COMPLEX NUMBERS 47

Complex Numbers and Solving Equations

What is the point? First, mathematicians proved that using the

set of complex numbers, which we call C, will not lead to any

contradictions. Even though you might not like the idea of a square

root of −1, pretending that it exists will not get you into logical

trouble.

Second, and this is really the key point, the job of adding on

square roots (or cube roots, or anything else that you might like

to add on) stops with C. You do not need to add on some other new

symbol j with j

2

= i, for example; there already is a square root of i

in C.

EXERCISE: Let α =

1

√

2

+

1

√

2

i

.Checkthatα

2

= i.

In fact, every root of a complex number is already in C. But even

more than that is true:

THEOREM 5.2: Let f (x) be a polynomial whose coefﬁcients

are any complex numbers. (For example, f (x) might have

integer coefﬁcients .) Then the equation f (x) = 0 has solutions

in C.

Digression: Theorem

A theorem is a mathematical statement that can be proved to be

true. Despite the similarity to the word “theory,” there is nothing

speculative about a theorem.

Modern mathematical usage distinguishes between various

terms, all of which refer to mathematical statements that are

proved. The word lemma refers to a minor “helping” statement

used along the way toward proving a theorem. A corollary is an

immediate and easy consequence of a theorem.

Algebraic Closure

The fancy way of conveying the content of Theorem 5.2 is to say

that C is algebraically closed. The ﬁrst really good proof was given

48 CHAPTER 5

by Gauss, though earlier mathematicians such as Euler knew this

fact as well.

5

We are not going to need to use all of C very often. It contains

numbers such as π, which cannot be understood algebraically. W e

will restrict ourselves mostly to that part of C which consists of

the solutions of all equations f (x) = 0wheref (x) is a polynomial

with integer coefﬁcients. That set is called Q

alg

, and it exists as a

subset of C.Infact,Q

alg

itself is a ﬁeld! Sample elements of Q

alg

:

0, 1, 3, −

1

4

,

√

11, i,

3

√

2 − 3i, and the solutions in C of the equation

x

5

+ x − 1 = 0.

5

Consult (Nahin, 1998) for a historical summary.

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