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 1 or
square root of negative 9.
And since any real number, when you square it, is either 0 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 1.
And then that simplifies the idea of taking a
negative square root.
Because then you could say, oh well the square root of
negative 9 is the same thing as i times the square root
of 9, which equals 3i.
And how can we say that?
Well, what happens when you square this thing?
So what's 3i squared?
3i squared is equal to 3 squared times i squared.
This is just exponent properties.
And that equals 9 times negative 1, which
equals negative 9.
So 3i squared is equal to negative 9.
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 3i is equal to the
square root of negative 9.
And 3i 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 5 plus 2i.
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 5, and an
imaginary part, like the 2i, 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 5 plus 2i, well it's real part is 5.
So one, two, three, four, five.
That's five.
And it's imaginary part is 2i, so you go along the imaginary
axis, or the i-axis you could even say, 2.
And then you would have this number here called 5 plus 2i.
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 1.
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 2.
So what is the complex number z sub 1 plus z sub 2?
Well that equals this, a plus bi, plus-- this is z sub
2 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 1 minus z sub 2.
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 1 times z sub 2?
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 8th grade algebra,
9th 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 1.
Right?
Times negative 1.
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 1,
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 1.
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 1 by z sub 2?
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 1, 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.