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

IMAGINARY NUMBERS 109
should thereby be led into more technical
processes of mathematics than falls within the
design of this book. But now we can immediately see that the associative law [numbered (y) above] for multiplication is satisfied.
Consider first the length of the resultant
vector; this is got by the ordinary process
of multiplication for real numbers; and thus
the associative law holds for it.
Again, the direction of the resultant vector
is got by the mere addition of angles, and the
associative law holds for this process also.
So much for multiplication. We have now
rapidly indicated, by considering addition and
multiplication, how an algebra or "calculus"
of vectors in one plane can be constructed,
which is such that any two vectors in the
plane can be added, or subtracted, and can
be multiplied, or divided one by the other.
We have not considered the technical details of all these processes because it would
lead us too far into mathematical details;
but we have shown the general mode of procedure. When we are interpreting our algebraic symbols in this way, we are said to be
employing "imaginary quantities" or "complex quantities." These terms are mere
details, and we have far too much to think
about to stop to enquire whether they are or
are not very happily chosen.
The net result of our investigations is that