Complex numbers
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Complex Numbers
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Complex Numbers (part 1)
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Complex Numbers (part 2)
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The complex plane
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Adding Complex Numbers
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Subtracting Complex Numbers
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Adding and subtracting complex numbers
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Multiplying Complex Numbers
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Multiplying complex numbers
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Dividing Complex Numbers
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Dividing complex numbers
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Complex Conjugates Example
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Absolute value of complex numbers
Complex Numbers (part 2) Dividing complex numbers. Complex conjugates.
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- So far we've learned what a complex number is; we've even
- learned how to graph it.
- And we learned how to add, subtract, and multiply it.
- Where I left off in the last video was, how do we divide
- two complex numbers.
- So I said, let's say I have one complex numbers, zone.
- And that equals a plus b i.
- i.
- And I want to divide that by ztwo.
- Which is, c plus d i.
- So let me ask you a question.
- And I touched on this in the last video.
- And let me do it in a different color over here.
- We know that a plus b times a minus b is equal to a
- squared minus b squared.
- And you can multiply it out, in case you're not sure.
- Remember, it's just a times a plus b times a minus
- a times b, plus a times a, and you'll get this.
- But you know how to do this, anyway.
- There's a review of it, if you need to do it.
- So, given that, what is c plus d?
- What happens if we do something very similar
- with a complex number?
- If we say c plus d i times c minus d i.
- Well, in this case a is c.
- And b is d i, right?
- So this is just going to be equal to c squared
- minus d i squared.
- d i squared.
- And that equals c squared minus d squared i squared.
- And that equals c squared minus d squared.
- And i squared is negative one, right?
- So this is going to be multiplied by negative one, so
- it cancels out this negative.
- So you get c squared plus d squared.
- That's interesting.
- When I multiply a complex number times this other number,
- which is very similar to it, but it's kind of the imaginary
- part, goes in the other direction.
- When I multiply the two, I get a completely real number.
- All of the i's disappear.
- And, in general, this number -- if we call this -- well, in our
- example this was ztwo, so if we say that ztwo equals c plus d i,
- the quantity c minus d i is called its conjugate.
- And that's just good terminology to know.
- And the sign for conjugate is that line over the top.
- So the conjugate of ztwo is c minus d i.
- Or you could say, the conjugate of c minus d i
- is equal to c plus d i.
- Or you could say it the other way around.
- The conjugate of c plus d i is equal to c minus d i.
- And notice, we're just switching the direction in
- the imaginary -- along the imaginary axis, when we take
- the conjugate of something.
- With that said, let me erase that and go back
- to our original problem.
- Because the conjugate is the tool we're going
- to use to divide this.
- So we know when we multiply an imaginary number times its
- conjugate, we get a real number.
- And we know, also, if we multiply -- we can multiply
- anything by one, and we get the same number.
- So let's multiply the numerator and denominator of this
- expression by the conjugate of the denominator.
- So let me do that.
- So the conjugate of the denominator is going
- to be c minus d i.
- So c minus d i over c minus d i.
- So this was c plud d i, so this is its conjugate.
- And so what do we get?
- So in the numerator, we get a c -- I don't want to run out of
- space, I always do -- a c, so a times c, minus a d i, minus a d
- i -- these i's are looking funny -- this is an i.
- Plus b c i; plus b c i.
- And then the last term, we have a plus b minus b.
- So it's minus b d i squared.
- Minus b d i squared.
- All of that.
- And this is a plus b times a minus b.
- So it's equal to a squared minus b squared.
- So this is going to be equal to -- and it this will become
- second nature to you after a while, but you might want
- to just multiply it out.
- This equals c squared plus d squared.
- And don't take my word for it.
- Actually, algebraically, multiply this out and just
- realize you can only add real parts to real parts and
- imaginary parts to imaginary parts.
- So let me simplify that.
- That equals -- let's see, the real parts.
- This is real, a c.
- And this is minus b d i squared.
- So the i squared is minus one.
- So it switches the sign here, so it becomes plus b d.
- And we can get rid of d i.
- So the real parts are, a c plus b d.
- That's that, and that.
- And then the imaginary parts are plus -- this one's
- positive, so I'll put one first -- b c minus a d i, all of that
- over c squared plus d squared.
- And that still might not look like a complex number to you.
- But then we can separate them out and we could say well,
- that equals a c plus b d over c squared plus d squared.
- And that's the real part.
- Plus b c minus a d over c squared plus d squared.
- And that times i, and that's the imaginary part.
- So you can't merge, when you're adding and subtracting, the
- real part to the imaginary part.
- But you can most definitely scale an imaginary
- number by a real number.
- And that's essentially what we're doing.
- We're multiplying one over c squared plus d
- squared times this.
- So, division might seem a little complicated when I
- write it all in variables.
- But let me give you an example and you will hopefully see that
- it -- with real numbers, and -- not real numbers, with
- actual numbers, I should be careful with what I say.
- Let's say I have one plus two i.
- And I want to divide that by, I don't know, let's divide it by,
- I'm going to pick a random number.
- two plus threei.
- And so what do we do?
- We multiply it times the conjugate of the denominator.
- two minus threei over -- over itself, right?
- Because then we're not changing the number.
- This is just one, this simplifies to one.
- It equals -- the bottom, we can multiply it out.
- But hopefully it's second nature to you.
- It equals four plus nine, right?
- Because that's just a squared plus b squared.
- Right?
- Well, I mean, it's a squared minus b squared, but then the
- i's, when you multiply, and it becomes a negative number.
- Try it out if you don't believe me.
- And then the top, we get one times two is two.
- one times minus threei is minus threei.
- And you have twoi times two, which is plus fouri.
- And then you have twoi times minus threei.
- So that's minus six.
- Minus sixi squared.
- Well, what does i squared equal?
- That equals negative one.
- So negative one times negative six.
- Get rid of the i squared and this becomes a positive.
- So then, what are our real parts?
- Our real parts are two and six.
- so two plus six is eight.
- And what are our imaginary parts?
- Minus threei plus fouri.
- So that's just plus onei, right?
- Minus three plus four is positive one.
- So it's just plus onei.
- Over thirteen.
- Or we could write that as -- if we wanted to write that in the
- traditional complex form -- is eight / thirteen plus one over thirteeni.
- So when I divided one complex number by another, I got
- another complex number.
- And an interesting exercise for you to do is, pick some
- random complex numbers.
- Plot them out on complex plane, and see what happens when you
- multiply them, when you divide them, when you add them,
- when you subtract them.
- And when you scale them.
- Or when you take the conjugate.
- And that'll give you a better intuition of what's going
- on with these numbers.
- Anyway, I will see you in the next video.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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