Composition / ring (∘) in LaTeX
\circ produces a small ring (∘), used above all for function composition, ()(x) = f(g(x)), and more generally for any abstract binary operation.
LaTeX command
\circNot the symbol you meant? Draw it to find the command →
Composition order matters: in , the function on the right, g, is applied first, and its output is fed into f. This reads opposite to left-to-right English, so it is worth double-checking which function acts first whenever \circ appears, especially since is generally a different function from .
In abstract algebra, \circ is also used as a placeholder symbol for a generic binary operation on a group or set, , when the author does not want to commit to a specific operation like + or \times. The meaning is then whatever the text defines it to be.
\circ is unrelated to the degree symbol despite looking similar; a superscripted \circ, as in 90^{\circ}, is how LaTeX writes degrees. Context (a lone superscript versus a symbol between two functions) is what distinguishes the two uses.
Other forms
\bulletbullet (∙)Examples
$(f \circ g)(x) = f(g(x))$function composition, g is applied first
$(g \circ f)(x) = g(f(x))$composition the other way round, generally not equal to
$a \circ b = b \circ a$a generic binary operation, here assumed commutative
Tips and common mistakes
- Read right to left: apply g first, then f.
\circsuperscripted after a number, like 90^{\circ}, is the degree symbol, a completely different meaning from composition; only the position (superscript versus between two symbols) tells them apart.- In group or ring theory,
\circis a common neutral stand-in for 'the' operation when the author hasn't fixed it to + or\timesyet. - Don't use
\circfor ordinary numeric multiplication;\timesor\cdotare the conventional choices there.
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What does \circ mean in LaTeX?
Most often function composition, , meaning apply g first then f. It is also used as a generic abstract binary operation, and, as a superscript, as the degree symbol.
How do I write function composition in LaTeX?
()(x) = f(g(x)), placing \circ between the two function names.
What's the difference between \circ and \cdot?
\circ is a small ring, typically composition or a generic operation; \cdot is a centred dot, typically multiplication or a vector dot product.
How do I write degrees using \circ?
As a bare superscript after the number, e.g. 90^{\circ}; there is no separate \degree command in core LaTeX.