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] CHAPTER VIII INFINITE CARDINAL NUMBERS