Kvotienten

Kvotienten of two numbers a and b, with b ≠ 0, is the result of dividing a by b, commonly written q = a/b. It expresses how many times b fits into a. The kvotienten may be a real number, a fraction, or an integer, depending on the values.

If a and b are integers and b divides a, the kvotienten is an integer; otherwise it is a rational number, and its decimal expansion may terminate or repeat. The kvotienten is often represented in reduced form by dividing the numerator and denominator by their greatest common divisor.

Division by zero is undefined; consequently kvotienten is defined only for nonzero denominators. In higher mathematics, the concept generalizes to quotient objects. In group theory, the quotient G/N is the set of cosets of a normal subgroup N; in ring theory, R/I is a quotient ring by an ideal I. The quotient map sends each element to its equivalence class and preserves structure.

Examples: 14 divided by 3 yields q ≈ 4.666..., whereas 12 divided by 4 yields q = 3. In modular arithmetic, one often considers quotients modulo a, where the division concept is replaced by multiplicative inverses.

See also: quotient, ratio, fraction, division, quotient group, quotient ring.