and size stay consistent throughout. See the mathtools page for details.