4 CHAPTER 1 • COMPLEX NUMBERSSince (2 + J=l)^3 = 2 + nJ=I, we clearly have 2 + A = {12 + UJ=l.
Similarly, Bombelli showed that 2 - A = {/2 - 11 J=I. But this means that
?/2+ uJ=I+ ?/2-11.J=l = (2+ J=l) + (2 - J=l) =4, (1-6)
which was a proverbial bombshell. Prior to Bombelli, mathematicians could
easily scoff at imaginary numbers when they arose as solutions to quadratic
equations. With cubic equations, they no longer had this luxury. That x = 4 was
a correct solution to the equation x^3 - 15x - 4 = 0 was indisputable, as it could
be checked easily. However, to arrive at this very real solution, mathematicians
had to take a detour through the uncharted territory of "imaginary numbers."
Thus, whatever else might have been said about these numbers (which, today,
we call complex numbers), their utility could no longer be ignored.
Admittedly, Bombelli's technique applies only to a few specialized cases, and
lots of work remained to be done even if Bombelli's results could be extended.
After all, today we represent real numbers geometrically on the number line.
What possible representation could complex numbers have? In 1673 John Wallis
made a stab at a geometric picture of complex numbers that comes close to what
we use today. He was interested in representing solutions to general quadratic
equations, which we can write as x^2 +2bx+c^2 = 0 to make the following discussion
easier to follow. \Vhen we use the quadratic formula with this equation, we get
x = -b - ,,/b^2 - c^2 and x = -b + ,,/b^2 - c^2.
Wallis imagined these solutions as displacements to the left and right from
the point -b. He saw each displacement, whose value is Jb^2 - c^2 , as the length
of the sides of the right triangles shown in Figure 1.1.
The points P 1 and P 2 represent the solutions to our equation, which is clearly
correct if b^2 - t? 2: O. But how should we picture P 1 and P2 when negative roots
arise (i.e., when b^2 - c^2 < O)? Wallis reasoned that, with negative roots, b would
be less than c, so the lines of length b in Figure 1.1 would no longer be able to
reach all the way to the x-axis. Instead, they would stop somewhere above it, as
Figure 1.2 shows. Wallis argued that P 1 and P 2 should represent the geometric
locations of the solutions x = - b - ~ and x = - b + ~ whenyF igure 1.1 Wallis's representation of real roots of quadratics.