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

IMAGINARY NUMBERS 103
assigned in terms of positive and negative
real numbers. We then found that all our
difficulties would vanish if we could interpret
the equation x2= -1, i.e., if we could so define (- 1) that /(-1) XV (-1) = -1.
Now let us consider the three special ordered couples * (0,0), (1,0), and (0,1).
We have already proved that
(x, y) +(0, 0)= (x, y).
Furthermore we now have
(x, y) x (0, 0) = (0, 0).
Hence both for addition and for multiplication the couple (0,0) plays the part of zero in
elementary arithmetic and algebra; compare the above equations with x +0 =x, and
xXO=0.
Again consider (1, 0): this plays the part
of 1 in elementary arithmetic and algebra.
In these elementary sciences the special characteristic of 1 is that xXl =x, for all values
of x. Now by our law of multiplication
(x, y) X(1, 0) = {(x -0), (y +O)} =(x, y).
Thus (1, O) is the unit couple.
* For the future we follow the custom of omitting the
+ sign wherever possible, thus (1,0) stands for (+ 1,0) and
(0,1) for (0,+1).