Conjugate and Modulus
In the previous section we looked at algebraic operations on
complex numbers. There are a couple of
other operations that we should take a look at since they tend to show up on
occasion. We’ll also take a look at
quite a few nice facts about these operations.
Complex Conjugate
The first one we’ll look at is the complex conjugate, (or just the conjugate). Given the complex number the complex conjugate is denoted by and is defined to be,


(1)

In other words, we just switch the sign on the imaginary
part of the number.
Here are some basic facts about conjugates.


(2)



(4)



(5)

The first one just says that if we conjugate twice we get
back to what we started with originally and hopefully this makes some
sense. The remaining three just say we
can break up sum, differences, products and quotients into the individual
pieces and then conjugate.
So, just so we can say that we worked a number example or
two let’s do a couple of examples illustrating the above facts.
There is another nice fact that uses conjugates that we
should probably take a look at. However,
instead of just giving the fact away let’s derive it. We’ll start with a complex number and then perform each of the following
operations.
Now, recalling that and we see that we have,
Modulus
The other operation we want to take a look at in this
section is the modulus of a complex
number. Given a complex number the modulus is denoted by and is defined by
Notice that the modulus of a complex number is always a real
number and in fact it will never be negative since square roots always return a
positive number or zero depending on what is under the radical.
Notice that if z
is a real number (i.e. ) then,
where the on the z
is the modulus of the complex number and the on the a
is the absolute value of a real number (recall that in general for any real
number a we have ).
So, from this we can see that for real numbers the modulus and absolute
value are essentially the same thing.
We can get a nice fact about the relationship between the
modulus of a complex numbers and its real and imaginary parts. To see this let’s square both sides of (7)
and use the fact that and . Doing this we arrive at
Since all three of these terms are positive we can drop the
Im z part on the left which gives the
following inequality,
If we then square root both sides of this we get,
where the on the z
is the modulus of the complex number and the on the Re z
are absolute value bars. Finally, for
any real number a we also know that (absolute value…) and so we get,
We can use a similar argument to arrive at,
There is a very nice relationship between the modulus of a
complex number and it’s conjugate. Let’s
start with a complex number and take a look at the following product.
From this product we can see that
This is a nice and convenient fact on occasion.
Notice as well that in computing the modulus the sign on the
real and imaginary part of the complex number won’t affect the value of the
modulus and so we can also see that,
and


(12)

We can also now formalize the process for division
from the previous section now that we have the modulus and conjugate
notations. In order to get the i out of the denominator of the quotient
we really multiplied the numerator and denominator by the conjugate of the
denominator. Then using (10)
we can simplify the notation a little.
Doing all this gives the following formula for division,
Here’s a quick example illustrating this,
Here are some more nice facts about the modulus of a complex
number.
Property (13) should make some sense
to you. If the modulus is zero then ,
but the only way this can be zero is if both a and b are zero.
To verify (14) consider the following,
So, from this we can see that
Finally, recall that we know that the modulus is always
positive so take the square root of both sides to arrive at
Property (15) can be verified using a
similar argument.
Triangle Inequality and Variants
Properties (14) and (15)
relate the modulus of a product/quotient of two complex numbers to the
product/quotient of the modulus of the individual numbers. We now need to take a look at a similar
relationship for sums of complex numbers.
This relationship is called the triangle
inequality and is,
We’ll also be able to use this to get a relationship for the
difference of complex numbers.
The triangle inequality is actually fairly simple to prove
so let’s do that. We'll start with the
left side squared and use (10) and (3)
to rewrite it a little.
Now multiply out the right side to get,
Next notice that,
and so using (6), (8)
and (11)
we can write middle two terms of the right side of (17) as
Also use (10) on the first and fourth
term in (17)
to write them as,
With the rewrite on the middle two terms we can now write (17)
as
So, putting all this together gives,
Now, recalling that the modulus is always positive we can
square root both sides and we’ll arrive at the triangle inequality.
There are several variations of the triangle inequality that
can all be easily derived.
Let’s first start by assuming that . This is not required for the derivation, but
will help to get a more general version of what we’re going to derive
here. So, let’s start with and do some work on it.
Now, rewrite things a little and we get,
If we now assume that we can go through a similar process as above
except this time switch and and we get,
Now, recalling the definition of absolute value we can
combine (18)
and (19)
into the following variation of the triangle inequality.
Also, if we replace with in (16)
and (20)
we arrive at two more variations of the triangle inequality.


(21)

On occasion you’ll see (22)
called the reverse triangle inequality.