A sum is a result of an operation called addition and denoted (binary), (nullary), or (arbitrary).
A group (or similar algebraic structure) is written additively if its operation is a sum: . Examples include addition of natural numbers (an abelian monoid) and the generalisation to other kinds of number?s (most of which form abelian groups or at least abelian monoids, although the ordinal numbers form a nonabelian monoid).
In the case of numbers or more generally any topological abelian group or topological vector space (and generalizations), we can consider sums of infinite series, and more generally integrals. (There are however also noncommutative integrals when the order of summation/multiplication of noncommuting quantities is taken into account.)
In a category the term sum may refer to the coproduct of two objects; in particular, the sum of two abstract sets is their disjoint union. The sum of objects in a preadditive category may refer to the biproduct or direct sum.