Negative numbers, fractions and irrational numbers
The simplest path towards understanding the so-called extensions of the number concept lies through the operations inverse to addition, multiplication and potentiation. Let us head our investigations with an observation by Russell that lays bare the basic mistake in the ingrained conception of these new "numbers': "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 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 which was a multiple of the square root of — 1) was zero. All these suppositions were erroneous, and must be discarded… if correct definitions are to be given."1
Kaufmann, F. (1978). Negative numbers, fractions and irrational numbers, in The infinite in mathematics, Dordrecht, Springer, pp. 91-113.
This document is unfortunately not available for download at the moment.