Hypercomplex Numbers

a generalization of the concept of numbers that is broader than the usual complex numbers. The meaning of the generalization consists in the fact that the ordinary arithmetic operations involving these numbers simultaneously expressed some geometric processes in multidimensional space or gave a quantitative description of some physical laws. Attempts to devise numbers that would play the same role for three-dimensional space as complex numbers do for a plane revealed that a complete analogy is impossible in this case. This led to the creation and development of systems of hypercomplex numbers.

Hypercomplex numbers are linear combinations (with real coefficients x₁, x₂,…, x_n) of a certain system e₁ e₂,…,e_n of “basis units”:

(1) x₁e₂ + x₂e₂ + …+ x_ne_n

in the same way as the complex numbers x + iy are linear combinations of two “basis units,” namely of the real unit 1 and of the imaginary unit i. In order to make use of hyper-complex numbers, it is, first of all, necessary to define the rules concerning the arithmetic operations for these numbers. The addition and subtraction of hypercomplex numbers are uniquely defined if the usual rules of arithmetic are retained for the new numbers, namely if the components x_l x₂,…, x_n of the “basis units” are added and subtracted, respectively.

The true significance of the problem, however, becomes clearly evident only on establishing the multiplication rule. In order to establish the term-by-term multiplication of the hypercomplex number of the form of equation (1), it is necessary to determine the values of the n² products of e_ie_k (i = 1, 2,…, n; k = 1, 2,…, n). The problem consists in ascribing to these products values of the form of equation (1) that preserve all of usual rules of arithmetic operations. This requirement is fulfilled (beyond the simplest case of real numbers) by only one system of hypercomplex numbers, namely the system of complex numbers. On devising any other hypercomplex number system, it is necessary to renounce one or another of the rules of arithmetic. The following rules are the ones that are usually violated: the single-valued nature of the result of division, the commutative nature of multiplication, the rule according to which the equality to zero of the product of two numbers implies that at least one of the factors is equal to zero, and so on. The most important system of hypercomplex numbers is that of quaternions, which results from the abandonment of the commutative properties of multiplication while preserving the remaining properties of addition and multiplication.