Introduction to Mathematical Philosophy

By Bertrand Russell | Go to book overview

CHAPTER VII
RATIONAL, REAL, AND COMPLEX NUMBERS

WE have now seen how to define cardinal numbers, and also relation-numbers, of which what are commonly called ordinal numbers are a particular species. It will be found that each of these kinds of number may be infinite just as well as finite. But neither is capable, as it stands, of the more familiar extensions of the idea of number, namely, the extensions to negative, fractional, irrational, and complex numbers. In the present chapter we shall briefly supply logical definitions of these various extensions.

One of the mistakes that have delayed the discovery of correct definitions in this region is the common idea that each extension of number included the previous sorts as special cases. It was thought that, in dealing with positive and negative integers, the positive integers might be identified with the original signless integers. Again it was thought that a fraction whose denominator is 1 may be identified with the natural number which is its numerator. And the irrational numbers, such as the square root of 2, were supposed to find their place among rational fractions, as being greater than some of them and less than the others, so that rational and irrational numbers could be taken together as one class, called "real numbers." And when the idea of number was further extended so as to include "complex" numbers, i.e. numbers involving the square root of -- 1, it was thought that real numbers could be regarded as those among complex numbers in which the imaginary part (i.e. the part

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