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,

In other words, we just switch the sign on the imaginary part of the number.

Here are some basic facts about conjugates.

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

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,