Basic Definitions

It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called `` The Complex Numbers." In this amazing number field every algebraic equation in z with complex coefficients

has a solution. To prove this fact we need Liouville's Theorem, but to get started using complex numbers all we need are the following basic rules.

Rules of Complex Arithmetic

1. 

2. Every complex number has the ``Standard Form''

   

   for some real a and b.

3. For real a and b,

   

4. 

5. 

Click on EXERCISES for some practice using these rules.

Division

Notice that rules 4 and 5 state that we can't get out of the complex numbers by adding (or subtracting) or multiplying two complex numbers together. What about dividing one complex number by another? Is the result another complex number? Let's ask the question in another way. If you are given four real numbers a,b,c and d, can you find two other real numbers x and y so that

As an exercise in using rules 1 through 5, multiply both sides of the above equation by c + di and then solve for x and y to prove that the answer to our question is yes.( Click on help for a solution.)

OK, so we can divide by c + di if c and d are not both zero. But there is a much easier way to do division.

Notice that

We say that c+di and c-di are complex conjugates. To simplify a complex fraction, multiply the numerator and the denominator by the complex conjugate of the denominator.

Examples of division

Real and Imaginary Parts

If z= a+bi is a complex number and a and b are real, we say that a is the real part of z and that b is the imaginary part of z and we write

Exercise:

Find and . (Solution)

Complex Conjugates

If z=a +bi is a complex number with real part a and imaginary part b, then we denote the complex conjugate of z by .

Exercise:

Write in standard form.

(Solution)

Exercise:

Prove that for any pair of complex numbers and similarly .

(Proof.)

Exercise:

Prove that for any integer n.

(Proof.)

Modulus of a Complex Number

The magnitude or modulus of a complex number z is denoted |z| and defined as

Notice that . (proof.)

Exercise:

Prove that . (Solution)

The Polar Form of a Complex Number

The unit circle

The fundamental trigonometric identity (i.e the Pythagorean theorem) is

From this we can see that the complex numbers

are points on the circle of radius one centered at the origin.

Think of the point moving counterclockwise around the circle as the real number moves from left to right. Similarly, the point moves clockwise if decreases. And whether increases or decreases, the point returns to the same position on the circle whenever changes by or by or by where k is any integer.

Exercise: Verify that

Exercise: Prove de Moivre's formula

Now picture a fixed complex number on the unit circle

Consider multiples of z by a real, positive number r.

As r grows from 1, our point moves out along the ray whose tail is at the origin and which passes through the point z. As r shrinks from 1 toward zero, our point moves inward along the same ray toward the origin. The modulus of the point is r. We call the angle which this ray makes with the x-axis, the argument of the number z. All the numbers rz have the same argument. We write

Just as a point in the plane is completely determined by its polar coordinates , a complex number is completely determined by its modulus and its argument.

Notice that the argument is not defined when r=0 and in any case is only determined up to an integer multiple of .

Why not just use polar coordinates? What's new about this way of thinking about points in the plane?

We now have a geometric interpretation of multiplication!