multiplication

a binary operation that associates to objects a, b an object c; a and b are called factors, and c is called their product. Multiplication is indicated by the symbol × or by the symbol. The first symbol was introduced by the English mathematician W. Oughtred in 1631, and the second by the German savant G. von Leibniz in 1698. When multiplying letters rather than numbers, we omit these symbols and write ab instead of a × b or a · b. The concrete sense of a multiplication depends on the nature of the factors and the definition of the multiplication. Multiplication of positive integers is the operation that associates to positive integers a and b the positive integer c = ab = a + a + . . . + a, where a is taken b times. Multiplication of fractions m/n and p/q is defined by the equation

The product of fractions is a fraction whose absolute value is the product of the absolute values of the factors. The product of fractions is positive if both factors have the same sign and is negative otherwise. Multiplication of irrational numbers is defined in terms of multiplication of rational approximations of these numbers. Multiplication of complex numbers α and β given as α = a + bi and β = c + di is defined by means of the equation

αβ = (ac – bd) + (ad + bc)i

If α and β are given in polar form,

α = r₁(cos φ₁) + i sin φ₁)

β = r₂(cos φ₂ + i sin φ₂)

then αβ is defined as

αβ = r₁,r₂ {cos (φ₁ + φ₂) + i sin (φ₁ + φ₂)}

that is, the modulus of the product is the product of the moduli of the factors and the argument of the product is the sum of the arguments of the factors.

Multiplication of numbers has the following properties: (1) ab = ba (commutativity), (2) a(bc) = (ab)c (associativity), and (3) a(b + c) = ab + ac (distributivity of multiplication over addition). We have a · 0 = 0 and a · 1 = a. The techniques for multiplying multivalued expressions rely on these properties.

Further generalization of multiplication relies on the possibility of viewing numbers as operators on vectors in the plane. Thus, to the complex number r(cos φ + i sin φ) we associate the operator of dilation of all vectors by a factor r and their rotation through an angle φ about the origin. Here, to the product of complex numbers there corresponds the product of the operators associated with these numbers, that is, the operator that is the result of successive application of the operators associated with the numbers in question. Such multiplication of operators can be extended to operators that cannot be represented by numbers, for example, to linear operators. In this way, we are led to define multiplication of matrices, of quarternions viewed as dilations and rotations in 3-space, and of kernels of integral operators. In these generalizations some of the properties of multiplication of numbers may not hold. The property that fails to hold most frequently is commutativity.

The study of the general properties of multiplication is part of algebra, in particular, group theory and ring theory.