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

IMAGINARY NUMBERS 89
solutions is usually written x = - / V2. So far
all is plain sailing, as it was in the previous
case. But now an analogous difficulty arises.
For the equation x2+3 =1 gives x2 = -2 and
there is no positive or negative number which,
when multiplied by itself, will give a negative
square. Hence, if our symbols are to mean
the ordinary positive or negative numbers,
there is no solution to x2= -2, and the equation is in fact nonsense. Thus, finally taking
the general form x2+a=b, we find the pair
of solutions x = -- V(b-a), when, and only
when, b is not less than a. Accordingly we
cannot say unrestrictedly that the "constants" a and b may be any numbers, that is,
the "constants" a and b are not, as they
ought to be, independent unrestricted "variables"; and so again a host of limitations
and restrictions will accumulate round our
work as we proceed.
The same task as before therefore awaits
us: we must give a new interpretation to our
symbols, so that the solutions -~ <(b-a) for
the equation x2+a=b always have meaning.
In other words, we require an interpretation
of the symbols so that Va always has meaning whether a be positive or negative. Of
course, the interpretation must be such that
all the ordinary formal laws for addition, subtraction, multiplication, and division hold
good; and also it must not interfere with the