- [Voiceover] In the last video we did a quick review
of the exponential and what it means.
And then we looked in and figured out
what the magnitude exponential is,
the magnitude is 92 is equal to one.
Now we're gonna look closely at this complex exponential
as it represents a cosine, a part of a cosine.
Now we're gonna keep combining some of our ideas
from the last couple of videos.
And you remember if one of the things we did
is we used Euler's formula
and inside out and developed an expression for cosine.
So if I say cosine of theta
I can say that equals one half
times E to the plus
J theta
plus E to the minus
J theta.
This is cosine of theta expressed as two
separate exponentials.
And now we're gonna take a really special step.
I'm gonna put in an argument right here of time.
I'm gonna say
cosine of omega T.
T is time.
N is shape here
this symbol here is the lowercase omega
from the Greek alphabet.
And that's the frequency.
So time is in units of seconds.
And omega is a frequency
so it's in
units of one over seconds.
That's the units of frequency are per seconds.
And when these multiply together
we get dimensionless number right here.
And we can take the cosine of the dimensionless number.
So what does this equal to?
This equals
one half times
E to the plus
J omega T
plus E to the
minus J omega T.
Well we make T the argument of the cosine here.
T is the stuff that keeps going up and up and up.
The number T gets just gets bigger all the time.
And so we ended up with
a cosine wave formula
I'll just to make a bad cosine looking thing here.
That's what a cosine looks like.
And it keeps going and going and going.
So we have an idea of what a cosine wave looks like.
The frequency determines how fast this goes up and down
or how often it goes up and down.
Now what I want to do is I want to look at
a really special thing.
I want to look at what is this thing right here?
What is this thing, E to the plus J omega T?
And what we see is
this cosine here is made of two of these things.
So whatever these things are
I can make a cosine out of them.
So now we're gonna really carefully add
E to the J omega T.
What we just reviewed was
that this is a complex number.
Let's draw that complex number.
So we're gonna put a number out here.
We know it falls on the unit circle.
We know its angle
is whatever is multiplying
the J up in this exponent.
Whatever's up in the exponent is the angle of this thing.
So this angle right here
is omega T.
And we know the magnitude of this is
as we decided before, the magnitude is one.
That's why it falls on the unit circle.
Okay, so now look at this.
Here's this number T that's determining the angle.
And that means, what?
That means that the angle is increasing with time.
If time is equal to zero
the point is right here.
At time equals zero.
Because angle is zero.
As time proceeds
the angle keep starts growing and growing
and it basically keeps growing and it keeps going.
It comes back to here after omega T equals two pi.
And then what happens?
It goes, it keeps going around it again.
And, this basically goes along
for as long as time goes along.
So here's this number.
Here's this complex number
moving along the unit circle
in time, over and over and over again.
So this is a number that is rotating,
the number is rotating.
So I can write here,
E to the J
omega T.
And I know that that because time's up here
I know it's rotating in time.
All right, now I'm gonna put a different number on there.
Let's put it over here say.
Actually let's start this number at zero.
And I'm gonna call this number
E to the minus
J omega T.
What does that number look like?
That's this guy here.
That's this one here.
We'll make him orange.
So at time equals zero is E to the zero
or one, just as we would expect.
Now as time gets bigger
the angle
the thing multiplying J
is minus omega T.
And so the angle is becoming more and more negative.
So after a little bit of time
it's here.
And after a little bit more time
it's here.
And what we notice is
it keeps it rotates this way.
This is what happens when you have
E to the minus J omega T.
You rotate in this direction.
And it keeps going and going and going.
So these two numbers are pretty similar in behavior
except one rotates counterclockwise
and the other rotates clockwise
in our coordinate system which is the complex plane.
So in summary, if you see either of these shapes
E to the minus plus J omega T
or E to the minus J omega T.
What pops into your head is
a number that spins.
So for me the simple idea is
I have a number here and I have a number here.
They both spin in a complex base.
And to represent those in mathematical notation
I need this kind of notation here
which is a little bit awkward
but as I get used to it
E to the J omega T is a spinning number.
E to the minus J omega T
is a spinning number.
This is a amazingly powerful idea
and we'll be able to describe every single
that happens using these kind of terms.