The Definition
As I’ve already stated, I am assuming that you have seen
complex numbers to this point and that you’re aware that and so . This is an idea that most people first see in
an algebra class (or wherever they first saw complex numbers) and is defined so that we can deal with square
roots of negative numbers as follows,
What I’d like to do is give a more mathematical definition of a complex numbers and show that (and hence ) can be thought of as a consequence of
this definition. We’ll also take a look
at how we define arithmetic for complex numbers.
What we’re going to do here is going to seem a little
backwards from what you’ve probably already seen but is in fact a more accurate
and mathematical definition of complex numbers.
Also note that this section is not really required to understand the
remaining portions of this document. It
is here solely to show you a different way to define complex numbers.
So, let’s give the definition of a complex number.
Given two real numbers a
and b we will define the complex
number z as,


(1)

Note that at this point we’ve not actually defined just what
i is at this point. The number a is called the real part
of z and the number b is called the imaginary part of z and
are often denoted as,


(2)

There are a couple of special cases that we need to look at
before proceeding. First, let’s take a
look at a complex number that has a zero real part,
In these cases, we call the complex number a pure imaginary number.
Next, let’s take a look at a complex number that has a zero
imaginary part,
In this case we can see that the complex number is in fact a
real number. Because of this we can
think of the real numbers as being a subset of the complex numbers.
We next need to define how we do addition and multiplication
with complex numbers. Given two complex
numbers and we define addition and multiplication as
follows,


(3)



(4)

Now, if you’ve seen complex numbers prior to this point you
will probably recall that these are the formulas that were given for addition and
multiplication of complex numbers at that point. However, the multiplication formula that you
were given at that point in time required the use of to completely derive and for this section we
don’t yet know that is true. In fact, as
noted previously will be a consequence of this definition as
we’ll see shortly.
Above we noted that we can think of the real numbers as a
subset of the complex numbers. Note that
the formulas for addition and multiplication of complex numbers give the
standard real number formulas as well.
For instance given the two complex numbers,
the formulas yield the correct formulas for real numbers as
seen below.
The last thing to do in this section is to show that is a consequence of the definition of
multiplication. However, before we do
that we need to acknowledge that powers of complex numbers work just as they do
for real numbers. In other words, if n is a positive integer we will define
exponentiation as,
So, let’s start by looking at ,
use the definition of exponentiation and the use the definition of
multiplication on that. Doing this
gives,
So, by defining multiplication as we’ve done above we get
that as a consequence of the definition instead of
just stating that this is a true fact.
If we now take to be the standard square root, i.e. what did we square to get the
quantity under the radical, we can see that .