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## Friday, 1 August 2014

### CHAPTER 1 - COMPLEX NUMBERS - 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. Every complex number has the ``Standard Form''
for some real a and b.
2. For real a and b,

## 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

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.

## Real and Imaginary Parts

If za+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

Find  and  .

So that  and

## 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.

so that  .

Prove that  for any integer n.
This follows from the remark made in the solution to the previous problem. Or we can give a simple proof by induction.
We have the result for n=1. Suppose that we know it for integers up to m. We have shown that  for any pair of complex numbers. So we may write

## Modulus of a Complex Number

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

Let zaib where a and b are real so that  . Then

Prove that  .
This is equivalent to  which is

So we just need to show that  . But if za + ib for real numbers a and b then  and  . We could, of course, have verified the identity directly and suggest that the reader do this.