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Properties of Modulus, so

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the modulus of a complex 
number z equals x plus y i

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is the non-negative real number 
defined by taking the length

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of the number in the plane, that’s 
basically what's happening here.

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The Properties of Modulus, so something 
to mention here, when we're dealing

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with modulus and actually with proofs of 
modulus things, it actually helps to take…

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to prove things about the
square of the modulus

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because it's a little bit easier to work with. Here
dealing with the square root’s a little bit annoying.

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Keep in mind that the modulus 
is a non-negative real number.

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So if we prove that the square of two…

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so for example let's say 
question 1: we have the

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the modulus of…or the absolute value, 
that’s another way to look at it,

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the absolute value of z bar is 
equal to the absolute value of z.

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So if we wanted to prove this one, one 
way we can do this is square both sides,

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prove that the squares are 
equal, and then justify that

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we can take square roots because 
it's a non-negative real number.

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So we don't have to deal with
like the multiple solution issue.

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If you do a little bit of algebra, z z bar is 
the same as the modulus of z squared.

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The modulus of z is equal to 
0 if and only if z equals 0.

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That's actually fairly quick.

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The modulus splits over multiplication, 
so the modulus of z times w

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is equal to the modulus of 
z times the modulus of w.

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And absolute value, or modulus, 
satisfies the triangle inequality.

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So the modulus of z plus 
w is less than or equal to

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the modulus z plus the modulus of w,

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and this is called the triangle inequality.

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if you actually draw a picture in
the plane of z plus w, it'll look like…

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so if I add the size of 
z plus the size of w,

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it'll look like a triangle when I 
compare it to the size of z plus w.

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I don't think I have a picture here. There's a 
picture in the notes, you can check that out.

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The last three properties define 
something called the norm.

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That's not something that we're going to say 
anything more about other than that sentence.

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You can check it out in real and complex 
analysis in a later course if you're interested.

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Of these, I'm going to prove only 5.

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Again a lot of these can be proved 
by introducing coordinates.

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If you want - they're not hard.

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The fact that length doesn't change if I reflect 
about the x-axis, that shouldn't surprise us.

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This is just an algebra proof, just crunch it in.

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This is very straightforward, and this, again,

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is an algebra proof, or you could 
square both sides and use

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property 2 and Properties of 
Conjugates that also works as well.

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Let's look at 5, okay, so here's the 
proof of the triangle inequality.

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It's the only one that's really involved.

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I'm going to use that trick 
that I said so to prove that

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the absolute value of z [plus w] less than or equal to 
the absolute value of z plus the absolute value of w,

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it suffices to show that their squares 
are less than or equal to each other

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because again, they're 
non-negative real numbers so

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if the squares or less than or equal to, then 
the original number is less than or equal to.

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So it suffices to show - I’m 
going to show that, in fact,

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the mod of z plus w squared 
is less than or equal to

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the size of z squared plus 2 times the size 
of z times w, plus the size of w squared.

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So using Properties of Modulus and 
Properties of Conjugate, what do we have?

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So we have z plus w 
squared, well that's equal to,

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by definition, the number 
times its conjugate.

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We're going to split it up by PCJ, so

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the conjugate of the sum is
the sum of the conjugates.

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Now we're going to do the FOIL-ing, 
we're going to expand it out,

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just like we normally would.
Distributive property, remember 

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complex numbers form a 
field so I can distribute,

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and then we're going to use Properties of Conjugates 
again and a little bit of Properties of Modulus.

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So the first value is the size of z squared,

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the last value is the size of w squared.

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Second value stays the same,

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and this last value, well what 
I'm going to do is I'm going

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to use the fact that w w bar

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is the same as w, and I'm going to

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use the fact that the product of the
conjugate is the conjugate of the products.

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So that gives us this value.

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Now what do we notice here? 
Here we have z times w bar,

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and here we have plus the 
conjugate of z times w bar. 

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So if I treat this as some…

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if we treat z times w as some

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complex number, then this is the complex 
number plus its complex conjugate.

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So by Properties of Conjugate, 
what does that mean?

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Well it means that that's equal 
to 2 times the real part, right?

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z plus z bar is equal to 
2 times the real part,

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and here z is z times w bar.

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So that's good, now if you’re 2
times the real part of z times w bar,

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well then you must be less than or equal 
to 2 times the length of z w, right, if we…

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so if you think about it, right, if you think 
about the real part of z times w bar,

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and then you're adding
a little component, then

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you're going to add a 
little bit to the length, okay,

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and that's what's happening here.

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I mean we can prove this by 
introducing coordinates as well,

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but I think that the idea is pretty clear, right? 
If you have just the real component, then your

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length must be smaller than if I had the
real and an imaginary component.

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It could be equal though, right? You 
could have a real number out of this.

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So then we get 2 times z times

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w bar and the length of that is the same 
as the length of 2 times z times w,

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you can use PCJ to get that,

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and this actually completes 
the proof, right, because

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now this middle term is 2 times 
the modulus of z times w,

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and that's exactly what we wanted to show.

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So an interesting little proof, I 
suggest playing around with it,

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maybe drawing a couple pictures 
is probably a good idea

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if you really want to understand 
the triangle inequality.

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It is a triangle inequality,

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if you draw pictures it
does look like a triangle,

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but again I'll leave it for you to 
check that out in the notes.