Subtraction

Subtraction is an arithmetic operation that computes the difference between two numbers. It is denoted by the minus sign (−) and is the inverse of addition: if a + b = c, then c − a = b and c − b = a. In standard notation, a is the minuend, b is the subtrahend, and a − b is the difference.

Rules of subtraction include that if the minuend is at least as large as the subtrahend, the difference is nonnegative. If the subtrahend is larger, the difference is negative. Subtraction is not commutative or associative: a − b ≠ b − a in general, and (a − b) − c ≠ a − (b − c). In base-10 arithmetic, subtraction often uses borrowing (regrouping) when digits in the minuend are smaller than those in the subtrahend; the concept generalizes to other numeral bases as well.

Interpretation of subtraction includes its geometric view on a number line, where it represents the distance from one value to another, with the sign indicating direction. The expression |a − b| is the absolute difference and measures the gap between a and b. Algebraically, subtraction extends to integers, fractions, and real numbers, obeying properties such as a − (b + c) = (a − b) − c, with the latter equivalent to a − b − c.

Applications and variants: Subtraction is used in everyday calculations, budgeting, measurement, and data analysis. In computing, it can be implemented by adding the additive inverse. In modular arithmetic, subtraction is performed modulo a given modulus. Subtraction also has historical roots across ancient numeral systems and remains fundamental in mathematics and applied sciences.