Properties of Modulus, so
the modulus of a complex number z equals x plus y i
is the non-negative real number defined by taking the length
of the number in the plane, that’s basically what's happening here.
The Properties of Modulus, so something to mention here, when we're dealing
with modulus and actually with proofs of modulus things, it actually helps to take…
to prove things about the square of the modulus
because it's a little bit easier to work with. Here dealing with the square root’s a little bit annoying.
Keep in mind that the modulus is a non-negative real number.
So if we prove that the square of two…
so for example let's say question 1: we have the
the modulus of…or the absolute value, that’s another way to look at it,
the absolute value of z bar is equal to the absolute value of z.
So if we wanted to prove this one, one way we can do this is square both sides,
prove that the squares are equal, and then justify that
we can take square roots because it's a non-negative real number.
So we don't have to deal with like the multiple solution issue.
If you do a little bit of algebra, z z bar is the same as the modulus of z squared.
The modulus of z is equal to 0 if and only if z equals 0.
That's actually fairly quick.
The modulus splits over multiplication, so the modulus of z times w
is equal to the modulus of z times the modulus of w.
And absolute value, or modulus, satisfies the triangle inequality.
So the modulus of z plus w is less than or equal to
the modulus z plus the modulus of w,
and this is called the triangle inequality.
if you actually draw a picture in the plane of z plus w, it'll look like…
so if I add the size of z plus the size of w,
it'll look like a triangle when I compare it to the size of z plus w.
I don't think I have a picture here. There's a picture in the notes, you can check that out.
The last three properties define something called the norm.
That's not something that we're going to say anything more about other than that sentence.
You can check it out in real and complex analysis in a later course if you're interested.
Of these, I'm going to prove only 5.
Again a lot of these can be proved by introducing coordinates.
If you want - they're not hard.
The fact that length doesn't change if I reflect about the x-axis, that shouldn't surprise us.
This is just an algebra proof, just crunch it in.
This is very straightforward, and this, again,
is an algebra proof, or you could square both sides and use
property 2 and Properties of Conjugates that also works as well.
Let's look at 5, okay, so here's the proof of the triangle inequality.
It's the only one that's really involved.
I'm going to use that trick that I said so to prove that
the absolute value of z [plus w] less than or equal to the absolute value of z plus the absolute value of w,
it suffices to show that their squares are less than or equal to each other
because again, they're non-negative real numbers so
if the squares or less than or equal to, then the original number is less than or equal to.
So it suffices to show - I’m going to show that, in fact,
the mod of z plus w squared is less than or equal to
the size of z squared plus 2 times the size of z times w, plus the size of w squared.
So using Properties of Modulus and Properties of Conjugate, what do we have?
So we have z plus w squared, well that's equal to,
by definition, the number times its conjugate.
We're going to split it up by PCJ, so
the conjugate of the sum is the sum of the conjugates.
Now we're going to do the FOIL-ing, we're going to expand it out,
just like we normally would. Distributive property, remember
complex numbers form a field so I can distribute,
and then we're going to use Properties of Conjugates again and a little bit of Properties of Modulus.
So the first value is the size of z squared,
the last value is the size of w squared.
Second value stays the same,
and this last value, well what I'm going to do is I'm going
to use the fact that w w bar
is the same as w, and I'm going to
use the fact that the product of the conjugate is the conjugate of the products.
So that gives us this value.
Now what do we notice here? Here we have z times w bar,
and here we have plus the conjugate of z times w bar.
So if I treat this as some…
if we treat z times w as some
complex number, then this is the complex number plus its complex conjugate.
So by Properties of Conjugate, what does that mean?
Well it means that that's equal to 2 times the real part, right?
z plus z bar is equal to 2 times the real part,
and here z is z times w bar.
So that's good, now if you’re 2 times the real part of z times w bar,
well then you must be less than or equal to 2 times the length of z w, right, if we…
so if you think about it, right, if you think about the real part of z times w bar,
and then you're adding a little component, then
you're going to add a little bit to the length, okay,
and that's what's happening here.
I mean we can prove this by introducing coordinates as well,
but I think that the idea is pretty clear, right? If you have just the real component, then your
length must be smaller than if I had the real and an imaginary component.
It could be equal though, right? You could have a real number out of this.
So then we get 2 times z times
w bar and the length of that is the same as the length of 2 times z times w,
you can use PCJ to get that,
and this actually completes the proof, right, because
now this middle term is 2 times the modulus of z times w,
and that's exactly what we wanted to show.
So an interesting little proof, I suggest playing around with it,
maybe drawing a couple pictures is probably a good idea
if you really want to understand the triangle inequality.
It is a triangle inequality,
if you draw pictures it does look like a triangle,
but again I'll leave it for you to check that out in the notes.