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So far we've learned what a complex number is; we've even

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learned how to graph it.

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And we learned how to add, subtract, and multiply it.

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Where I left off in the last video was, how do we divide

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two complex numbers.

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So I said, let's say I have one complex numbers, z1.

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And that equals a plus b i.

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i.

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And I want to divide that by z2.

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Which is, c plus d i.

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So let me ask you a question.

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And I touched on this in the last video.

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And let me do it in a different color over here.

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We know that a plus b times a minus b is equal to a

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squared minus b squared.

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And you can multiply it out, in case you're not sure.

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Remember, it's just a times a plus b times a minus

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a times b, plus a times a, and you'll get this.

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But you know how to do this, anyway.

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There's a review of it, if you need to do it.

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So, given that, what is c plus d?

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What happens if we do something very similar

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with a complex number?

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If we say c plus d i times c minus d i.

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Well, in this case a is c.

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And b is d i, right?

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So this is just going to be equal to c squared

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minus d i squared.

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d i squared.

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And that equals c squared minus d squared i squared.

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And that equals c squared minus d squared.

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And i squared is negative 1, right?

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So this is going to be multiplied by negative 1, so

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it cancels out this negative.

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So you get c squared plus d squared.

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That's interesting.

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When I multiply a complex number times this other number,

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which is very similar to it, but it's kind of the imaginary

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part, goes in the other direction.

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When I multiply the two, I get a completely real number.

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All of the i's disappear.

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And, in general, this number -- if we call this -- well, in our

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example this was z2, so if we say that z2 equals c plus d i,

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the quantity c minus d i is called its conjugate.

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And that's just good terminology to know.

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And the sign for conjugate is that line over the top.

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So the conjugate of z2 is c minus d i.

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Or you could say, the conjugate of c minus d i

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is equal to c plus d i.

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Or you could say it the other way around.

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The conjugate of c plus d i is equal to c minus d i.

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And notice, we're just switching the direction in

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the imaginary -- along the imaginary axis, when we take

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the conjugate of something.

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With that said, let me erase that and go back

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to our original problem.

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Because the conjugate is the tool we're going

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to use to divide this.

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So we know when we multiply an imaginary number times its

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conjugate, we get a real number.

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And we know, also, if we multiply -- we can multiply

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anything by 1, and we get the same number.

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So let's multiply the numerator and denominator of this

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expression by the conjugate of the denominator.

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So let me do that.

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So the conjugate of the denominator is going

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to be c minus d i.

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So c minus d i over c minus d i.

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So this was c plud d i, so this is its conjugate.

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And so what do we get?

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So in the numerator, we get a c -- I don't want to run out of

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space, I always do -- a c, so a times c, minus a d i, minus a d

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i -- these i's are looking funny -- this is an i.

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Plus b c i; plus b c i.

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And then the last term, we have a plus b minus b.

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So it's minus b d i squared.

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Minus b d i squared.

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All of that.

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And this is a plus b times a minus b.

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So it's equal to a squared minus b squared.

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So this is going to be equal to -- and it this will become

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second nature to you after a while, but you might want

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to just multiply it out.

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This equals c squared plus d squared.

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And don't take my word for it.

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Actually, algebraically, multiply this out and just

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realize you can only add real parts to real parts and

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imaginary parts to imaginary parts.

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So let me simplify that.

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That equals -- let's see, the real parts.

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This is real, a c.

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And this is minus b d i squared.

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So the i squared is minus 1.

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So it switches the sign here, so it becomes plus b d.

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And we can get rid of d i.

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So the real parts are, a c plus b d.

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That's that, and that.

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And then the imaginary parts are plus -- this one's

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positive, so I'll put one first -- b c minus a d i, all of that

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over c squared plus d squared.

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And that still might not look like a complex number to you.

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But then we can separate them out and we could say well,

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that equals a c plus b d over c squared plus d squared.

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And that's the real part.

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Plus b c minus a d over c squared plus d squared.

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And that times i, and that's the imaginary part.

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So you can't merge, when you're adding and subtracting, the

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real part to the imaginary part.

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But you can most definitely scale an imaginary

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number by a real number.

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And that's essentially what we're doing.

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We're multiplying 1 over c squared plus d

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squared times this.

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So, division might seem a little complicated when I

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write it all in variables.

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But let me give you an example and you will hopefully see that

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it -- with real numbers, and -- not real numbers, with

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actual numbers, I should be careful with what I say.

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Let's say I have 1 plus 2 i.

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And I want to divide that by, I don't know, let's divide it by,

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I'm going to pick a random number.

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2 plus 3i.

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And so what do we do?

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We multiply it times the conjugate of the denominator.

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2 minus 3i over -- over itself, right?

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Because then we're not changing the number.

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This is just 1, this simplifies to 1.

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It equals -- the bottom, we can multiply it out.

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But hopefully it's second nature to you.

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It equals 4 plus 9, right?

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Because that's just a squared plus b squared.

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Right?

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Well, I mean, it's a squared minus b squared, but then the

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i's, when you multiply, and it becomes a negative number.

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Try it out if you don't believe me.

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And then the top, we get 1 times 2 is 2.

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1 times minus 3i is minus 3i.

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And you have 2i times 2, which is plus 4i.

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And then you have 2i times minus 3i.

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So that's minus 6.

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Minus 6i squared.

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Well, what does i squared equal?

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That equals negative 1.

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So negative 1 times negative 6.

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Get rid of the i squared and this becomes a positive.

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So then, what are our real parts?

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Our real parts are 2 and 6.

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so 2 plus 6 is 8.

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And what are our imaginary parts?

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Minus 3i plus 4i.

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So that's just plus 1i, right?

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Minus 3 plus 4 is positive 1.

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So it's just plus 1i.

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Over 13.

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Or we could write that as -- if we wanted to write that in the

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traditional complex form -- is 8/13 plus 1 over 13i.

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So when I divided one complex number by another, I got

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another complex number.

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And an interesting exercise for you to do is, pick some

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random complex numbers.

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Plot them out on complex plane, and see what happens when you

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multiply them, when you divide them, when you add them,

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when you subtract them.

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And when you scale them.

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Or when you take the conjugate.

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And that'll give you a better intuition of what's going

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on with these numbers.

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Anyway, I will see you in the next video.