subtracting

Subtracting is an arithmetic operation that computes the difference between two numbers. It is written as a - b, and is defined as the operation of adding the additive inverse: a - b = a + (-b). The result represents how much remains after removing b from a. If a ≥ b the result is nonnegative; if a < b the result is negative.

Subtraction is defined on standard number systems such as integers, rational numbers, real numbers, and complex numbers; it can be extended component-wise to vectors and matrices, and more generally to any algebraic structure that has an additive inverse.

Properties: Subtraction is not commutative, since a - b ≠ b - a in general. It is also not associative: (a - b) - c ≠ a - (b - c). It has identity element 0, in the sense that a - 0 = a. It interacts with addition and negation through a - b = a + (-b).

In decimal arithmetic, subtraction is performed column by column with borrowing when needed. Computers implement subtraction using various representations, such as two's complement for signed integers.

Differences include the absolute difference |a - b|, which measures distance between numbers. Subtraction can be generalized to vectors via component-wise subtraction, producing a vector a - b, and to matrices by matrix difference A - B = A + (-B).

Common contexts include linear equations, change over time, and time series differences.