Complex
Numbers Primer
Before I get started on this let me first make it clear that
this document is not intended to teach you everything there is to know about
complex numbers. That is a subject that
can (and does) take a whole course to cover.
The purpose of this document is to give you a brief overview of complex
numbers, notation associated with complex numbers, and some of the basic
operations involving complex numbers.
This document has been written with the assumption that
you’ve seen complex numbers at some point in the past, know (or at least knew
at some point in time) that complex numbers can be solutions to quadratic
equations, know (or recall) ,
and that you’ve seen how to do basic arithmetic with complex numbers. If you don’t remember how to do arithmetic I
will show an example or two to remind you how to do arithmetic, but I’m going
to assume that you don’t need more than that as a reminder.
For most students the assumptions I’ve made above about
their exposure to complex numbers is the extent of their exposure. Problems tend to arise however because most
instructors seem to assume that either students will see beyond this exposure
in some later class or have already seen beyond this in some earlier
class. Students are then all of a sudden
expected to know more than basic arithmetic of complex numbers but often
haven’t actually seen it anywhere and have to quickly pick it up on their own
in order to survive in the class.
That is the purpose of this document. We will go beyond the
basics that most students have seen at some point and show you some of the
notation and operations involving complex numbers that many students don’t ever
see once they learn how to deal with complex numbers as solutions to quadratic equations. We’ll also be seeing a slightly different way
of looking at some of the basics that you probably didn’t see when you were
first introduced to complex numbers and proving some of the basic facts.
The first section is a more mathematical definition of
complex numbers and is not really required for understanding the remainder of
the document. It is presented solely for
those who might be interested.
The second section (arithmetic) is assumed to be mostly a
review for those reading this document and can be read if you need a quick
refresher on how to do basic arithmetic with complex numbers. Also included in this section is a more
precise definition of subtraction and division than is normally given when a person
is first introduced to complex numbers.
Again, understanding these definitions is not required for the remainder
of the document they are only presented so you can say you’ve seen it.
The remaining sections are the real point of this document
and involve the topics that are typically not taught when students are first
exposed to complex numbers.
So, with that out of the way, let’s get started…
The Definition – In this section we'll take a look at a more precise and mathematical definition of a complex number. This section is not required for understanding the remainder of this document.
Arithmetic – Here we will take a look at arithmetic of complex numbers. We will also provide a more mathematical definition of subtraction and division of complex numbers that is generally given in your first exposure to complex numbers.
Conjugate and Modulus – We will look at the complex conjugate and modulus of a complex number in this section. We will also look at some nice facts involving these.
Polar & Exponential Form – In this section we'll take a look at two alternate forms of the complex numbers and some nice facts that can be derived form them.
Powers and Roots – We'll take a look at how to quickly compute powers and roots of complex numbers in this section.