Binary operators

Symbols such as \pm (±), \times (×), and \oplus (⊕) belong to a class that TeX internally calls binary operators. This class matters because TeX chooses the spacing around a symbol according to its class: a binary operator gets a medium space on each side, while a relation (=, <, …) gets a wider one. This page explains what a binary operator is, then gathers the commands into lookup tables — arithmetic, circled/boxed, set and lattice, and miscellaneous — marking which ones require amssymb.

What a binary operator is

TeX sorts every symbol in a formula into a handful of classes according to its role. The two most common are binary operators (+, \times, \cup, …), which join two things, and relations (=, <, \in, …), which state a relationship between two things. Each class has its own spacing: TeX automatically puts a medium space around a binary operator and a wider space around a relation. That is why = sits more loosely than + in a+b versus a=b.

This spacing is decided independently of whatever whitespace you type in the source. a\times b and a\times b produce the same output, because \times is a binary operator and gets a medium space. Conversely, getting a symbol’s class wrong upsets the spacing. If a would-be binary operator has nothing to “join” on one side, TeX treats it from context as a unary sign and tightens the space — which is why the - in -1 is set snugly as a negation rather than as subtraction.

% 二項演算子は中くらいの空き、関係子はより広い空きが自動で入る
\[ a \pm b \qquad a \times b \qquad A \cup B \]
% ソースの空白の数は出力に影響しない（クラスで決まる）
\[ a\otimes b \quad = \quad a \otimes b \]

Unless noted, every symbol in the tables below is math-mode only — use it inside $...$ or \[...\]. Most are standard LaTeX (no extra package); a few, such as some circled/boxed forms, need the amssymb package. Only the entries whose Notes column says “needs amssymb” require it; everything else works out of the box. Relations themselves are covered on the separate “Relations” page.

Arithmetic and basic operators

Start with the everyday operators. \pm (±) and \mp (∓) are the plus-or-minus signs, as in x = a \pm b (\mp is the reversed “minus-or-plus”). For multiplication, choose by context: \times (×) for numbers or cross products, \cdot (⋅) for scalar products or a dot product. The division sign \div (÷) is mostly for elementary teaching; research writing normally uses a fraction with \frac.

Note that \cdot (⋅) is not the same as a plain period .. Typing a.b treats the period as punctuation — no operator spacing, and it sits on the baseline — so it does not read as multiplication. For a multiplication dot, always use \cdot. Likewise \ast (∗) is a star that takes binary-operator spacing, which differs from the ordinary * character you might place as a superscript.

Circled and boxed operators

Circled operators — direct sum \oplus (⊕), tensor product \otimes (⊗), the Hadamard-style \odot (⊙), and so on — turn up constantly in algebra and category theory. These are all standard LaTeX. The boxed forms \boxplus (⊞) and \boxtimes (⊠), however, **require the amssymb package** (\usepackage{amssymb} in the preamble). Among the circled family, extras like \circledast (⊛) are AMS additions and also need amssymb.

% 丸囲みは標準。角囲みは amssymb が必要
\usepackage{amssymb}
% ...
\[ V \oplus W, \qquad u \otimes v, \qquad A \boxplus B \]

Note that the big, sum-like versions of these — \bigoplus (⨁), \bigotimes, \bigodot, \biguplus, and so on — are not binary operators but big operators (variable-sized operators) that take limits above and below. They are covered on the separate “Sums, integrals & big operators” page.

Set and lattice operators

For set operations, use intersection \cap (∩), union \cup (∪), multiset union \uplus (⊎), and set difference \setminus (∖). In lattice or order contexts you meet the square \sqcap (⊓) and \sqcup (⊔), and the meet/join \wedge (∧) and \vee (∨), which double as logical and/or. \wedge has the alias \land, and \vee the alias \lor (handy when you want to stress the logic); both are standard and give the same glyph. All of these are standard LaTeX.

Set difference \setminus (∖) is a dedicated symbol with binary-operator spacing — not a bare backslash — written A \setminus B. With amssymb loaded you also get a thinner \smallsetminus (∖). \sqcap and \sqcup are common in posets and domain theory, and pairing them with the rounded \cap/\cup consistently keeps a formula readable.

Other operators

A few more common binary operators. There is the coproduct/amalgamation \amalg (⨿); the dagger \dagger (†) and double dagger \ddagger (‡), also familiar as footnote marks; and the triangles \triangleleft (◁) and \triangleright (▷), used for partial orders and group actions. Everything listed here is standard LaTeX.

The lower half of the table above — \dotplus (∔), \intercal (⊺), \ltimes (⋉), \rtimes (⋊) — are AMS additions that **need amssymb**. The semidirect products \ltimes/\rtimes are common in group theory, so remember \usepackage{amssymb} when you reach for them. (Some authors use \triangleleft for a normal subgroup; if you prefer the heavier \lhd (◁), that needs amssymb or similar.)

One practical note to close. Sometimes you want a symbol to behave like a binary operator, but TeX assigns it a different class. Wrapping it in \mathbin{...} forces TeX to treat the contents as a binary operator and gives the correct medium spacing (use \mathrel{...} to force a relation instead). For the full picture of classes and spacing, see also “Math mode basics.”