DIVISION

Division is a mathematical operation that combines two numbers, the dividend and the divisor, to produce a quotient. It answers how many times the divisor is contained in the dividend. It is typically written as a ÷ b or a/b. In arithmetic over real numbers or complex numbers, division by zero is undefined.

In fields and most number systems, division is the inverse of multiplication: a/b × b = a for b ≠ 0; equivalently a × b^{-1} = a/b when b has a multiplicative inverse.

Dividing integers yields a quotient and remainder: for integers a and b ≠ 0, there exist unique q and r such that a = b q + r with 0 ≤ r < |b|. The remainder r measures how close a is to a multiple of b. If r = 0, b divides a.

Divisibility: a divides b, written a|b, if ∃ k with b = a k.

Division in algebra: a division ring is a ring where every nonzero element has a multiplicative inverse; a field is a commutative division ring. Some algebraic structures use division in an informal sense.

Computing: many programming languages distinguish between true division (producing a floating result) and integer division (quotient truncating toward zero). The remainder operator yields the modulus.

Representations: rational numbers are ratios p/q; performing long division yields terminating or repeating decimals.

Special cases: 0 / a = 0 for a ≠ 0; a / 0 is undefined.