Complex Plane

Definition of i

We make the following definition

The number

is a number that lies nowhere on the real number line. It behaves in a very non-real way since the above definition

immediately implies that

The letter

was chosen for this number because our ancestors had a hard time accepting it and therefore considered it somewhat

fanciful or "imaginary". But imaginary numbers are now universally accepted by the mathematical community as essential to our

understanding of the universe.

Using i

The square root of any negative number is a multiple of

. Consider each number below written in its unsimplified and its proper

imaginary format

Unsimplifed ........becomes.......... proper form

We will solve an equation using

Example: Solve

Solution: Our strategy is to isolate the part that contains the square, then take the square root of both sides.

then

and finally

Any number with an

in it is called an imaginary number. Any number without in it is called a real number. Any number which

is the sum of a real part and an imaginary part is called a complex number.

a real number an imaginary number a complex number

-0.23

(Side note for purists: Since any real number can be said to have an imaginary part of 0i and any imaginary number can be

said to have a real part of 0, the pure real and pure imaginary numbers are really subsets of the complex numbers.)

The most important facts to remember about

when manipulating it are the following two facts:

and

Example: Simplify

Solution:

Notice that since multiplication is associative and commutative, we can move the objects around and multiply the roots together,

the integers together and the i 's together.

Example: Simplify

Solution:

Notice that we treat the symbol i like an x would be treated and combine like terms.

Example: Simplify

.Solution: Here we have the multiplication of two binomials (objects with two tems separated by a plus or minus sign) so we must FOIL:

since

and then

So a complex number added to, subtracted from or multiplied by another complex number equals a complex number. No surprise there.

Conjugates and division

Example: Multiply

Solution:

This last example is perhaps a bit surprising. What it showed us is that two complex numbers with

in them can sometimes multiply

together to give a pure real number! Here's another example:

Example: Multiply

Solution:

It happened again. Under what circumstances does this happen? You may have noticed in the last two examples that the two

complex numbers I chose to multiply only differed in their imaginary parts. The first number was

and the second number was

. These numbers are called conjugates.

Whenever you multiply conjugates together you get a pure real number.

We will now employ this fact to complete the picture we have of complex numbers by showing how division is accomplished.

Example: Simplify

.Solution: My strategy will be to multiply the entire fraction by

. Notice that I chose this multiplying factor carefully. It is made up of the conjugate of the denominator of my original fraction.

Some notation

It is customary to use

for a complex number instead of . Any complex number can be written as

where

and are pure real numbers.

We can refer to the components individually as

and

we define the conjugate as

although many British writers are fond of calling this

One observation we've made already is that if

then

The Geometry of Complex Numbers

Each complex number has a real and an imaginary part that can be independently chosen. Therefore, in order to visualize complex

numbers, we would need two independent axes. The most common method in use is to mimic the organization of the xy-plane

and to use a horizontal axis for the real component and the vertical axis for the imaginary component. The set-up is referred to

as the complex plane or, sometimes, as the Argand diagram.

On the complex plane at left you will see the following points graphed:

From the diagram at left showing a single complex number you can see several relationships.If you imagine a line drawn from the origin to then there is an angle that this arrow makes with the positive real axis. We define this angle as the argument of