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We learned in the imaginary numbers video, that hopefully

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you've watched, that every now and then in certain equations

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you end up with a square root of a negative number.

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You know, you end up with square root of negative 1 or

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square root of negative 9.

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And since any real number, when you square it, is either 0 or

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positive, this was undefined for us.

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And then in order to make it defined in that video, people

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came up with this thing called i.

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And you can call that the imaginary unit,

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or just the number i.

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And you define i as saying, well if i squared-- we'll

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just make a definition.

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This thing called i when you square it,

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it equals negative 1.

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And then that simplifies the idea of taking a

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negative square root.

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Because then you could say, oh well the square root of

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negative 9 is the same thing as i times the square root

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of 9, which equals 3i.

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And how can we say that?

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Well, what happens when you square this thing?

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So what's 3i squared?

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3i squared is equal to 3 squared times i squared.

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This is just exponent properties.

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And that equals 9 times negative 1, which

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equals negative 9.

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So 3i squared is equal to negative 9.

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And then if we are kind of extending the definition of

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square roots to negative numbers, then we can go the

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other way around and we could say 3i is equal to the

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square root of negative 9.

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And 3i we can call an imaginary number.

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And the word is "imaginary," but it's as real as anything.

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I mean to some degree, do negative numbers even exist?

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They're just kind of a way of us-- when we put this sign in

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front of things, it just tells us something about how it

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relates to this magnitude.

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Anyway, I don't want to confuse you.

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But imaginary numbers tend to get a bad rap because they're

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called imaginary numbers, so some people think that they

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exist less than other things.

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But with that said, any number times this imaginary unit

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i is an imaginary number.

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When you do the quadratic equation you realize that

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sometimes you end up with a number that has a

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little bit of both.

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It has a real part and an imaginary part.

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So let me give you an example of that.

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Let's say I have the number 5 plus 2i.

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That number you might say, oh, well maybe I can simplify it.

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But you really can't.

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You can't add a real number plus an imaginary number.

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You can almost kind of imagine them-- and I don't want to use

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the word "imagine" too much-- as in different dimensions.

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And so a number that has a real part, like the 5, and an

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imaginary part, like the 2i, this is called a

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complex number.

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And you could, if you want, even graph a complex number.

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Let me see.

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You could make the vertical axis.

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The vertical axis you could call it the imaginary axis.

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And you could make the horizontal axis the real axis.

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This is a symbol for the set of real numbers.

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And 5 plus 2i, well it's real part is 5.

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So one, two, three, four, five.

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That's five.

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And it's imaginary part is 2i, so you go along the imaginary

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axis, or the i-axis you could even say, 2.

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And then you would have this number here called 5 plus 2i.

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In future videos I'll actually do examples where we'll make

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more use of complex numbers.

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But now that we've defined what a complex number is, let's see

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how we can operate with it.

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So what happens when we add two complex numbers?

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Clear image.

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So let's say we have two complex numbers.

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One is a plus bi.

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So the real part is a, the imaginary part is bi.

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And let's say that I have another complex

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number, c plus di.

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And in general the symbol that people tend to use-- you know,

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you always use x for when you're dealing with equations.

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Right?

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Like x could be any general real number when you're

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dealing with an equation.

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In complex numbers the convention is to use

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z, the letter z.

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So, for example, we could call this the complex

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number z sub 1.

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And it could have been any.

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I mean, this choice of z is arbitrary.

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This is an i right here.

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This could be z sub 2.

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So what is the complex number z sub 1 plus z sub 2?

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Well that equals this, a plus bi, plus-- this is z sub

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2 right here-- c plus di.

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Let me switch colors.

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This is getting monotonous.

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So when you add two complex numbers, all you do is you add

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the real parts to each other and you add the complex

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parts to each other.

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So that equals a plus b-- oh sorry.

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The real parts.

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So it's a and c.

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Those are the real parts.

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So that equals a plus c plus-- and then you add

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the imaginary parts.

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b plus d times i.

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So you have b i's and d i's.

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So when you add them together you have b plus d i's.

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b plus d in the imaginary direction, you can

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almost imagine.

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I'm using the word imagine too much.

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So it's pretty easy.

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You just add the real and you add the imaginary parts.

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So what happens if you multiply two complex numbers?

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And actually, let's just go in order.

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What happens when you subtract them?

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Well, it's the same thing.

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Nothing fancy here. z sub 1 minus z sub 2.

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That's just going to be equal to-- you subtract

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the real parts.

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a minus c.

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And that's the new real part.

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And then you subtract the imaginary parts.

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b minus d times i.

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And this will be the new imaginary part.

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So this will be the new complex number.

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What happens when I multiply the two numbers?

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So what is z sub 1 times z sub 2?

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Well that equals a plus bi times c plus di.

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And we essentially can just FOIL it out.

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I don't know if you've learned FOIL in 8th grade algebra,

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9th grade algebra.

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I don't like it.

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I actually just think of it as the distributive

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property twice.

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So what we could do is we can take the c plus di

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and distribute it on each of these terms.

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Right?

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So this should be equal to a times c plus di.

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Right?

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Plus bi times c plus di.

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Right?

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All I did is I took the c plus di and I multiplied it by this

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to get this, and multiplied it by this to get this.

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And then we distribute again.

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We get ac plus adi-- this is an i.

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I know my i's aren't looking good.

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adi plus-- now what's bi times c?

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Well that's cbi.

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And then we have bi times di.

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So you get b times d, which is bd.

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And then what's i times i?

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It's i squared, or negative 1.

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Right?

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Times negative 1.

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So what does this simplify to?

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Well this simplifies to-- I'm going to go back here.

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So let's see what the real parts are.

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We have this term, a times c, that's real.

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So ac.

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And then this last term, which was bi times di.

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Because we're multiplying i times i we get a negative 1,

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but it becomes real again.

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Right?

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So we get negative bd minus bd.

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So that's our new real part.

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So that was this term and that term are real.

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And now what's our new imaginary part?

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Well it's going to be these two terms added together.

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So it's-- I'm running out of space again-- ad plus

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cb, all of that times i.

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It's a little easier with real numbers, and maybe in the next

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video I'll use real numbers.

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But the important thing to realize is essentially you just

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use the distributive property and realize that you can only

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add real terms to each other.

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You can't add a real to an imaginary.

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And you can only add imaginary terms to each other.

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And just remember, when you have two imaginary numbers

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times each other, the i's, when multiplied times each other,

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and you get negative 1.

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Now the last operation, when you divide complex numbers.

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This gets a little bit interesting, and maybe

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a little unintuitive.

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So what happens if I divide z sub 1 by z sub 2?

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So once again that equals a plus bi, divided by c plus di.

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How do I divide this?

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Well, we're going to use a property that hopefully

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you learned in algebra.

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That if I multiply-- let me do this in this corner.

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a plus b times a minus b.

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What does that equal?

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That equals a squared minus b squared.

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And so if I multiply a complex number.

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Let's say, in theory, I multiply a complex number c

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plus di times c minus di.

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What do I get?

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I get c squared minus di squared.

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And what's di squared going to be?

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It's d squared times negative 1, so it becomes c squared--

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oh, I'm out of time.

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Actually, let me continue the division in the next video,

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because it can get involved.

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See you soon.