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I assume from high school you
know how to add and multiply

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complex numbers using the
relation i squared equals

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negative one.
I'm a little less certain that

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you remember how to divide them.
I hope you read last night by

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way of preparation for that,
but since that's something

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we're going to have to do a lot
of a differential equations,

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so remember that the division
is done by making use of the

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complex conjugate.
So, if z is equal to a plus bi,

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some people write a plus ib,

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and sometimes I'll do
that too if it's more

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convenient.
Then, the complex conjugate is

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what you get by changing i to
negative i.

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And, the important thing is
that the product of those two is

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a real number.
The product of these is a

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squared minus the quantity ib
all squared,

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which makes a squared plus b
squared because i

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squared is negative one.

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So, the product of those,
that's what you multiply if you

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want to multiply this by
something to make it real.

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You always multiplied by its
complex conjugate.

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And that's the trick that
underlines the doing of the

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division.
So, for example,

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I better hang onto these or
I'll never remember all the

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examples.
Suppose, for example,

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we wanted to calculate (two
plus i) divided by (one minus 3

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i).
To calculate it means I want to

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do the division;
I want to express the answer in

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the form a plus bi.
What you do is multiply the top

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and bottom by the complex
conjugate of the denominator in

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order to make it real.
So, it's (one plus 3i) divided

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by (one plus 3i),
as they taught you in

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elementary school,
that is one,

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in a rather odd notation;
therefore, multiplying doesn't

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change the value of the
fraction.

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And so, the denominator now
becomes 1 squared plus 3

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squared, which is ten.

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And, the numerator is,
learn to do this without

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multiplying out four terms.
You must be able to do this in

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your head.
And, you always do it by the

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grouping, or post office method,
whatever you want to call it,

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namely, first put down the real
part, which is made out of two

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times one minus three times one.
So, that's negative one.

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And then, the imaginary part,
which is i times one.

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That's one, coefficient one,
plus 6i.

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So, that makes 7i.
Now, some people feel this

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still doesn't look right,
if you wish,

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and for some places and
differential equations,

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it will be useful to write that
as minus one tenth plus seven

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tenths i.
And, now it's perfectly clear

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that it's in the form a plus bi.
So, learn to do that if you

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don't know already.
It's going to be important.

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Now, the main thing today is
the polar representation,

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which sometimes they don't get
to in high school.

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And if they do,
it's usually not in a grown

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up-enough in a form for us to be
able to use it.

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So, I have to worry about that
little bit.

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The polar representation,
of course, is nominally just

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the switch to polar coordinates.
If here's a plus bi,

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then this is r,
and that's theta.

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And therefore,
this can be written as,

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in the polar form,
that would be r cosine theta

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plus i, or r cosine theta.

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That's the A part.

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And, the B part is,
the imaginary part is r

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sin(theta) times i.

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Now, it would be customary,
at this point,

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to put the i in front,
just because it looks better.

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The complex numbers are
commutative, satisfied to

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commutative law of
multiplication,

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which means it doesn't matter
in multiplication whether you

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put i in front or behind.
It's still the same answer.

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So, this would be r cosine
theta plus i times 

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r sine theta,

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which, of course,
will factor out,

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and will make it cosine theta
plus i sine theta.

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Now, it was Euler who took the

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decisive step and said,
hey, look, I'm going to call

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that e to the i theta.

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Now, why did he do that?
Because everything seemed to

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indicate that it should.
But that's certainly worth the

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best color we have,
which is what?

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We are getting low here.
Okay, nonetheless,

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it's worth pink.
I will even give him his due,

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Euler.
Sometimes it's called Euler's

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formula, but it really shouldn't
be.

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It's not a formula.
It's a definition.

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So, in some sense,
you can't argue with it.

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If you want to call putting a
complex number in a power,

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and calling it that,
you can.

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But, one can certainly ask why
he did it.

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And the answer,
I guess, is that all the

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evidence seemed to point to the
fact that it was the thing to

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do.
Now, I think it's important to

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talk about a little bit because
I think it's,

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in my opinion,
if you're seeing this for the

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first time, even if you read
about it last night,

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it's a mysterious thing,
and one needs to see it from

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every possible point of view.
It's something you get used to.

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You will never see it in a
sudden flash of insight.

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It will just get as familiar to
you as more common arithmetic,

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and algebraic,
and calculus processes are.

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But, look.
What is it we demand?

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If you're going to call
something an exponential,

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what is it we want an
exponential to do,

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what gives an expression like
this the right to be called e to

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the i theta?
The answer is I can't creep

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inside Euler's mind.
It must have been a very big

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day of his life.
He had a lot of big days,

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but when he realized that that
was the thing to write down as

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the definition of e to the i
theta.

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But, what is it one wants of an
exponential?

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Well, the high school answer
surely is you want it to satisfy

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the exponential law.
Now, to my shock,

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I realize a lot of people don't
know.

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In my analysis class,
these are some math majors,

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or graduate engineers in
various subjects,

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and if I say prove such and
such using the exponential law,

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I'm sure to get at least half a
dozen e-mails asking me,

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what's the exponential law?
Okay, the exponential law is a

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to the x times a to the y equals
a to the x plus y:

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the law of exponents.

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That's the most important
reason why, that's the single

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most important thing about
exponents, are the way one uses

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them.
And, this is the exponential

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function, called the exponential
function because all this

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significant stuff is in the
exponents.

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All right, so it should
satisfy-- we want,

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first of all,
the exponential law to be true.

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But that's not all.
That's a high school answer.

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An MIT answer would be,
I mean, why is e to the x such

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a popular function?
Well, of course,

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it does satisfy the exponential
law, but for us,

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an even more reasonable thing.
It's the function,

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which, when you differentiate
it, you get the same thing you

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started with.
And, it's apart from a constant

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factor, the only such function.
Now, in terms of differential

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equations, it means that it's
the solution that e to the,

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let's be a little generous,
make it e to the ax.

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No, better not to use x because
complex numbers tend to be

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called x plus iy.
Let's use t as a more neutral

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variable, which is standing
outside the fray,

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as it were.
It satisfies the relationship

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that it's the solution,
if you like,

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to the differential equation.
That's a fancy way of saying

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it.
dy / dt equals a times y.

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Now, of course,

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that is not unique.
We could make it unique by

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putting in an initial value.
So, if I want to get this

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function and not a constant
times it, I should make this an

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initial value problem and say
that y of zero should be one.

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And now, I will get only the

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function, e to the at.
So, in other words,

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that characterizes this
function.

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It's the only function in the
whole world that has that

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property.
Now, if you're going to call

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something e to the i theta,
we want that to be true.

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So, here are my questions.
Is it true that e to the i

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theta one, let's use that,
times e to the i theta two,

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see, I'm on a collision course

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here, but that's easily fixed.
Is that equal to e to the i

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(theta one plus theta two)?

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If that turns out to be so,
that's a big step.

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What would we like to be true
here?

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Well, will it be true that the
derivative, with respect to t of

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e to the i theta,
I would like that to be equal

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to i times e to the i theta.

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So, question,

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question.
I think those are the two most

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significant things.
Now, the nodes do a third

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thing, talk about infinite
series.

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Since we haven't done infinite
series, anyway,

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it's not officially part of the
syllabus, the kind of power

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series that are required.
But, I will put it down for the

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sake of completeness,
as people like to say.

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So, it should behave right.
The infinite series should be

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nice.
The infinite series should work

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out.
There is no word for this,

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should work out,
let's say.

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I mean, what's the little
music?

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Is that some weird music idea,
or is it only me that hears it?

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[LAUGHTER] Yes,
Lord.

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I feel I'm being watched up
there.

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This is terrible.
So, there's one guy.

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Here's another guy.
And, I won't put a box around

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the infinite series,
since I'm not going to say

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anything about it.
Now, these things,

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in fact, are both true.
Otherwise, why would I be

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saying them, and why would Euler
have made the formula?

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But, what's interesting to see
is what's behind them.

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And, that gives you little
practice also in calculating

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with the complex numbers.
So, let's look at the first

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one.
What will it say?

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It is asking the question.
It says, please,

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calculate the product of these
two things.

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Okay, I do it,
I'm told.

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I will calculate the product of
cosine theta one plus i cosine

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theta two-- Sine.
Sine theta one.

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That's e to the i

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theta one, right?

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So, that corresponds to this.
The other factor times the

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other factor,
cosine theta two plus i sine

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theta two.

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Okay, what does that come out
to be?

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Well, again,
we will use the method of

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grouping.
What's the real part of it?

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The real part of it is cosine
theta one cosine theta two.

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And then, there's a real part,

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which comes from these two
factors.

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It's going to occur with a
minus sign because of the i

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squared.
And, what's left is sine theta

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one sine theta two.

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And then, the imaginary part,
I'll factor out the i.

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And then, what's left,
I won't have to keep repeating

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the i.
So, it will have to be sine

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theta one cosine theta two.

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And, the other factor will be
cosine theta one sine theta

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two-- plus sine theta two cosine
theta one.

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Well, it looks like a mess,

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but, again, high school to the
rescue.

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What is this?
The top thing is nothing in

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disguise, but it's a disguised
form of cosine (theta one plus

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theta two).

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And the bottom is sine of
(theta one plus theta two).

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So, the product of these two

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things is this,
and that's exactly the formula.

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In other words,
this formula is a way of

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writing those two trigonometric
identities for the cosine of the

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sum and the sine of the sum.
Instead of the two identities

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taking up that much space,
written one after the other,

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they take up as much space,
and they say exactly the same

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thing.
Those two trigonometric

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identities are exactly the same
as saying that e to the i theta

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satisfies the
exponential law.

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Now, people ask,
you know, what's beautiful in

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mathematics?
To me, that's beautiful.

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I think that's great.
Something long turns into

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something short,
and it's just as good,

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and moreover,
connects with all these other

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things in the world,
differential equations,

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infinite series,
blah, blah, blah,

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blah, blah.
Okay, I don't have to sell

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Euler.
He sells himself.

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Now, how about the other one?
How about the other one?

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Now, that's obviously,
I haven't said something

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because for one thing,
how do you differentiate if

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there's theta here,
and t down there.

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Okay, that's easily fixed.
But, how do I differentiate

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this?
What kind of a guy is e to the

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i theta?
Well, if I write it out,

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take a look at what it is.
It's cosine theta plus i sine

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theta.

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As theta varies,
it's a function.

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The variable is real.
Theta is a real variable.

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Its angle in radians,
but it runs from negative

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infinity to infinity.
So, if you think of functions

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as a black box,
what's going in is a real

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number.
But, what's coming out is a

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complex number.
So, schematically,

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here is the e to the i theta
box,

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if you like to think that way,
theta goes in,

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and that's real,
and a complex number,

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this particular complex number
goes out.

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So, one, we'd call it,
I'm not going to write this

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down because it's sort of
pompous and takes too long.

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But, it is a complex valued
function of a real variable.

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00:17:33 --> 00:17:39
You got that?
Up to now, we studied real

268
00:17:36 --> 00:17:42
functions of real variables.
But now, real valued functions

269
00:17:41 --> 00:17:47
of real variables,
those are the kind calculus is

270
00:17:45 --> 00:17:51
concerned with.
But now, it's a complex-valued

271
00:17:48 --> 00:17:54
function because the variable is
real.

272
00:17:51 --> 00:17:57
But, the output,
the value of the function is a

273
00:17:55 --> 00:18:01
complex number.
Now, in general,

274
00:17:58 --> 00:18:04
such a function,
well, maybe a better say,

275
00:18:01 --> 00:18:07
complex-valued,
how about complex-valued

276
00:18:04 --> 00:18:10
function of a real variable,
let's change the name of the

277
00:18:09 --> 00:18:15
variable.
t is always a real variable.

278
00:18:14 --> 00:18:20
I don't think we have complex
time yet, although I'm sure

279
00:18:19 --> 00:18:25
there will be someday.
But, the next Einstein appears.

280
00:18:24 --> 00:18:30
A complex-valued function of a
real variable,

281
00:18:28 --> 00:18:34
t, in general,
would look like this.

282
00:18:32 --> 00:18:38
t goes in, and what comes out?
Well: a complex number,

283
00:18:35 --> 00:18:41
which I would then have to
write this way.

284
00:18:38 --> 00:18:44
In other words,
the real part depends on t,

285
00:18:41 --> 00:18:47
and the imaginary part depends
upon t.

286
00:18:44 --> 00:18:50
So, a general function looks
like this, a general

287
00:18:47 --> 00:18:53
complex-valued function.
This is just a special case of

288
00:18:51 --> 00:18:57
it, where the variable has a
different name.

289
00:18:54 --> 00:19:00
But, the first function would
be cosine t, and the second

290
00:18:57 --> 00:19:03
function would be sine t.
So, my only question is,

291
00:19:01 --> 00:19:07
how do you differentiate such a
thing?

292
00:19:03 --> 00:19:09
Well, I'm not going to fuss
over this.

293
00:19:08 --> 00:19:14
The general definition is,
with deltas and whatnot,

294
00:19:11 --> 00:19:17
but the end result of a
perfectly fine definition is,

295
00:19:14 --> 00:19:20
you differentiate it by
differentiating each component.

296
00:19:18 --> 00:19:24
The reason you don't have to
work so very hard is because

297
00:19:22 --> 00:19:28
this is a real variable,
and I already know what it

298
00:19:25 --> 00:19:31
means to differentiate a
function of a real variable.

299
00:19:30 --> 00:19:36
So, I could write it this way,
that the derivative of u plus

300
00:19:34 --> 00:19:40
iv, I'll abbreviate it that way,
this means the derivative,

301
00:19:37 --> 00:19:43
with respect to whatever
variable, since I didn't tell

302
00:19:41 --> 00:19:47
you what the variable in these
functions were,

303
00:19:44 --> 00:19:50
well, I don't have to tell you
what I'm differentiating with

304
00:19:48 --> 00:19:54
respect to.
It's whatever was there because

305
00:19:51 --> 00:19:57
you can't see.
And the answer is,

306
00:19:53 --> 00:19:59
it would be the derivative of u
plus i times the derivative of

307
00:19:57 --> 00:20:03
v.
You differentiate it just the

308
00:20:01 --> 00:20:07
way you would if these were the
components of a motion vector.

309
00:20:05 --> 00:20:11
You would get the velocity by
differentiating each component

310
00:20:09 --> 00:20:15
separately.
And, that's what you're doing

311
00:20:12 --> 00:20:18
here.
Okay, now, the importance of

312
00:20:15 --> 00:20:21
that is that it at least tells
me what it is I have to check

313
00:20:19 --> 00:20:25
when I check this formula.
So, let's do it now that we

314
00:20:23 --> 00:20:29
know what this is.
We know how to differentiate

315
00:20:26 --> 00:20:32
the function.
Let's actually differentiate

316
00:20:29 --> 00:20:35
it.
That's fortunately,

317
00:20:32 --> 00:20:38
by far, the easiest part of the
whole process.

318
00:20:37 --> 00:20:43
So, let's do it.
So, what's the derivative?

319
00:20:41 --> 00:20:47
Let's go back to t,
our generic variable.

320
00:20:45 --> 00:20:51
I want to emphasize that these
functions, when we write them as

321
00:20:51 --> 00:20:57
functions, that theta will
almost never be the variable

322
00:20:56 --> 00:21:02
outside of these notes on
complex numbers.

323
00:21:02 --> 00:21:08
It will normally be time or
something like that,

324
00:21:05 --> 00:21:11
or x, a neutral variable like
x.

325
00:21:07 --> 00:21:13
So, what's the derivative of e
to the i theta?

326
00:21:12 --> 00:21:18
I'm hoping that it will turn
out to be i e to the i theta,

327
00:21:16 --> 00:21:22
and that the yellow law may be

328
00:21:19 --> 00:21:25
true just as the green one was.
Okay, let's calculate it.

329
00:21:23 --> 00:21:29
It's the derivative,
with respect to,

330
00:21:26 --> 00:21:32
unfortunately I can convert t's
to thetas, but not thetas to

331
00:21:30 --> 00:21:36
t's.
C'est la vie,

332
00:21:33 --> 00:21:39
okay.
Times cosine t plus i sine t,

333
00:21:36 --> 00:21:42
and what's that?

334
00:21:40 --> 00:21:46
Well, the derivative of cosine
t, differentiating the real and

335
00:21:46 --> 00:21:52
imaginary parts separately,
and adding them up.

336
00:21:51 --> 00:21:57
It's negative sine t,
plus i times cosine t.

337
00:21:56 --> 00:22:02
Now, let's factor out at the i,

338
00:22:01 --> 00:22:07
because it says if I factor out
the i, what do I get?

339
00:22:08 --> 00:22:14
Well, now, the real part of
what's left would be cosine t.

340
00:22:12 --> 00:22:18
And, how about the imaginary
part?

341
00:22:15 --> 00:22:21
Do you see, it will be i sine t
because i times i

342
00:22:21 --> 00:22:27
gives me that negative one.

343
00:22:24 --> 00:22:30
And, what's that?
e to the it.

344
00:22:27 --> 00:22:33
i times e to the i t.

345
00:22:30 --> 00:22:36
So, that works too.
What about the initial

346
00:22:35 --> 00:22:41
condition?
No problem.

347
00:22:38 --> 00:22:44
What is y of zero?
What's the function at zero?

348
00:22:43 --> 00:22:49
Well, don't say right away,
i times zero is zero,

349
00:22:49 --> 00:22:55
so it must be one.
That's illegal because,

350
00:22:54 --> 00:23:00
why is that illegal?
It's because in that formula,

351
00:23:00 --> 00:23:06
you are not multiplying i times
theta.

352
00:23:06 --> 00:23:12
I mean, sort of,
you are, but that formula is

353
00:23:09 --> 00:23:15
the meaning of e to 
the i theta.

354
00:23:12 --> 00:23:18
Now, it would be very nice if
this is like,

355
00:23:16 --> 00:23:22
well, anyway,
you can't do that.

356
00:23:18 --> 00:23:24
So, you have to do it by saying
it's the cosine of zero plus i

357
00:23:23 --> 00:23:29
times the sine of zero.

358
00:23:26 --> 00:23:32
And, how much is that?
The sine of zero is zero.

359
00:23:31 --> 00:23:37
Now, it's okay to say i times
zero is zero because that's the

360
00:23:35 --> 00:23:41
way complex numbers multiply.
What is the cosine of zero?

361
00:23:40 --> 00:23:46
That's one.
So, the answer,

362
00:23:42 --> 00:23:48
indeed, turns out to be one.
So, this checks,

363
00:23:46 --> 00:23:52
really, from every conceivable
standpoint down as I indicated,

364
00:23:51 --> 00:23:57
also from the standpoint of
infinite series.

365
00:23:54 --> 00:24:00
So, we are definitely allowed
to use this.

366
00:23:58 --> 00:24:04
Now, the more general
exponential law is true.

367
00:24:03 --> 00:24:09
I'm not going to say much about
it.

368
00:24:05 --> 00:24:11
So, in other words,
e to the a, this is really a

369
00:24:09 --> 00:24:15
definition.
e to the (a plus ib)

370
00:24:13 --> 00:24:19
is going to be,
in order for the general

371
00:24:17 --> 00:24:23
exponential law to be true,
this is really a definition.

372
00:24:21 --> 00:24:27
It's e to the a times e to the
ib.

373
00:24:26 --> 00:24:32
Now, notice when I look at
the-- at any complex number,

374
00:24:30 --> 00:24:36
--
-- so, in terms of this,

375
00:24:34 --> 00:24:40
the polar form of a complex
number, to draw the little

376
00:24:38 --> 00:24:44
picture again,
if here is our complex number,

377
00:24:42 --> 00:24:48
and here is r,
and here is the angle theta,

378
00:24:46 --> 00:24:52
so the nice way to write this
complex number is r e to the i

379
00:24:51 --> 00:24:57
theta.
The e to the i theta

380
00:24:56 --> 00:25:02
is, now, why is that?

381
00:25:00 --> 00:25:06
What is the magnitude of this?
This is r.

382
00:25:04 --> 00:25:10
The length of the absolute
value, I didn't talk about

383
00:25:10 --> 00:25:16
magnitude in argument.
I guess I should have.

384
00:25:14 --> 00:25:20
But, it's in the notes.
So, r is called the modulus.

385
00:25:20 --> 00:25:26
Well, the fancy word is the
modulus.

386
00:25:24 --> 00:25:30
And, we haven't given the
complex number a name.

387
00:25:29 --> 00:25:35
Let's call it alpha,
modulus of alpha,

388
00:25:33 --> 00:25:39
and theta is called,
it's the angle.

389
00:25:39 --> 00:25:45
It's called the argument.
I didn't make up these words.

390
00:25:44 --> 00:25:50
There, from a tradition of
English that has long since

391
00:25:49 --> 00:25:55
vanished, when I was a kid,
and you wanted to know what a

392
00:25:55 --> 00:26:01
play was about,
you looked in the playbill,

393
00:25:59 --> 00:26:05
and it said the argument of the
play, it's that old-fashioned

394
00:26:05 --> 00:26:11
use of the word argument.
Argument means the angle,

395
00:26:11 --> 00:26:17
and sometimes that's
abbreviated by arg alpha.

396
00:26:16 --> 00:26:22
And, this is abbreviated,

397
00:26:21 --> 00:26:27
of course, as absolute value of
alpha, its length.

398
00:26:26 --> 00:26:32
Okay, the notes give you a
little practice changing things

399
00:26:33 --> 00:26:39
to a polar form.
I think we will skip that in

400
00:26:39 --> 00:26:45
favor of doing a couple of other
things because that's pretty

401
00:26:46 --> 00:26:52
easy.
But let me, you should at least

402
00:26:50 --> 00:26:56
realize when you should look at
polar form.

403
00:26:55 --> 00:27:01
The great advantage of polar
form is, particularly once

404
00:27:01 --> 00:27:07
you've mastered the exponential
law, the great advantage of

405
00:27:08 --> 00:27:14
polar form is it's good for
multiplication.

406
00:27:15 --> 00:27:21
Now, of course,
you know how to multiply

407
00:27:17 --> 00:27:23
complex numbers,
even when they are in the

408
00:27:20 --> 00:27:26
Cartesian form.
That's the first thing you

409
00:27:23 --> 00:27:29
learn in high school,
how to multiply a plus bi times

410
00:27:27 --> 00:27:33
c plus di.
But, as you will see,

411
00:27:31 --> 00:27:37
when push comes to shove,
you will see this very clearly

412
00:27:35 --> 00:27:41
on Friday when we talk about
trigonometric inputs to

413
00:27:39 --> 00:27:45
differential equations,
--

414
00:27:42 --> 00:27:48
-- that the changing to complex
numbers makes all sorts of

415
00:27:46 --> 00:27:52
things easy to calculate,
and the answers come out

416
00:27:49 --> 00:27:55
extremely clear,
whereas if we had to do it any

417
00:27:52 --> 00:27:58
other way, it's a lot more work.
And worst of all,

418
00:27:56 --> 00:28:02
when you finally slog through
to the end, you fear you are

419
00:28:00 --> 00:28:06
none the wiser.
It's good for multiplication

420
00:28:03 --> 00:28:09
because the product,
so here's any number in its

421
00:28:07 --> 00:28:13
polar form.
That's a general complex

422
00:28:09 --> 00:28:15
number.
It's modulus times e to the i

423
00:28:12 --> 00:28:18
theta times r two e to the i
theta two-- 

424
00:28:16 --> 00:28:22
Well,
you just multiply them as

425
00:28:19 --> 00:28:25
ordinary numbers.
So, the part out front will be

426
00:28:22 --> 00:28:28
r1 r2, and the e to the i theta
parts gets

427
00:28:26 --> 00:28:32
multiplied by the exponential
law and becomes e to the i

428
00:28:30 --> 00:28:36
(theta one plus theta two) --

429
00:28:36 --> 00:28:42
-- which makes very clear that
the multiply geometrically two

430
00:28:42 --> 00:28:48
complex numbers,
you multiply the moduli,

431
00:28:46 --> 00:28:52
the r's, the absolute values,
how long the arrow is from zero

432
00:28:52 --> 00:28:58
to the complex number,
multiply the moduli,

433
00:28:56 --> 00:29:02
and add the arguments.
So the new number,

434
00:29:02 --> 00:29:08
its modulus is the product of
r1 and r2.

435
00:29:07 --> 00:29:13
And, its argument,
its angle, polar angle,

436
00:29:12 --> 00:29:18
is the sum of the old two
angles.

437
00:29:15 --> 00:29:21
And, you add the angles.
And, you put down in your books

438
00:29:22 --> 00:29:28
angles, but I'm being
photographed,

439
00:29:26 --> 00:29:32
so I'm going to write
arguments.

440
00:29:31 --> 00:29:37
In other words,
it makes the geometric content

441
00:29:34 --> 00:29:40
of multiplication clear,
in a sense in which this is

442
00:29:38 --> 00:29:44
extremely unclear.
From this law,

443
00:29:40 --> 00:29:46
blah, blah, blah,
blah, blah, whatever it turns

444
00:29:44 --> 00:29:50
out to be, you have not the
slightest intuition that this is

445
00:29:48 --> 00:29:54
true about the complex numbers.
That first thing is just a

446
00:29:52 --> 00:29:58
formula, whereas this thing is
insightful representation of

447
00:29:57 --> 00:30:03
complex multiplication.
Now, I'd like to use it for

448
00:30:02 --> 00:30:08
something, but before we do
that, let me just indicate how

449
00:30:08 --> 00:30:14
just the exponential notation
enables you to do things in

450
00:30:14 --> 00:30:20
calculus, formulas that are
impossible to remember from

451
00:30:19 --> 00:30:25
calculus.
It makes them very easy to

452
00:30:23 --> 00:30:29
derive.
A typical example of that is,

453
00:30:27 --> 00:30:33
suppose you want to,
for example,

454
00:30:30 --> 00:30:36
integrate (e to the negative x)
cosine x.

455
00:30:38 --> 00:30:44
Well, number one,
you spend a few minutes running

456
00:30:41 --> 00:30:47
through a calculus textbook and
try to find out the answer

457
00:30:45 --> 00:30:51
because you know you are not
going to remember how to do it.

458
00:30:49 --> 00:30:55
Or, you run to a computer,
and type in Matlab and

459
00:30:53 --> 00:30:59
something.
Or, you fish out your little

460
00:30:55 --> 00:31:01
pocket calculator,
which will give you a formula,

461
00:30:59 --> 00:31:05
and so on.
So, you have aides for doing

462
00:31:03 --> 00:31:09
that.
But, the way to do it if you're

463
00:31:06 --> 00:31:12
on a desert island,
and the way I always do it

464
00:31:10 --> 00:31:16
because I never have any of
these little aides around,

465
00:31:14 --> 00:31:20
and I cannot trust my memory,
probably a certain number of

466
00:31:19 --> 00:31:25
you remember how you did it at
high school, or how you did it

467
00:31:24 --> 00:31:30
in 18.01, if you took it here.
You have to use integration by

468
00:31:29 --> 00:31:35
parts.
But, it's one of the tricky

469
00:31:33 --> 00:31:39
things that's not required on an
exam because you had to use

470
00:31:37 --> 00:31:43
integration by parts twice in
the same direction,

471
00:31:40 --> 00:31:46
and then suddenly by comparing
the end product with the initial

472
00:31:45 --> 00:31:51
product and writing an equation.
Somehow, the value falls out.

473
00:31:50 --> 00:31:56
Well, that's tricky.
And it's not the sort of thing

474
00:31:53 --> 00:31:59
you can waste time stuffing into
your head, unless you are going

475
00:31:58 --> 00:32:04
to be the integration bee during
IAP or something like that.

476
00:32:04 --> 00:32:10
Instead, using complex numbers
is the way to do this.

477
00:32:09 --> 00:32:15
How do I think of this,
cosine x?

478
00:32:12 --> 00:32:18
What I do, is I think of that e
to the negative x cosine x

479
00:32:18 --> 00:32:24
is the real
part, the real part of what?

480
00:32:24 --> 00:32:30
Well, cosine x is the real part
of e to the ix.

481
00:32:29 --> 00:32:35
So, this thing,
this is real.

482
00:32:32 --> 00:32:38
This is real,
too.

483
00:32:34 --> 00:32:40
But I'm thinking of it as the
real part of e to the ix.

484
00:32:39 --> 00:32:45
Now, if I multiply these two

485
00:32:45 --> 00:32:51
together, this is going to turn
out to be, therefore,

486
00:32:49 --> 00:32:55
the real part of e to the minus
x.

487
00:32:53 --> 00:32:59
I'll write it out very
pompously, and then I will fix

488
00:32:57 --> 00:33:03
it.
I would never write this,

489
00:33:00 --> 00:33:06
you are you.
Okay, it's e to the minus x

490
00:33:04 --> 00:33:10
times, when I write cosine x
plus i sine x,

491
00:33:09 --> 00:33:15
so it is the real part of that
is cosine x.

492
00:33:14 --> 00:33:20
So, it's the real part of,
write it this way for real part

493
00:33:20 --> 00:33:26
of e to the, factor out the x,
and what's up there is

494
00:33:26 --> 00:33:32
(negative one plus i) times x.

495
00:33:33 --> 00:33:39
Okay, and now,
so, the idea is the same thing

496
00:33:36 --> 00:33:42
is going to be true for the
integral.

497
00:33:39 --> 00:33:45
This is going to be the real
part of that,

498
00:33:43 --> 00:33:49
the integral of e to the (minus
one plus i) times x dx.

499
00:33:48 --> 00:33:54
In other words,

500
00:33:51 --> 00:33:57
what you do is,
this procedure is called

501
00:33:54 --> 00:34:00
complexifying the integral.
Instead of looking at the

502
00:33:58 --> 00:34:04
original real problem,
I'm going to turn it into a

503
00:34:03 --> 00:34:09
complex problem by turning this
thing into a complex

504
00:34:07 --> 00:34:13
exponential.
This is the real part of that

505
00:34:12 --> 00:34:18
complex exponential.
Now, what's the advantage of

506
00:34:15 --> 00:34:21
doing that?
Simple.

507
00:34:16 --> 00:34:22
It's because nothing is easier
to integrate than an

508
00:34:20 --> 00:34:26
exponential.
And, though you may have some

509
00:34:23 --> 00:34:29
doubts as to whether the
familiar laws work also with

510
00:34:26 --> 00:34:32
complex exponentials,
I assure you they all do.

511
00:34:30 --> 00:34:36
It would be lovely to sit and
prove them.

512
00:34:34 --> 00:34:40
On the other hand,
I think after a while,

513
00:34:37 --> 00:34:43
you find it rather dull.
So, I'm going to do the fun

514
00:34:41 --> 00:34:47
things, and assume that they are
true because they are.

515
00:34:46 --> 00:34:52
So, what's the integral of e to
the (minus one plus i) x dx?

516
00:34:51 --> 00:34:57

517

518

519
00:34:54 --> 00:35:00
Well, if there is justice in
heaven, it must be e to the

520
00:34:58 --> 00:35:04
(minus one plus i) times x
divided by minus one plus i.

521
00:35:03 --> 00:35:09
In some sense,

522
00:35:08 --> 00:35:14
that's the answer.
This does, in fact,

523
00:35:12 --> 00:35:18
give that.
That's correct.

524
00:35:15 --> 00:35:21
I want the real part of this.
I want the real part because

525
00:35:22 --> 00:35:28
that's the way the original
problem was stated.

526
00:35:27 --> 00:35:33
I want the real part only.
So, I want the real part of

527
00:35:34 --> 00:35:40
this.
Now, this is what separates the

528
00:35:38 --> 00:35:44
girls from the women.
[LAUGHTER] This is why you have

529
00:35:44 --> 00:35:50
to know how to divide complex
numbers.

530
00:35:48 --> 00:35:54
So, watch how I find the real
part.

531
00:35:52 --> 00:35:58
I write it this way.
Normally when I do the

532
00:35:56 --> 00:36:02
calculations for myself,
I would skip a couple of these

533
00:36:02 --> 00:36:08
steps.
But this time,

534
00:36:05 --> 00:36:11
I will write everything out.
You're going to have to do this

535
00:36:09 --> 00:36:15
a lot in this course,
by the way, both over the

536
00:36:12 --> 00:36:18
course of the next few weeks,
and especially towards the end

537
00:36:16 --> 00:36:22
of the term where we get into a
complex systems,

538
00:36:19 --> 00:36:25
which involve complex numbers.
There's a lot of this.

539
00:36:22 --> 00:36:28
So, now is the time to learn to
do it, and to feel skillful at

540
00:36:26 --> 00:36:32
it.
So, it's this times e to the

541
00:36:28 --> 00:36:34
negative x times e to the ix,

542
00:36:31 --> 00:36:37
which is cosine x plus 
i sine x.

543
00:36:36 --> 00:36:42
Now, I'm not ready,
yet, to do the calculation to

544
00:36:39 --> 00:36:45
find the real part because I
don't like the way this looks.

545
00:36:43 --> 00:36:49
I want to go back to the thing
I did right at the very

546
00:36:46 --> 00:36:52
beginning of the hour,
and turn it into an a plus bi

547
00:36:50 --> 00:36:56
type of complex
number.

548
00:36:52 --> 00:36:58
In other words,
what we have to do is the

549
00:36:55 --> 00:37:01
division.
So, the division is going to

550
00:36:57 --> 00:37:03
be, now, I'm going to ask you to
do it in your head.

551
00:37:02 --> 00:37:08
I multiply the top and bottom
by negative one minus I.

552
00:37:06 --> 00:37:12
What does that put in the
denominator?

553
00:37:09 --> 00:37:15
One squared plus one squared:
Two.

554
00:37:13 --> 00:37:19
And in the numerator,
negative one minus i.

555
00:37:17 --> 00:37:23
This is the same as that.

556
00:37:20 --> 00:37:26
But now, it looks at the form a
+ bi.

557
00:37:24 --> 00:37:30
It's negative one over two
minus i times one half.

558
00:37:28 --> 00:37:34
So, this is multiplied by e to

559
00:37:33 --> 00:37:39
the minus x and cosine x.

560
00:37:36 --> 00:37:42
So, if you are doing it,
and practice a little bit,

561
00:37:40 --> 00:37:46
please don't put in all these
steps.

562
00:37:42 --> 00:37:48
Go from here;
well, I would go from here to

563
00:37:46 --> 00:37:52
here by myself.
Maybe you shouldn't.

564
00:37:48 --> 00:37:54
Practice a little before you do
that.

565
00:37:51 --> 00:37:57
And now, what do we do with
this?

566
00:37:53 --> 00:37:59
Now, this is in a form to pick
out the real part.

567
00:37:57 --> 00:38:03
We want the real part of this.
So, you don't have to write the

568
00:38:03 --> 00:38:09
whole thing out as a complex
number.

569
00:38:05 --> 00:38:11
In other words,
you don't have to do all the

570
00:38:08 --> 00:38:14
multiplications.
You only have to find the real

571
00:38:11 --> 00:38:17
part of it, which is what?
Well, e to the negative x

572
00:38:14 --> 00:38:20
will be simply a factor.

573
00:38:16 --> 00:38:22
That's a real factor,
which I don't have to worry

574
00:38:20 --> 00:38:26
about.
And, in that category,

575
00:38:21 --> 00:38:27
I can include the two also.
So, I only have to pick out the

576
00:38:25 --> 00:38:31
real part of this times that.
And, what's that?

577
00:38:30 --> 00:38:36
It's negative cosine x.

578
00:38:32 --> 00:38:38
And, the other real part comes
from the product of these two

579
00:38:37 --> 00:38:43
things.
I times negative i is one.

580
00:38:40 --> 00:38:46
And, what's left is sine x.

581
00:38:42 --> 00:38:48
So, that's the answer to the

582
00:38:45 --> 00:38:51
question.
That's the integral of e to the

583
00:38:48 --> 00:38:54
negative x * cosine x.

584
00:38:52 --> 00:38:58
Notice, it's a completely
straightforward process.

585
00:38:56 --> 00:39:02
It doesn't involve any tricks,
unless you call going to the

586
00:39:00 --> 00:39:06
complex domain a trick.
But, I don't.

587
00:39:04 --> 00:39:10
As soon as you see in this
course the combination of e to

588
00:39:08 --> 00:39:14
ax times cosine bx or sine bx,

589
00:39:11 --> 00:39:17
you should immediately think,

590
00:39:14 --> 00:39:20
and you're going to get plenty
of it in the couple of weeks

591
00:39:18 --> 00:39:24
after the exam,
you are going to get plenty of

592
00:39:21 --> 00:39:27
it, and you should immediately
think of passing to the complex

593
00:39:25 --> 00:39:31
domain.
That will be the standard way

594
00:39:27 --> 00:39:33
we solve such problems.
So, you're going to get lots of

595
00:39:32 --> 00:39:38
practice doing this.
But, this was the first time.

596
00:39:37 --> 00:39:43
Now, I guess in the time
remaining, I'm not going to talk

597
00:39:42 --> 00:39:48
about in the notes,
i, R, at all,

598
00:39:44 --> 00:39:50
but I would like to talk a
little bit about the extraction

599
00:39:49 --> 00:39:55
of the complex roots,
since you have a problem about

600
00:39:54 --> 00:40:00
that and because it's another
beautiful application of this

601
00:39:59 --> 00:40:05
polar way of writing complex
numbers.

602
00:40:04 --> 00:40:10
Suppose I want to calculate.
So, the basic problem is to

603
00:40:09 --> 00:40:15
calculate the nth roots of one.
Now, in the real domain,

604
00:40:15 --> 00:40:21
of course, the answer is,
sometimes there's only one of

605
00:40:21 --> 00:40:27
these, one itself,
and sometimes there are two,

606
00:40:26 --> 00:40:32
depending on whether n is an
even number or an odd number.

607
00:40:34 --> 00:40:40
But, in the complex domain,
there are always n answers as

608
00:40:40 --> 00:40:46
complex numbers.
One always has n nth roots.

609
00:40:45 --> 00:40:51
Now, where are they?
Well, geometrically,

610
00:40:50 --> 00:40:56
it's easy to see where they
are.

611
00:40:54 --> 00:41:00
Here's the unit circle.
Here's the unit circle.

612
00:41:01 --> 00:41:07
One of the roots is right here
at one.

613
00:41:04 --> 00:41:10
Now, where are the others?
Well, do you see that if I

614
00:41:09 --> 00:41:15
place, let's take n equal five
because that's a nice,

615
00:41:14 --> 00:41:20
dramatic number.
If I place these peptides

616
00:41:18 --> 00:41:24
equally spaced points around the
unit circle, so,

617
00:41:23 --> 00:41:29
in other words,
this angle is alpha.

618
00:41:26 --> 00:41:32
Alpha should be the angle.
What would be the expression

619
00:41:32 --> 00:41:38
for that?
If there were five such equally

620
00:41:37 --> 00:41:43
spaced, it would be one fifth of
all the way around the circle.

621
00:41:43 --> 00:41:49
All the way around the circle
is two pi.

622
00:41:47 --> 00:41:53
So, it will be one fifth of two
pi in radians.

623
00:41:52 --> 00:41:58
Now, it's geometrically clear
that those are the five fifth

624
00:41:58 --> 00:42:04
roots because,
how do I multiply complex

625
00:42:02 --> 00:42:08
numbers?
I multiply the moduli.

626
00:42:06 --> 00:42:12
Well, they all have moduli one.
So, if I take this guy,

627
00:42:11 --> 00:42:17
let's call that complex number,
oh, I hate to give you,

628
00:42:16 --> 00:42:22
they are always giving you
Greek notation.

629
00:42:20 --> 00:42:26
All right, why not torture you?
Zeta.

630
00:42:23 --> 00:42:29
At least you will learn how to
make a zeta in this period,

631
00:42:28 --> 00:42:34
small zeta, so that's zeta.
There's our fifth root of

632
00:42:34 --> 00:42:40
unity.
It's the first one that occurs

633
00:42:36 --> 00:42:42
on the circle that isn't the
trivial one, one.

634
00:42:40 --> 00:42:46
Now, do you see that,
how would I calculate zeta to

635
00:42:44 --> 00:42:50
the fifth?
Well, if I write zeta in polar

636
00:42:47 --> 00:42:53
notation, what would it be?
The modulus would be one,

637
00:42:51 --> 00:42:57
and therefore it will be
simply, the r will be one

638
00:42:56 --> 00:43:02
for it because its length is
one.

639
00:42:59 --> 00:43:05
Its modulus is one.
What's up here?

640
00:43:03 --> 00:43:09
I times that angle,
and that angle is two pi over

641
00:43:06 --> 00:43:12
five. So, there's just,

642
00:43:09 --> 00:43:15
geometrically I see where zeta
is.

643
00:43:11 --> 00:43:17
And, if I translate that
geometry into the e to the i

644
00:43:15 --> 00:43:21
theta form for
the formula, I see that it must

645
00:43:20 --> 00:43:26
be that number.
Now, let's say somebody gives

646
00:43:23 --> 00:43:29
you that number and says,
hey, is this the fifth root of

647
00:43:27 --> 00:43:33
one?
I forbid you to draw any

648
00:43:30 --> 00:43:36
pictures.
What would you do?

649
00:43:33 --> 00:43:39
You say, well,
I'll raise it to the fifth

650
00:43:36 --> 00:43:42
power.
What's zeta to the fifth power?

651
00:43:39 --> 00:43:45
Well, it's e to the i two pi /
five,

652
00:43:43 --> 00:43:49
and now, if I think of raising
that to the fifth power,

653
00:43:48 --> 00:43:54
by the exponential law,
that's the same thing as

654
00:43:51 --> 00:43:57
putting a five in front of the
exponent.

655
00:43:54 --> 00:44:00
So, this times five,
and what's that?

656
00:43:57 --> 00:44:03
That's e to the i times two pi.

657
00:44:01 --> 00:44:07
And, what is that?
Well, it's the angle.

658
00:44:06 --> 00:44:12
If the angle is two pi,
I've gone all the way around

659
00:44:13 --> 00:44:19
the circle and come back here
again.

660
00:44:17 --> 00:44:23
I've got the number one.
So, this is one.

661
00:44:22 --> 00:44:28
Since the argument,
two pi, is the same as an

662
00:44:28 --> 00:44:34
angle, it's the same as,
well, let's not write it that

663
00:44:35 --> 00:44:41
way. It's wrong.

664
00:44:39 --> 00:44:45
It's just wrong since two pi
and zero are the same angle.

665
00:44:52 --> 00:44:58
So, I could replace this by
zero.

666
00:45:01 --> 00:45:07
Oh dear.
Well, I guess I have to stop

667
00:45:09 --> 00:45:15
right in the middle of things.
So, you're going to have to

668
00:45:22 --> 00:45:28
read a little bit about how to
find roots in order to do that

669
00:45:36 --> 00:45:42
problem.
And, we will go on from that

670
00:45:44 --> 00:45:50
point Friday.