Arithmetic
Before proceeding in this section let me first say that I’m
assuming that you’ve seen arithmetic with complex numbers at some point before
and most of what is in this section is going to be a review for you. I am also going to be introducing subtraction
and division in a way that you probably haven’t seen prior to this point, but
the results will be the same and aren’t important for the remaining sections of
this document.
In the previous section we
defined addition and multiplication of complex numbers and showed that 
is a consequence of how we defined
multiplication. However, in practice, we
generally don’t multiply complex numbers using the definition. In practice we tend to just multiply two
complex numbers much like they were polynomials and then make use of the fact
that we now know that .
Just so we can say that we’ve worked an example let’s do a
quick addition and multiplication of complex numbers.
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Example 1 Compute
each of the following.
(a)
(b)
(c)
Solution
As noted above, I’m assuming that this is a review for you
and so won’t be going into great detail here.
(a)
(b)
(c)
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It is important to recall that sometimes when adding or
multiplying two complex numbers the result might be a real number as shown in
the third part of the previous example!
The third part of the previous example also gives a nice
property about complex numbers.
We’ll be using this fact with division and looking at it in
slightly more detail in the next section.
Let’s now take a look at the subtraction and division of two
complex numbers. Hopefully, you recall
that if we have two complex numbers, and then you subtract them as,
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(2)
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And that division of two complex numbers,
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(3)
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can be thought of as simply a process for eliminating the i from the denominator and writing the
result as a new complex number .
Let’s take a quick look at an example of both to remind us
how they work.
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Example 2 Compute
each of the following.
(a)
(b)
(c)
Solution
(a) There
really isn’t too much to do here so here is the work,
(b) Recall that
with division we just need to eliminate the i from the denominator and using (1)
we know how to do that. All we need to
do is multiply the numerator and denominator by and we will eliminate the i from the denominator.
(c) We’ll do
this one a little quicker.
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Now, for the most part this is all that you need to know
about subtraction and multiplication of complex numbers for this rest of this
document. However, let’s take a look at
a more precise and mathematical definition of both of these. If you aren’t interested in this then you can
skip this and still be able to understand the remainder of this document.
The remainder of this document involves topics that are
typically first taught in a Abstract/Modern Algebra class. Since we are going to be applying them to the
field of complex variables we won’t be going into great detail about the
concepts. Also note that we’re going to
be skipping some of the ideas and glossing over some of the details that don’t
really come into play in complex numbers.
This will especially be true with the “definitions” of inverses. The definitions I’ll be giving below are
correct for complex numbers, but in a more general setting are not quite
correct. You don’t need to worry about
this in general to understand what were going to be doing below. I just wanted to make it clear that I’m
skipping some of the more general definitions for easier to work with
definitions that are valid in complex numbers.
Okay, now that I’ve got the warnings/notes out of the way
let’s get started on the actual topic…
Technically, the only arithmetic operations that are defined
on complex numbers are addition and multiplication. This means that both subtraction and division
will, in some way, need to be defined in terms of these two operations. We’ll start with subtraction since it is
(hopefully) a little easier to see.
We first need to define something called an additive inverse. An additive inverse is some element typically
denoted by so that
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(4)
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Now, in the general field of abstract algebra, is just the notation for the additive inverse
and in many cases is NOT give by ! Luckily for us however, with complex
variables that is exactly how the additive inverse is defined and so for a
given complex number the additive inverse, ,
is given by,
It is easy to see that this does meet the definition of the
additive inverse and so that won’t be shown.
With this definition we can now officially define the
subtraction of two complex numbers.
Given two complex numbers and we define the subtraction of them as,
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(5)
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Or, in other words, when subtracting from we are really just adding the additive inverse
of (which is denoted by ) to . If we further use the definition of the
additive inverses for complex numbers we can arrive at the formula given above
for subtraction.
So, that wasn’t too bad I hope. Most of the problems that students have with
these kinds of topics is that they need to forget
some notation and ideas that they are very used to working with. Or, to put it another way, you’ve always been
taught that is just a shorthand notation for ,
but in the general topic of abstract algebra this does not necessarily have to
be the case. It’s just that in all of
the examples where you are liable to run into the notation in “real life”, whatever that means, we really
do mean .
Okay, now that we have subtraction out of the way, let’s
move on to division. As with subtraction
we first need to define an inverse. This
time we’ll need a multiplicative inverse. A multiplicative inverse for a non-zero
complex number z is an element
denoted by such that
Now, again, be careful not to make the assumption that the
“exponent” of -1 on the notation is in fact an exponent. It isn’t!
It is just a notation that is used to denote the multiplicative
inverse. With real (non-zero) numbers
this turns out to be a real exponent and we do have that
for instance.
However, with complex numbers this will not be the case! In fact, let’s see just what the
multiplicative inverse for a complex number is.
Let’s start out with the complex number and let’s call its multiplicative inverse . Now, we know that we must have
so, let’s actual do the multiplication.
This tells us that we have to have the following,
Solving this system of two equations for the two unknowns u and v (remember a and b are known quantities from the original
complex number) gives,
Therefore, the multiplicative inverse of the complex number z is,
As you can see, in this case, the “exponent” of -1 is not in
fact an exponent! Again, you really need
to forget some notation that you’ve
become familiar with in other math courses.
So, now that we have the definition of the multiplicative
inverse we can finally define division of two complex numbers. Suppose that we have two complex numbers and then the division of these two is defined to
be,
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(7)
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In other words, division is defined to be the multiplication
of the numerator and the multiplicative inverse of the denominator. Note as well that this actually does match
with the process that we used above.
Let’s take another look at one of the examples that we looked at earlier
only this time let’s do it using multiplicative inverses. So, let’s start out with the following division.
We now need the multiplicative inverse of the denominator
and using (6)
this is,
Now, we can do the multiplication,
Notice that the second to last step is identical to one of
the steps we had in the original working of this problem and, of course, the
answer is the same.
As a final topic let’s note that if we don’t want to
remember the formula for the multiplicative inverse we can get it by using the
process we used in the original multiplication.
In other words, to get the multiplicative inverse we can do the
following
As you can see this is essentially the process we used in
doing the division initially.