Adding

Addition is a mathematical operation that combines two or more numbers, called addends, to produce a sum. In arithmetic, the plus symbol + denotes the operation. Addition is defined for many number systems, including integers, rationals, reals, and complex numbers, as well as for vectors, matrices, and functions, where the result is interpreted in a corresponding structure (component-wise for vectors, element-wise for matrices).

Basic properties include commutativity (a + b = b + a), associativity ((a + b) + c = a + (b + c)), and the identity element 0, since a + 0 = a. Subtraction serves as the inverse operation: a + (-a) = 0. For nonnegative numbers, addition corresponds to combining quantities to increase value; with negative numbers it can increase or decrease the total depending on signs.

Techniques and variations: in decimal arithmetic, addition proceeds from right to left, carrying when the sum of digits exceeds 9. In binary arithmetic (used by computers), digits are 0 or 1 and addition uses carries to higher bits. In modular arithmetic, addition is performed modulo n, wrapping around after n. Vector addition adds corresponding components; matrix addition adds corresponding entries.

Applications: addition is a foundational operation in mathematics and science, used to combine measurements, quantities, probabilities, and datasets. It underpins algebraic structures, analysis, and computer algorithms. It generalizes to more complex objects such as functions, where addition is defined pointwise, and to higher-dimensional entities through component-wise addition.