206 8. NUMBERS AND NUMBER THEORY IN MODERN MATHEMATICSMathematicians have accepted the need for Dedekind's rigor in the teaching of
mathematics majors, although the idea of defining real numbers as partitions of the
rational numbers (Dedekind cuts) is no longer the most popular approach to that
rigor. More often, students are now given a set of axioms for the real numbers and
asked to accept on faith that those axioms are consistent and that they characterize
a set that has the properties of a geometric line. Only a few books attempt to start
with the rational numbers and construct the real numbers. Those that do tend to
follow an alternative approach, defining a real number to be a sequence of rational
numbers (more precisely, an equivalence class of such sequences, one of which is the
sequence of successive decimal approximations to the number).
2.3. Imaginary and complex numbers. Although imaginary numbers seem
more abstract to moderns than irrational numbers, that is because their physical
interpretation is more remote from everyday experience. One interpretation of i —
for example, is as a rotation through a right angle (the effect of multiplying by
i in the complex plane). We have an intuitive concept of the length of a line segment
and decimal approximations to describe that length as a number; that is what gives
us confidence that irrational numbers really are numbers. But it is difficult to think
of a rotation as a number. On the other hand, the rules for multiplying complex
numbers—at least those whose real and imaginary parts are rational—are much
simpler and easier to understand than the definition just given for irrationals. In
fact, complex numbers were understood before real numbers were properly defined;
mathematicians began trying to make sense of them as soon as there was a clear
need to do so. That need came not, as one might expect, from trying to solve
quadratic equations such as x^2 — 2x + 2 = 0. where the quadratic formula produces
.r = — 1 ± y/—T. It was possible in this case simply to say that the equation had
no solution. On the other hand, as discussed in Chapter 14, the sixteenth-century
Italian mathematicians succeeded in giving an arithmetic solution of the general
cubic equation. However, the algorithm for finding the solution had the peculiar
property that it involved taking the square root of a negative number precisely
when there were three real solutions. Looking at their algorithm as a formula, one
would find that the solution of the equation x^3 — 7x + 6 = 0 is
We cannot say that the equation has no roots, since it obviously has 1, 2, and —3
as roots. Thus the challenge arose: Make sense of this formula. Make it say "1, 2,
and —3."
This challenge was taken up by Rafael Bombelli (1526- 1572), an engineer in
the service of an Italian nobleman. Bombelli was the author of a treatise on algebra
which he wrote in 1560, but which was not published until 1572. In that treatise
he invented the name "plus of minus" to denote a square root of —1 and "minus of
minus" for its negative. He did not think of these two concepts as different numbers,
but rather as the same number being added in the first case and subtracted in the
second. What is most important is that he realized what rules must apply to them
in computation: plus of minus times plus of minus makes minus and minus of minus
times minus of minus makes minus, while plus of minus times minus of minus makes
plus. Bombelli had no systematic way of taking the cube root of a complex number.
In considering the equation x^3 = 15x + 4, he found by applying the formula that