Additions

Additions refers to the act or operation of combining quantities to produce a total. In mathematics, addition is a binary operation that takes two elements from a set and returns another element within the same set, typically denoted by the plus sign.

In arithmetic with numbers, addition is associative and commutative and has an identity element, zero. Every element has an additive inverse with respect to addition under suitable sets, enabling subtraction as an inverse operation. Beyond numbers, addition applies to vectors (component-wise), matrices (entrywise), polynomials (coefficients added), and functions (pointwise). In many algebraic contexts, addition is defined to be closed, associative, and, in familiar settings, commutative, which makes the underlying structure an abelian group or vector space when combined with other operations.

The concept of addition goes back to ancient mathematics, with the practice of adding quantities present in early civilizations. The modern plus symbol (+) was introduced in medieval and early modern Europe and became standard in mathematical notation by the early modern period, alongside broader developments in arithmetic and algebra.

Applications of addition are widespread. It is a foundational operation in computation, science, and finance for calculating totals, sums, and cumulative quantities. In linear algebra, addition is a basic operation on vectors and matrices. In calculus and analysis, addition extends to series and functions, forming the basis for many constructions in mathematics and applied disciplines.