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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
n1
x
n1
+···+
a
1
x + a
0
= 0, where the coefﬁcients a
n
, a
n1
, ..., 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 =
1andi =−
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
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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