Introduction to Mathematical Philosophy

of the two real numbers. (In all such definitions, the series of ratios is to be defined as excluding 0 and infinity.) There is no difficulty in extending our definitions to positive and negative real numbers and their addition and multiplication. It remains to give the definition of complex numbers. Complex numbers, though capable of a geometrical interpretation, are not demanded by geometry in the same imperative way in which irrationals are demanded. A "complex" number means a number involving the square root of a negative number, whether integral, fractional, or real. Since the square of a negative number is positive, a number whose square is to be negative has to be a new sort of number. Using the letter for the square root of , any number involving the square root of a negative number can be expressed in the form , where and are real. The part is called the "imaginary" part of this number, being the "real" part. (The reason for the phrase "real numbers" is that they are contrasted with such as are "imaginary.") Complex numbers have been for a long time habitually used by mathematicians, in spite of the absence of any precise definition. It has been simply assumed that they would obey the usual arithmetical rules, and on this assumption their employment has been found profitable. They are required less for geometry than for algebra and analysis. We desire, for example, to be able to say that every quadratic equation has two roots, and every cubic equation has three, and so on. But if we are confined to real numbers, such an equation as has no roots, and such an equation as has only one. Every generalisation of number has first presented itself as needed for some simple problem: negative numbers were needed in order that subtraction might be always possible, since otherwise would be meaningless if were less than ; fractions were needed [Pg 74] in order that division might be always possible; and complex numbers are needed in order that extraction of roots and solution of equations may be always possible. But extensions of number are not created by the mere need for them: they are created by the definition, and it is to the definition of complex numbers that we must now turn our attention. A complex number may be regarded and defined as simply an ordered couple of real numbers. Here, as elsewhere, many definitions are possible. All that is necessary is that the definitions adopted shall lead to certain properties. In the case of complex numbers, if they are defined as ordered couples of real numbers, we secure at once some of the properties required, namely, that two real numbers are required to determine a complex number, and that among these we can distinguish a first and a second, and that two complex numbers are only identical when the first real number involved in the one is equal to the first involved in the other, and the second to the second. What is needed further can be secured by defining the rules of addition and multiplication. We are to have Thus we shall define that, given two ordered couples of real numbers, and , their sum is to be the couple , and their product is to be the couple . By these definitions we shall secure that our ordered couples shall have the properties we desire. For example, take the product of the two couples and . This will, by the above rule, be the couple . Thus the square of the couple will be the couple . Now those couples in which the second term is 0 are those which, according to the usual nomenclature, have their imaginary part zero; in the notation , they are , which it is natural to write simply . Just as it is natural (but erroneous) to identify ratios whose denominator is unity with integers, so it is natural (but erroneous) [Pg 75] to identify complex numbers whose imaginary part is zero with real numbers. Although this is an error in theory, it is a convenience in practice; "" may be replaced simply by "" and "" by "," provided we remember that the "" is not really a real number, but a special case of a complex number. And when is 1, "" may of course be replaced by "." Thus the couple is represented by , and the couple is represented by -1. Now our rules of multiplication make the square of equal to , i.e. the square of is -1. This is what we desired to secure. Thus our definitions serve all necessary purposes. It is easy to give a geometrical interpretation of complex numbers in the geometry of the plane. This subject was agreeably expounded by W. K. Clifford in his Common Sense of the Exact Sciences, a book of great merit, but written before the importance of purely logical definitions had been realised. Complex numbers of a higher order, though much less useful and important than those what we have been defining, have certain uses that are not without importance in geometry, as may be seen, for example, in Dr Whitehead's Universal Algebra. The definition of complex numbers of order is obtained by an obvious extension of the definition we have given. We define a complex number of order as a one-many relation whose domain consists of certain real numbers and whose converse domain consists of the integers from 1 to .[19] This is what would ordinarily be indicated by the notation , where the suffixes denote correlation with the integers used as suffixes, and the correlation is one-many, not necessarily one-one, because and may be equal when and are not equal. The above definition, with a suitable rule of multiplication, will serve all purposes for which complex numbers of higher orders are needed. [19]Cf. Principles of Mathematics, § 360, p. 379. We have now completed our review of those extensions of number which do not involve infinity. The application of number to infinite collections must be our next topic. [Pg 76]