An introduction to mathematics, by A. N. Whitehead.

104 INTRODUCTION TO MATHEMATICS
Finally consider (0,1): this will interpret
for us the symbol V/( -1). The symbol must
therefore possess the characteristic property
that V/(-1) XV(-1)= -1. Now by the
law of multiplication for ordered couples
(0,1) X (0,1) = {(0 -1), (0+0)} = ( -1, 0).
But (1,0) is the unit couple, and (-1, 0)
is the negative unit couple; so that (0,1) has
the desired property. There are, however,
two roots of -1 to beprovided for, namely
~/(- 1). Consider (0, - 1); here again remembering that (- 1)2 = 1, we find, (0, -1)
X(0, -1)=(-1, 0).
Thus (0, -1) is the other square root of
V(-1l). Accordingly the ordered couples
(0,1) and (0, -1) are the interpretations of
— /i(-1) in terms of ordered couples. But
which corresponds to which? Does (0,1)
correspond to +V/(-1) and (0, -1) to
-(-1), or (0,1) to -V/(-1), and (0, -1)
to +/(- 1) The answer is that it is perfectly indifferent which symbolism we adopt.
The ordered couples can be divided into
three types, (i) the "complex imaginary"
type (x,y), in which neither x nor y is zero;
(ii) the "real" type (x,0); (iii) the "pure
imaginary" type (0,y). Let us consider the
relations of these types to each other. First
multiply together the "complex imaginary"