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 1) Introduction to complex numbers. Adding, subtracting and multiplying complex numbers.
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- We learned in the imaginary numbers video, that hopefully
- you've watched, that every now and then in certain equations
- you end up with a square root of a negative number.
- You know, you end up with square root of negative one or
- square root of negative nine.
- And since any real number, when you square it, is either zero or
- positive, this was undefined for us.
- And then in order to make it defined in that video, people
- came up with this thing called i.
- And you can call that the imaginary unit,
- or just the number i.
- And you define i as saying, well if i squared-- we'll
- just make a definition.
- This thing called i when you square it,
- it equals negative one.
- And then that simplifies the idea of taking a
- negative square root.
- Because then you could say, oh well the square root of
- negative nine is the same thing as i times the square root
- of nine, which equals threei.
- And how can we say that?
- Well, what happens when you square this thing?
- So what's threei squared?
- threei squared is equal to three squared times i squared.
- This is just exponent properties.
- And that equals nine times negative one, which
- equals negative nine.
- So threei squared is equal to negative nine.
- And then if we are kind of extending the definition of
- square roots to negative numbers, then we can go the
- other way around and we could say threei is equal to the
- square root of negative nine.
- And threei we can call an imaginary number.
- And the word is "imaginary," but it's as real as anything.
- I mean to some degree, do negative numbers even exist?
- They're just kind of a way of us-- when we put this sign in
- front of things, it just tells us something about how it
- relates to this magnitude.
- Anyway, I don't want to confuse you.
- But imaginary numbers tend to get a bad rap because they're
- called imaginary numbers, so some people think that they
- exist less than other things.
- But with that said, any number times this imaginary unit
- i is an imaginary number.
- When you do the quadratic equation you realize that
- sometimes you end up with a number that has a
- little bit of both.
- It has a real part and an imaginary part.
- So let me give you an example of that.
- Let's say I have the number five plus twoi.
- That number you might say, oh, well maybe I can simplify it.
- But you really can't.
- You can't add a real number plus an imaginary number.
- You can almost kind of imagine them-- and I don't want to use
- the word "imagine" too much-- as in different dimensions.
- And so a number that has a real part, like the five, and an
- imaginary part, like the twoi, this is called a
- complex number.
- And you could, if you want, even graph a complex number.
- Let me see.
- You could make the vertical axis.
- The vertical axis you could call it the imaginary axis.
- And you could make the horizontal axis the real axis.
- This is a symbol for the set of real numbers.
- And five plus twoi, well it's real part is five.
- So one, two, three, four, five.
- That's five.
- And it's imaginary part is twoi, so you go along the imaginary
- axis, or the i-axis you could even say, two.
- And then you would have this number here called five plus twoi.
- In future videos I'll actually do examples where we'll make
- more use of complex numbers.
- But now that we've defined what a complex number is, let's see
- how we can operate with it.
- So what happens when we add two complex numbers?
- Clear image.
- So let's say we have two complex numbers.
- One is a plus bi.
- So the real part is a, the imaginary part is bi.
- And let's say that I have another complex
- number, c plus di.
- And in general the symbol that people tend to use-- you know,
- you always use x for when you're dealing with equations.
- Right?
- Like x could be any general real number when you're
- dealing with an equation.
- In complex numbers the convention is to use
- z, the letter z.
- So, for example, we could call this the complex
- number z sub one.
- And it could have been any.
- I mean, this choice of z is arbitrary.
- This is an i right here.
- This could be z sub two.
- So what is the complex number z sub one plus z sub two?
- Well that equals this, a plus bi, plus-- this is z sub
- two right here-- c plus di.
- Let me switch colors.
- This is getting monotonous.
- So when you add two complex numbers, all you do is you add
- the real parts to each other and you add the complex
- parts to each other.
- So that equals a plus b-- oh sorry.
- The real parts.
- So it's a and c.
- Those are the real parts.
- So that equals a plus c plus-- and then you add
- the imaginary parts.
- b plus d times i.
- So you have b i's and d i's.
- So when you add them together you have b plus d i's.
- b plus d in the imaginary direction, you can
- almost imagine.
- I'm using the word imagine too much.
- So it's pretty easy.
- You just add the real and you add the imaginary parts.
- So what happens if you multiply two complex numbers?
- And actually, let's just go in order.
- What happens when you subtract them?
- Well, it's the same thing.
- Nothing fancy here. z sub one minus z sub two.
- That's just going to be equal to-- you subtract
- the real parts.
- a minus c.
- And that's the new real part.
- And then you subtract the imaginary parts.
- b minus d times i.
- And this will be the new imaginary part.
- So this will be the new complex number.
- What happens when I multiply the two numbers?
- So what is z sub one times z sub two?
- Well that equals a plus bi times c plus di.
- And we essentially can just FOIL it out.
- I don't know if you've learned FOIL in eightth grade algebra,
- nineth grade algebra.
- I don't like it.
- I actually just think of it as the distributive
- property twice.
- So what we could do is we can take the c plus di
- and distribute it on each of these terms.
- Right?
- So this should be equal to a times c plus di.
- Right?
- Plus bi times c plus di.
- Right?
- All I did is I took the c plus di and I multiplied it by this
- to get this, and multiplied it by this to get this.
- And then we distribute again.
- We get ac plus adi-- this is an i.
- I know my i's aren't looking good.
- adi plus-- now what's bi times c?
- Well that's cbi.
- And then we have bi times di.
- So you get b times d, which is bd.
- And then what's i times i?
- It's i squared, or negative one.
- Right?
- Times negative one.
- So what does this simplify to?
- Well this simplifies to-- I'm going to go back here.
- So let's see what the real parts are.
- We have this term, a times c, that's real.
- So ac.
- And then this last term, which was bi times di.
- Because we're multiplying i times i we get a negative one,
- but it becomes real again.
- Right?
- So we get negative bd minus bd.
- So that's our new real part.
- So that was this term and that term are real.
- And now what's our new imaginary part?
- Well it's going to be these two terms added together.
- So it's-- I'm running out of space again-- ad plus
- cb, all of that times i.
- It's a little easier with real numbers, and maybe in the next
- video I'll use real numbers.
- But the important thing to realize is essentially you just
- use the distributive property and realize that you can only
- add real terms to each other.
- You can't add a real to an imaginary.
- And you can only add imaginary terms to each other.
- And just remember, when you have two imaginary numbers
- times each other, the i's, when multiplied times each other,
- and you get negative one.
- Now the last operation, when you divide complex numbers.
- This gets a little bit interesting, and maybe
- a little unintuitive.
- So what happens if I divide z sub one by z sub two?
- So once again that equals a plus bi, divided by c plus di.
- How do I divide this?
- Well, we're going to use a property that hopefully
- you learned in algebra.
- That if I multiply-- let me do this in this corner.
- a plus b times a minus b.
- What does that equal?
- That equals a squared minus b squared.
- And so if I multiply a complex number.
- Let's say, in theory, I multiply a complex number c
- plus di times c minus di.
- What do I get?
- I get c squared minus di squared.
- And what's di squared going to be?
- It's d squared times negative one, so it becomes c squared--
- oh, I'm out of time.
- Actually, let me continue the division in the next video,
- because it can get involved.
- See you soon.
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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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