Complex Numbers

1 Introduction

1.1 Real or Complex ?

As you might recall, the Real Number System is made
up of rational numbers (those that can be written as
p/q , with p and q integers) and irrational numbers (like
etc.).

The Complex Number System is made up of objects
that look like:

z = a + bi (1)

where a, b are Real Numbers and Some
examples of complex numbers: 4.23
(which is 4.23 + 0i), etc. Notice that by taking b = 0
in the definition in Equation (1), we can get any Real
Number. Therefore, the set of Complex Numbers is
"bigger" than the set of Real Numbers.

1.2 Visualizing Complex Numbers

A complex number is defined by it\'s two real numbers.
If we have z = a + bi, then we say that the real part
of z is the number "a", and the imaginary part of z is
the number b (without the i). To visualize a complex
number, we can plot it on the plane. The horizontal
axis is for the Real part, and the vertical axis is for the
Imaginary part. For example, the complex number 3-
5i is plotted as the ordered pair (3,-5). The complex
number i is plotted as (0, 1), and the complex number
5 is plotted as (5, 0).

1.3 Operations on Complex Numbers

1.3.1 The Conjugate of a Complex Number

If z = a + bi is a complex number, then its conjugate,
denoted by is a - bi. For example,

Graphically, the conjugate of a complex number is it\'s
mirror image across the horizontal axis.

1.3.2 The Size of a Complex Number

The size, or absolute value , of a complex number, denoted
by |z|, is graphically it \'s distance to the origin.
Therefore, if z = a+bi, then it\'s plotted as (a, b), and
it\'s distance to the origin is

1.3.3 Addition, Subtraction and Multiplication

To add (or subtract) two complex numbers, add (or
subtract) the real parts and the imaginary parts separately:

To multiply complex numbers (recall that i² = -1):

which is just the "FOIL" trick we learn for multiplying
things like (3x - 2)(x + 5).

To multiply a real number times a complex number:

a(c + di) = ac + adi

1.3.4 Division by a Complex Number

Let z = a + bi and w = c + di. The we interpret z/w in
the fol lowing way :

In this way, we change division by a complex number
into division by a real number (|w|²).
Example:

1.4 Alternate Representations of
Complex Numbers

We\'ve shown that we can represent a complex number
as a + bi or graphically as the ordered pair (a, b). We
can also represent a complex number in it\'s polar form.
In this case,

where r = |z|, and is the angle that the point (a, b)
makes with the positive real axis.

1.4.1 Interpretation of

Definition:

Examples of working with this definition:

From the second example above, we see that the laws
of exponents still work when the exponent is complex!

2 Real Polynomials and Complex
Numbers

Real polynomials are polynomials with real numbers
as coefficients. For example, quadratic polynomials
look like:

with a, b, c real numbers. We solve for the roots of the
quadratic by using the quadratic formula. If ax² +
bx + c = 0, then we obtain the roots by:

In the past, we only took real roots. Now we can use
complex roots. For example, the roots of x² + 1 = 0
are x = i and x = -i. We can check by multiplying
the factors together:

Some "Facts" about polynomials where we allow complex
roots:

1. An n^th degree polynomial can always be factored
into n roots. (Un like if we only have real roots!)
This is the Fundamental Theorem of Algebra .

2. If a+bi is a root to a real polynomial, then a-bi
must also be a root. This is sometimes referred
to as "roots must come in conjugate pairs". For
example, if we are finding roots to a third degree
polynomial, then we only have the choices that:

(a) One root is real, the other two are complex.
(b) All roots are real.
And in the case of the quadratic polynomial, either:
(a) Both roots are real.
(b) Both roots are complex (and are conjugates
of each other).

3 Exercises

1. Suppose the roots to a cubic polynomial are a =
3, b = 1-2i and c = 1+2i. What is (x-a)(x-
b)(x - c)? Will the coe cients be real?

2. Find the roots to x² - 2x + 10. Write them in
polar form.

3. Let z = a + bi. Show that This shows
that if we multiply a complex number by it\'s conjugate,
we get a real number!

4. For the following, let

(a) Compute
(b) Write in the form a + bi.
(c) Write in polar form.
(d) Suppose that z is written in polar form,
How would be written?

5. Compute etc. Do you see a pattern?

6. In each problem, rewrite each of the following in
the form a + bi: