Multiplication

of numbers

One of the basic arithmetic operations. Multiplication consists in assigning to two numbers $a,b$ (called the factors) a third number $c$ (called the product). Multiplication is denoted by the sign $\times$ or $\cdot$; in notations using letters the sign is, as a rule, omitted.

Multiplication of positive integers is defined in the following way by means of addition: the product of a number $a$ by a number $b$ is the number $c$ equal to the sum of $b$ summands each of which is equal to $a$; thus,

\[ab=a+\dots+a\quad(b\text{ times}).\]

The number $a$ is called the multiplicand and $b$ the multiplier. Multiplication of two positive rational numbers $m/n$ and $p/q$ is defined by the equation

\[\frac mn\cdot\frac pq=\frac{mp}{nq}\]

(cf. Fraction). The product of two negative fractions is positive, and that of a positive and a negative fraction is negative, where the modulus of the product in both cases is equal to the product of the moduli of the factors. The product of irrational numbers is defined as the limit of the products of rational approximations of them. Multiplication of two complex numbers $\alpha=a+bi$ and $\beta=c+di$ is given by the formula

\[\alpha\beta=(a+bi)(c+di)=ac-bd+(ad+bc)i,\]

or, in trigonometric form $(\alpha=r_1(\cos\phi_1+i\sin\phi_1),\ \beta=r_2(\cos\phi_2+i\sin\phi_2))$, by

\[\alpha\beta=r_1r_2(\cos(\phi_1+\phi_2)+i\sin(\phi_1+\phi_2)).\]

Multiplication of numbers is commutative, associative and distributive on the left and right relative to addition (cf. Commutativity; Associativity; Distributivity). Further, $a\cdot0=0$, $a\cdot1=a$.

In general algebra, multiplication may be any algebraic operation ($n$-ary, $n>2$); most frequently a binary operation (cf. Magma). In certain cases this operation is a generalization of the usual multiplication of numbers. E.g. multiplication of quaternions, multiplication of matrices and multiplication of transformations. However, properties of the multiplication of numbers (e.g. commutativity) may be lost in these cases.