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

CHAPTER VIII
IMAGINARY NUMBERS (Continued)
THE definition of the multiplication of
ordered couples is guided by exactly the same
considerations as is that of their addition.
The interpretation of multiplication must be
such that
(a) the result is another ordered couple,
(,8) the operation is commutative, so that
(x, y) X (', y') = (', y') X (x, y),
(r) the operation is associative, so that
i(x, y)X (x', y')}X (u, v)
= (x, y){(x', y')X(u, v)},
(8) must make the result of division unique
[with an exception for the case of the zero
couple (0, 0)], so that when we seek to determine the unknown couple (x, y) so as to
satisfy the equation
(x, y)X (a, b)=(c, d),
there is one and only one answer, which we
can represent by
(x, y) = (c, d)- (a, b), or by (x, y) =, d)
101(a, b)
101