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

90 INTRODUCTION TO MATHEMATICS
generality which we have attained by the use
of the positive and negative numbers. In
fact, it must in a sense include them as
special cases. When a is negative we may
write -c2 for it, so that c2 is positive. Then
a= V (i( - 2) /= c(-1) X 2
=VVÏ) VC2 =cV(.
Hence, if we can so interpret our symbols that
(- 1) has a meaning, we have attained our
object. Thus V(-1) has come to be looked
on as the head and forefront of all the
imaginary quantities.
This business of finding an interpretation
for (-1) is a much tougher job than the
analogous one of interpreting -1. In fact,
while the easier problem was solved almost
instinctively as soon as it arose, it at first
hardly occurred, even to the greatest mathematicians, that a problem existed which was
perhaps capable of solution. Equations like
x2= -3, when they arose, were simply ruled
aside as nonsense.
However, it came to be gradually perceived
during the eighteenth century, and even
earlier, how very convenient it would be if
an interpretation could be assigned to these
nonsensical symbols. Formal reasoning with
these symbols was gone through, merely
assuming that they obeyed the ordinary