We introduce (or review) the complex numbers, an exten-

sion of the real numbers useful for solving equations. The

set of complex numbers is another example of a ﬁeld. It

is handy because every polynomial in one variable with

integer coefﬁcients can be factored into linear factors if

we use complex numbers. Equivalently, every such poly-

nomial has a complex root. This gives us a standard place

to keep track of the solutions to polynomial equations.

As with the ﬁnite ﬁelds of chapter 4, we will be working

with complex numbers in most of the chapters to follow.

In this chapter, we also introduce an important subset

of the complex numbers, namely, the set of all “algebraic

numbers”—those numbers that are roots of polynomial

equations with integer coefﬁcients. This set is also a ﬁeld,

and will be important when we study the structure of

solutions of polynomial equations.

Overture to Complex Numbers

We do not absolutely need to use complex numbers to solve poly-

nomial equations with integer coefﬁcients. Instead, we can use a

complicated algebraic recipe for inventing solutions as we need

them and then keep track of them as we continue to add more

solutions. If we wish to ignore the complex numbers, we c an simply

COMPLEX NUMBERS 43

assume the existence of a large number system that contains all the

solutions to all equations of the form f (x) = a

n

x

n

+ a

n−1

x

n−1

+···+

a

1

x + a

0

= 0, where the coefﬁcients a

n

, a

n−1

, ..., a

1

, a

0

are integers.

In this case, you must remember that the same root can occur for

many different polynomials. A sophisticated method is required for

keeping track of these solutions, but it can be done.

One advantage of working with complex numbers is that each

solution comes with its own personality. For example, the solutions

of the equation of x

2

+ 1 = 0arei =

√

−1and−i =−

√

−1. These

two complex numbers have all the same algebraic properties, but

they are not equal to each other—they are “twins.” (For more about

this mystery, you can refer to Imagining Numbers (Mazur, 2003).)

But if we have the complex numbers sitting before us, we can call

one of them i and the other −i. How, you might ask, do you tell i

and −i apart? Easy. Multiplication by i rotates the complex plane

1

by 90

◦

counterclockwise as we look down upon it, while −i rotates it

the same amount clockwise.

Now, we deﬁne complex numbers, starting with the real

numbers.

DEFINITION: A real number is any number that can be

expressed a s a decimal.

For example, 0 = 0.0, 1 = 1.0,

3

5

= 0.6, and −

√

2 =−1.4121356 ...

are real numbers. A real number can be expressed as a terminating

or repeating decimal if and only if it is the ratio of two integers. The

set of all real numbers is usually denoted by the symbol R.

Complex numbers were forced upon the world when mathemati-

cians began solving cubic

2

equations. You may think that complex

numbers would have shown up as soon as someone tried to solve

the quadratic equation x

2

+ 1 = 0, but in fact mathematicians were

quite happy to proclaim that this equation simply had no solutions.

After all, the equation 0x = 1 has no solutions, so why should it be

a problem if x

2

+ 1 = 0 has no solutions?

1

The complex plane is the (x, y)-plane on which we plot the complex number x + iy as the

point (x, y). See below for how to multiply complex numbers.

2

Degree 3: For example, x

3

+ x − 1 = 0.

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