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 which was a multiple of the square root of -1) was zero. All these suppositions were erroneous, and must be discarded, as we shall find, if correct definitions are to be given.
Let us begin with positive and negative integers. It is obvious on a moment's consideration that +1 and -1 must both be relations, and in fact must be each other's converses. The obvious and sufficient definition is that +1 is the relation of
to
, and -1 is the relation of
to
. Generally, if