Chapter I

Complex Numbers

The complex number system, C, possesses both an algebraic and a topologi-

cal structure. In algebraic terms C is a ﬁeld: C is equipped with two binary

operations, addition and multiplication, satisfying certain axioms (listed be-

low). It is the ﬁeld one obtains by adjoining to R, the ﬁeld of real numbers, a

square root of −1. Remarkably, the ﬁeld so created contains not only square

roots of each of its elements, but even n-th roots for every positive integer n.

All the more remarkable is that adjoining a solution of the single equation

x2

+ 1 = 0 to R results in an algebraically closed ﬁeld: every nonconstant

polynomial with coeﬃcients in C can be factored over C into linear factors.

This is the “Fundamental Theorem of Algebra,” ﬁrst established by C. F.

Gauss (1777–1855) in 1799. Many diﬀerent proofs are now known—over

one hundred by one estimate. Gauss himself discovered four. Despite the

theorem’s name, most of the proofs, including the simplest ones, are not

purely algebraic. One of the standard ones is presented in Chapter VII.

When complex numbers were ﬁrst introduced, in the 16th century, and

for many years thereafter, they were viewed with suspicion, a feeling the

reader perhaps has shared. In high school one learns how to add and multiply

two complex numbers, a + ib and c + id, by treating them as binomials, with

the extra rule

i2

= −1 to be used in forming the product. According to this

recipe,

(a + ib) + (c + id) = (a + c) + i (b + d)

(a + ib)(c + id) = (ac − bd) + i (ad + bc) .

Everything seems to work out, but where does this number i come from?

How can one just “make up” a square root of −1?

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http://dx.doi.org/10.1090/mbk/049/01