WEBVTT
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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
00:13:09.000 --> 00:13:15.000
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?
00:13:32.000 --> 00:13:38.000
Well, again,
we will use the method of
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grouping.
What's the real part of it?
00:13:36.000 --> 00:13:42.000
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.
00:14:01.000 --> 00:14:07.000
And then, what's left,
I won't have to keep repeating
00:14:05.000 --> 00:14:11.000
the i.
So, it will have to be sine
00:14:08.000 --> 00:14:14.000
theta one cosine theta two.
00:14:13.000 --> 00:14:19.000
And, the other factor will be
cosine theta one sine theta
00:14:18.000 --> 00:14:24.000
two-- plus sine theta two cosine
theta one.
00:14:22.000 --> 00:14:28.000
Well, it looks like a mess,
00:14:27.000 --> 00:14:33.000
but, again, high school to the
rescue.
00:14:30.000 --> 00:14:36.000
What is this?
The top thing is nothing in
00:14:35.000 --> 00:14:41.000
disguise, but it's a disguised
form of cosine (theta one plus
00:14:41.000 --> 00:14:47.000
theta two).
00:14:44.000 --> 00:14:50.000
And the bottom is sine of
(theta one plus theta two).
00:14:49.000 --> 00:14:55.000
So, the product of these two
00:14:54.000 --> 00:15:00.000
things is this,
and that's exactly the formula.
00:14:58.000 --> 00:15:04.000
In other words,
this formula is a way of
00:15:02.000 --> 00:15:08.000
writing those two trigonometric
identities for the cosine of the
00:15:07.000 --> 00:15:13.000
sum and the sine of the sum.
Instead of the two identities
00:15:13.000 --> 00:15:19.000
taking up that much space,
written one after the other,
00:15:17.000 --> 00:15:23.000
they take up as much space,
and they say exactly the same
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thing.
Those two trigonometric
00:15:22.000 --> 00:15:28.000
identities are exactly the same
as saying that e to the i theta
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satisfies the
exponential law.
00:15:30.000 --> 00:15:36.000
Now, people ask,
you know, what's beautiful in
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mathematics?
To me, that's beautiful.
00:15:36.000 --> 00:15:42.000
I think that's great.
Something long turns into
00:15:39.000 --> 00:15:45.000
something short,
and it's just as good,
00:15:42.000 --> 00:15:48.000
and moreover,
connects with all these other
00:15:45.000 --> 00:15:51.000
things in the world,
differential equations,
00:15:49.000 --> 00:15:55.000
infinite series,
blah, blah, blah,
00:15:51.000 --> 00:15:57.000
blah, blah.
Okay, I don't have to sell
00:15:54.000 --> 00:16:00.000
Euler.
He sells himself.
00:15:56.000 --> 00:16:02.000
Now, how about the other one?
How about the other one?
00:16:02.000 --> 00:16:08.000
Now, that's obviously,
I haven't said something
00:16:09.000 --> 00:16:15.000
because for one thing,
how do you differentiate if
00:16:17.000 --> 00:16:23.000
there's theta here,
and t down there.
00:16:22.000 --> 00:16:28.000
Okay, that's easily fixed.
But, how do I differentiate
00:16:30.000 --> 00:16:36.000
this?
What kind of a guy is e to the
00:16:33.000 --> 00:16:39.000
i theta?
Well, if I write it out,
00:16:37.000 --> 00:16:43.000
take a look at what it is.
It's cosine theta plus i sine
00:16:41.000 --> 00:16:47.000
theta.
00:16:44.000 --> 00:16:50.000
As theta varies,
it's a function.
00:16:47.000 --> 00:16:53.000
The variable is real.
Theta is a real variable.
00:16:51.000 --> 00:16:57.000
Its angle in radians,
but it runs from negative
00:16:55.000 --> 00:17:01.000
infinity to infinity.
So, if you think of functions
00:16:59.000 --> 00:17:05.000
as a black box,
what's going in is a real
00:17:02.000 --> 00:17:08.000
number.
But, what's coming out is a
00:17:06.000 --> 00:17:12.000
complex number.
So, schematically,
00:17:09.000 --> 00:17:15.000
here is the e to the i theta
box,
00:17:12.000 --> 00:17:18.000
if you like to think that way,
theta goes in,
00:17:15.000 --> 00:17:21.000
and that's real,
and a complex number,
00:17:18.000 --> 00:17:24.000
this particular complex number
goes out.
00:17:20.000 --> 00:17:26.000
So, one, we'd call it,
I'm not going to write this
00:17:24.000 --> 00:17:30.000
down because it's sort of
pompous and takes too long.
00:17:28.000 --> 00:17:34.000
But, it is a complex valued
function of a real variable.
00:17:33.000 --> 00:17:39.000
You got that?
Up to now, we studied real
00:17:36.000 --> 00:17:42.000
functions of real variables.
But now, real valued functions
00:17:41.000 --> 00:17:47.000
of real variables,
those are the kind calculus is
00:17:45.000 --> 00:17:51.000
concerned with.
But now, it's a complex-valued
00:17:48.000 --> 00:17:54.000
function because the variable is
real.
00:17:51.000 --> 00:17:57.000
But, the output,
the value of the function is a
00:17:55.000 --> 00:18:01.000
complex number.
Now, in general,
00:17:58.000 --> 00:18:04.000
such a function,
well, maybe a better say,
00:18:01.000 --> 00:18:07.000
complex-valued,
how about complex-valued
00:18:04.000 --> 00:18:10.000
function of a real variable,
let's change the name of the
00:18:09.000 --> 00:18:15.000
variable.
t is always a real variable.
00:18:14.000 --> 00:18:20.000
I don't think we have complex
time yet, although I'm sure
00:18:19.000 --> 00:18:25.000
there will be someday.
But, the next Einstein appears.
00:18:24.000 --> 00:18:30.000
A complex-valued function of a
real variable,
00:18:28.000 --> 00:18:34.000
t, in general,
would look like this.
00:18:32.000 --> 00:18:38.000
t goes in, and what comes out?
Well: a complex number,
00:18:35.000 --> 00:18:41.000
which I would then have to
write this way.
00:18:38.000 --> 00:18:44.000
In other words,
the real part depends on t,
00:18:41.000 --> 00:18:47.000
and the imaginary part depends
upon t.
00:18:44.000 --> 00:18:50.000
So, a general function looks
like this, a general
00:18:47.000 --> 00:18:53.000
complex-valued function.
This is just a special case of
00:18:51.000 --> 00:18:57.000
it, where the variable has a
different name.
00:18:54.000 --> 00:19:00.000
But, the first function would
be cosine t, and the second
00:18:57.000 --> 00:19:03.000
function would be sine t.
So, my only question is,
00:19:01.000 --> 00:19:07.000
how do you differentiate such a
thing?
00:19:03.000 --> 00:19:09.000
Well, I'm not going to fuss
over this.
00:19:08.000 --> 00:19:14.000
The general definition is,
with deltas and whatnot,
00:19:11.000 --> 00:19:17.000
but the end result of a
perfectly fine definition is,
00:19:14.000 --> 00:19:20.000
you differentiate it by
differentiating each component.
00:19:18.000 --> 00:19:24.000
The reason you don't have to
work so very hard is because
00:19:22.000 --> 00:19:28.000
this is a real variable,
and I already know what it
00:19:25.000 --> 00:19:31.000
means to differentiate a
function of a real variable.
00:19:30.000 --> 00:19:36.000
So, I could write it this way,
that the derivative of u plus
00:19:34.000 --> 00:19:40.000
iv, I'll abbreviate it that way,
this means the derivative,
00:19:37.000 --> 00:19:43.000
with respect to whatever
variable, since I didn't tell
00:19:41.000 --> 00:19:47.000
you what the variable in these
functions were,
00:19:44.000 --> 00:19:50.000
well, I don't have to tell you
what I'm differentiating with
00:19:48.000 --> 00:19:54.000
respect to.
It's whatever was there because
00:19:51.000 --> 00:19:57.000
you can't see.
And the answer is,
00:19:53.000 --> 00:19:59.000
it would be the derivative of u
plus i times the derivative of
00:19:57.000 --> 00:20:03.000
v.
You differentiate it just the
00:20:01.000 --> 00:20:07.000
way you would if these were the
components of a motion vector.
00:20:05.000 --> 00:20:11.000
You would get the velocity by
differentiating each component
00:20:09.000 --> 00:20:15.000
separately.
And, that's what you're doing
00:20:12.000 --> 00:20:18.000
here.
Okay, now, the importance of
00:20:15.000 --> 00:20:21.000
that is that it at least tells
me what it is I have to check
00:20:19.000 --> 00:20:25.000
when I check this formula.
So, let's do it now that we
00:20:23.000 --> 00:20:29.000
know what this is.
We know how to differentiate
00:20:26.000 --> 00:20:32.000
the function.
Let's actually differentiate
00:20:29.000 --> 00:20:35.000
it.
That's fortunately,
00:20:32.000 --> 00:20:38.000
by far, the easiest part of the
whole process.
00:20:37.000 --> 00:20:43.000
So, let's do it.
So, what's the derivative?
00:20:41.000 --> 00:20:47.000
Let's go back to t,
our generic variable.
00:20:45.000 --> 00:20:51.000
I want to emphasize that these
functions, when we write them as
00:20:51.000 --> 00:20:57.000
functions, that theta will
almost never be the variable
00:20:56.000 --> 00:21:02.000
outside of these notes on
complex numbers.
00:21:02.000 --> 00:21:08.000
It will normally be time or
something like that,
00:21:05.000 --> 00:21:11.000
or x, a neutral variable like
x.
00:21:07.000 --> 00:21:13.000
So, what's the derivative of e
to the i theta?
00:21:12.000 --> 00:21:18.000
I'm hoping that it will turn
out to be i e to the i theta,
00:21:16.000 --> 00:21:22.000
and that the yellow law may be
00:21:19.000 --> 00:21:25.000
true just as the green one was.
Okay, let's calculate it.
00:21:23.000 --> 00:21:29.000
It's the derivative,
with respect to,
00:21:26.000 --> 00:21:32.000
unfortunately I can convert t's
to thetas, but not thetas to
00:21:30.000 --> 00:21:36.000
t's.
C'est la vie,
00:21:33.000 --> 00:21:39.000
okay.
Times cosine t plus i sine t,
00:21:36.000 --> 00:21:42.000
and what's that?
00:21:40.000 --> 00:21:46.000
Well, the derivative of cosine
t, differentiating the real and
00:21:46.000 --> 00:21:52.000
imaginary parts separately,
and adding them up.
00:21:51.000 --> 00:21:57.000
It's negative sine t,
plus i times cosine t.
00:21:56.000 --> 00:22:02.000
Now, let's factor out at the i,
00:22:01.000 --> 00:22:07.000
because it says if I factor out
the i, what do I get?
00:22:08.000 --> 00:22:14.000
Well, now, the real part of
what's left would be cosine t.
00:22:12.000 --> 00:22:18.000
And, how about the imaginary
part?
00:22:15.000 --> 00:22:21.000
Do you see, it will be i sine t
because i times i
00:22:21.000 --> 00:22:27.000
gives me that negative one.
00:22:24.000 --> 00:22:30.000
And, what's that?
e to the it.
00:22:27.000 --> 00:22:33.000
i times e to the i t.
00:22:30.000 --> 00:22:36.000
So, that works too.
What about the initial
00:22:35.000 --> 00:22:41.000
condition?
No problem.
00:22:38.000 --> 00:22:44.000
What is y of zero?
What's the function at zero?
00:22:43.000 --> 00:22:49.000
Well, don't say right away,
i times zero is zero,
00:22:49.000 --> 00:22:55.000
so it must be one.
That's illegal because,
00:22:54.000 --> 00:23:00.000
why is that illegal?
It's because in that formula,
00:23:00.000 --> 00:23:06.000
you are not multiplying i times
theta.
00:23:06.000 --> 00:23:12.000
I mean, sort of,
you are, but that formula is
00:23:09.000 --> 00:23:15.000
the meaning of e to
the i theta.
00:23:12.000 --> 00:23:18.000
Now, it would be very nice if
this is like,
00:23:16.000 --> 00:23:22.000
well, anyway,
you can't do that.
00:23:18.000 --> 00:23:24.000
So, you have to do it by saying
it's the cosine of zero plus i
00:23:23.000 --> 00:23:29.000
times the sine of zero.
00:23:26.000 --> 00:23:32.000
And, how much is that?
The sine of zero is zero.
00:23:31.000 --> 00:23:37.000
Now, it's okay to say i times
zero is zero because that's the
00:23:35.000 --> 00:23:41.000
way complex numbers multiply.
What is the cosine of zero?
00:23:40.000 --> 00:23:46.000
That's one.
So, the answer,
00:23:42.000 --> 00:23:48.000
indeed, turns out to be one.
So, this checks,
00:23:46.000 --> 00:23:52.000
really, from every conceivable
standpoint down as I indicated,
00:23:51.000 --> 00:23:57.000
also from the standpoint of
infinite series.
00:23:54.000 --> 00:24:00.000
So, we are definitely allowed
to use this.
00:23:58.000 --> 00:24:04.000
Now, the more general
exponential law is true.
00:24:03.000 --> 00:24:09.000
I'm not going to say much about
it.
00:24:05.000 --> 00:24:11.000
So, in other words,
e to the a, this is really a
00:24:09.000 --> 00:24:15.000
definition.
e to the (a plus ib)
00:24:13.000 --> 00:24:19.000
is going to be,
in order for the general
00:24:17.000 --> 00:24:23.000
exponential law to be true,
this is really a definition.
00:24:21.000 --> 00:24:27.000
It's e to the a times e to the
ib.
00:24:26.000 --> 00:24:32.000
Now, notice when I look at
the-- at any complex number,
00:24:30.000 --> 00:24:36.000
--
-- so, in terms of this,
00:24:34.000 --> 00:24:40.000
the polar form of a complex
number, to draw the little
00:24:38.000 --> 00:24:44.000
picture again,
if here is our complex number,
00:24:42.000 --> 00:24:48.000
and here is r,
and here is the angle theta,
00:24:46.000 --> 00:24:52.000
so the nice way to write this
complex number is r e to the i
00:24:51.000 --> 00:24:57.000
theta.
The e to the i theta
00:24:56.000 --> 00:25:02.000
is, now, why is that?
00:25:00.000 --> 00:25:06.000
What is the magnitude of this?
This is r.
00:25:04.000 --> 00:25:10.000
The length of the absolute
value, I didn't talk about
00:25:10.000 --> 00:25:16.000
magnitude in argument.
I guess I should have.
00:25:14.000 --> 00:25:20.000
But, it's in the notes.
So, r is called the modulus.
00:25:20.000 --> 00:25:26.000
Well, the fancy word is the
modulus.
00:25:24.000 --> 00:25:30.000
And, we haven't given the
complex number a name.
00:25:29.000 --> 00:25:35.000
Let's call it alpha,
modulus of alpha,
00:25:33.000 --> 00:25:39.000
and theta is called,
it's the angle.
00:25:39.000 --> 00:25:45.000
It's called the argument.
I didn't make up these words.
00:25:44.000 --> 00:25:50.000
There, from a tradition of
English that has long since
00:25:49.000 --> 00:25:55.000
vanished, when I was a kid,
and you wanted to know what a
00:25:55.000 --> 00:26:01.000
play was about,
you looked in the playbill,
00:25:59.000 --> 00:26:05.000
and it said the argument of the
play, it's that old-fashioned
00:26:05.000 --> 00:26:11.000
use of the word argument.
Argument means the angle,
00:26:11.000 --> 00:26:17.000
and sometimes that's
abbreviated by arg alpha.
00:26:16.000 --> 00:26:22.000
And, this is abbreviated,
00:26:21.000 --> 00:26:27.000
of course, as absolute value of
alpha, its length.
00:26:26.000 --> 00:26:32.000
Okay, the notes give you a
little practice changing things
00:26:33.000 --> 00:26:39.000
to a polar form.
I think we will skip that in
00:26:39.000 --> 00:26:45.000
favor of doing a couple of other
things because that's pretty
00:26:46.000 --> 00:26:52.000
easy.
But let me, you should at least
00:26:50.000 --> 00:26:56.000
realize when you should look at
polar form.
00:26:55.000 --> 00:27:01.000
The great advantage of polar
form is, particularly once
00:27:01.000 --> 00:27:07.000
you've mastered the exponential
law, the great advantage of
00:27:08.000 --> 00:27:14.000
polar form is it's good for
multiplication.
00:27:15.000 --> 00:27:21.000
Now, of course,
you know how to multiply
00:27:17.000 --> 00:27:23.000
complex numbers,
even when they are in the
00:27:20.000 --> 00:27:26.000
Cartesian form.
That's the first thing you
00:27:23.000 --> 00:27:29.000
learn in high school,
how to multiply a plus bi times
00:27:27.000 --> 00:27:33.000
c plus di.
But, as you will see,
00:27:31.000 --> 00:27:37.000
when push comes to shove,
you will see this very clearly
00:27:35.000 --> 00:27:41.000
on Friday when we talk about
trigonometric inputs to
00:27:39.000 --> 00:27:45.000
differential equations,
--
00:27:42.000 --> 00:27:48.000
-- that the changing to complex
numbers makes all sorts of
00:27:46.000 --> 00:27:52.000
things easy to calculate,
and the answers come out
00:27:49.000 --> 00:27:55.000
extremely clear,
whereas if we had to do it any
00:27:52.000 --> 00:27:58.000
other way, it's a lot more work.
And worst of all,
00:27:56.000 --> 00:28:02.000
when you finally slog through
to the end, you fear you are
00:28:00.000 --> 00:28:06.000
none the wiser.
It's good for multiplication
00:28:03.000 --> 00:28:09.000
because the product,
so here's any number in its
00:28:07.000 --> 00:28:13.000
polar form.
That's a general complex
00:28:09.000 --> 00:28:15.000
number.
It's modulus times e to the i
00:28:12.000 --> 00:28:18.000
theta times r two e to the i
theta two--
00:28:16.000 --> 00:28:22.000
Well,
you just multiply them as
00:28:19.000 --> 00:28:25.000
ordinary numbers.
So, the part out front will be
00:28:22.000 --> 00:28:28.000
r1 r2, and the e to the i theta
parts gets
00:28:26.000 --> 00:28:32.000
multiplied by the exponential
law and becomes e to the i
00:28:30.000 --> 00:28:36.000
(theta one plus theta two) --
00:28:36.000 --> 00:28:42.000
-- which makes very clear that
the multiply geometrically two
00:28:42.000 --> 00:28:48.000
complex numbers,
you multiply the moduli,
00:28:46.000 --> 00:28:52.000
the r's, the absolute values,
how long the arrow is from zero
00:28:52.000 --> 00:28:58.000
to the complex number,
multiply the moduli,
00:28:56.000 --> 00:29:02.000
and add the arguments.
So the new number,
00:29:02.000 --> 00:29:08.000
its modulus is the product of
r1 and r2.
00:29:07.000 --> 00:29:13.000
And, its argument,
its angle, polar angle,
00:29:12.000 --> 00:29:18.000
is the sum of the old two
angles.
00:29:15.000 --> 00:29:21.000
And, you add the angles.
And, you put down in your books
00:29:22.000 --> 00:29:28.000
angles, but I'm being
photographed,
00:29:26.000 --> 00:29:32.000
so I'm going to write
arguments.
00:29:31.000 --> 00:29:37.000
In other words,
it makes the geometric content
00:29:34.000 --> 00:29:40.000
of multiplication clear,
in a sense in which this is
00:29:38.000 --> 00:29:44.000
extremely unclear.
From this law,
00:29:40.000 --> 00:29:46.000
blah, blah, blah,
blah, blah, whatever it turns
00:29:44.000 --> 00:29:50.000
out to be, you have not the
slightest intuition that this is
00:29:48.000 --> 00:29:54.000
true about the complex numbers.
That first thing is just a
00:29:52.000 --> 00:29:58.000
formula, whereas this thing is
insightful representation of
00:29:57.000 --> 00:30:03.000
complex multiplication.
Now, I'd like to use it for
00:30:02.000 --> 00:30:08.000
something, but before we do
that, let me just indicate how
00:30:08.000 --> 00:30:14.000
just the exponential notation
enables you to do things in
00:30:14.000 --> 00:30:20.000
calculus, formulas that are
impossible to remember from
00:30:19.000 --> 00:30:25.000
calculus.
It makes them very easy to
00:30:23.000 --> 00:30:29.000
derive.
A typical example of that is,
00:30:27.000 --> 00:30:33.000
suppose you want to,
for example,
00:30:30.000 --> 00:30:36.000
integrate (e to the negative x)
cosine x.
00:30:38.000 --> 00:30:44.000
Well, number one,
you spend a few minutes running
00:30:41.000 --> 00:30:47.000
through a calculus textbook and
try to find out the answer
00:30:45.000 --> 00:30:51.000
because you know you are not
going to remember how to do it.
00:30:49.000 --> 00:30:55.000
Or, you run to a computer,
and type in Matlab and
00:30:53.000 --> 00:30:59.000
something.
Or, you fish out your little
00:30:55.000 --> 00:31:01.000
pocket calculator,
which will give you a formula,
00:30:59.000 --> 00:31:05.000
and so on.
So, you have aides for doing
00:31:03.000 --> 00:31:09.000
that.
But, the way to do it if you're
00:31:06.000 --> 00:31:12.000
on a desert island,
and the way I always do it
00:31:10.000 --> 00:31:16.000
because I never have any of
these little aides around,
00:31:14.000 --> 00:31:20.000
and I cannot trust my memory,
probably a certain number of
00:31:19.000 --> 00:31:25.000
you remember how you did it at
high school, or how you did it
00:31:24.000 --> 00:31:30.000
in 18.01, if you took it here.
You have to use integration by
00:31:29.000 --> 00:31:35.000
parts.
But, it's one of the tricky
00:31:33.000 --> 00:31:39.000
things that's not required on an
exam because you had to use
00:31:37.000 --> 00:31:43.000
integration by parts twice in
the same direction,
00:31:40.000 --> 00:31:46.000
and then suddenly by comparing
the end product with the initial
00:31:45.000 --> 00:31:51.000
product and writing an equation.
Somehow, the value falls out.
00:31:50.000 --> 00:31:56.000
Well, that's tricky.
And it's not the sort of thing
00:31:53.000 --> 00:31:59.000
you can waste time stuffing into
your head, unless you are going
00:31:58.000 --> 00:32:04.000
to be the integration bee during
IAP or something like that.
00:32:04.000 --> 00:32:10.000
Instead, using complex numbers
is the way to do this.
00:32:09.000 --> 00:32:15.000
How do I think of this,
cosine x?
00:32:12.000 --> 00:32:18.000
What I do, is I think of that e
to the negative x cosine x
00:32:18.000 --> 00:32:24.000
is the real
part, the real part of what?
00:32:24.000 --> 00:32:30.000
Well, cosine x is the real part
of e to the ix.
00:32:29.000 --> 00:32:35.000
So, this thing,
this is real.
00:32:32.000 --> 00:32:38.000
This is real,
too.
00:32:34.000 --> 00:32:40.000
But I'm thinking of it as the
real part of e to the ix.
00:32:39.000 --> 00:32:45.000
Now, if I multiply these two
00:32:45.000 --> 00:32:51.000
together, this is going to turn
out to be, therefore,
00:32:49.000 --> 00:32:55.000
the real part of e to the minus
x.
00:32:53.000 --> 00:32:59.000
I'll write it out very
pompously, and then I will fix
00:32:57.000 --> 00:33:03.000
it.
I would never write this,
00:33:00.000 --> 00:33:06.000
you are you.
Okay, it's e to the minus x
00:33:04.000 --> 00:33:10.000
times, when I write cosine x
plus i sine x,
00:33:09.000 --> 00:33:15.000
so it is the real part of that
is cosine x.
00:33:14.000 --> 00:33:20.000
So, it's the real part of,
write it this way for real part
00:33:20.000 --> 00:33:26.000
of e to the, factor out the x,
and what's up there is
00:33:26.000 --> 00:33:32.000
(negative one plus i) times x.
00:33:33.000 --> 00:33:39.000
Okay, and now,
so, the idea is the same thing
00:33:36.000 --> 00:33:42.000
is going to be true for the
integral.
00:33:39.000 --> 00:33:45.000
This is going to be the real
part of that,
00:33:43.000 --> 00:33:49.000
the integral of e to the (minus
one plus i) times x dx.
00:33:48.000 --> 00:33:54.000
In other words,
00:33:51.000 --> 00:33:57.000
what you do is,
this procedure is called
00:33:54.000 --> 00:34:00.000
complexifying the integral.
Instead of looking at the
00:33:58.000 --> 00:34:04.000
original real problem,
I'm going to turn it into a
00:34:03.000 --> 00:34:09.000
complex problem by turning this
thing into a complex
00:34:07.000 --> 00:34:13.000
exponential.
This is the real part of that
00:34:12.000 --> 00:34:18.000
complex exponential.
Now, what's the advantage of
00:34:15.000 --> 00:34:21.000
doing that?
Simple.
00:34:16.000 --> 00:34:22.000
It's because nothing is easier
to integrate than an
00:34:20.000 --> 00:34:26.000
exponential.
And, though you may have some
00:34:23.000 --> 00:34:29.000
doubts as to whether the
familiar laws work also with
00:34:26.000 --> 00:34:32.000
complex exponentials,
I assure you they all do.
00:34:30.000 --> 00:34:36.000
It would be lovely to sit and
prove them.
00:34:34.000 --> 00:34:40.000
On the other hand,
I think after a while,
00:34:37.000 --> 00:34:43.000
you find it rather dull.
So, I'm going to do the fun
00:34:41.000 --> 00:34:47.000
things, and assume that they are
true because they are.
00:34:46.000 --> 00:34:52.000
So, what's the integral of e to
the (minus one plus i) x dx?
00:34:54.000 --> 00:35:00.000
Well, if there is justice in
heaven, it must be e to the
00:34:58.000 --> 00:35:04.000
(minus one plus i) times x
divided by minus one plus i.
00:35:03.000 --> 00:35:09.000
In some sense,
00:35:08.000 --> 00:35:14.000
that's the answer.
This does, in fact,
00:35:12.000 --> 00:35:18.000
give that.
That's correct.
00:35:15.000 --> 00:35:21.000
I want the real part of this.
I want the real part because
00:35:22.000 --> 00:35:28.000
that's the way the original
problem was stated.
00:35:27.000 --> 00:35:33.000
I want the real part only.
So, I want the real part of
00:35:34.000 --> 00:35:40.000
this.
Now, this is what separates the
00:35:38.000 --> 00:35:44.000
girls from the women.
[LAUGHTER] This is why you have
00:35:44.000 --> 00:35:50.000
to know how to divide complex
numbers.
00:35:48.000 --> 00:35:54.000
So, watch how I find the real
part.
00:35:52.000 --> 00:35:58.000
I write it this way.
Normally when I do the
00:35:56.000 --> 00:36:02.000
calculations for myself,
I would skip a couple of these
00:36:02.000 --> 00:36:08.000
steps.
But this time,
00:36:05.000 --> 00:36:11.000
I will write everything out.
You're going to have to do this
00:36:09.000 --> 00:36:15.000
a lot in this course,
by the way, both over the
00:36:12.000 --> 00:36:18.000
course of the next few weeks,
and especially towards the end
00:36:16.000 --> 00:36:22.000
of the term where we get into a
complex systems,
00:36:19.000 --> 00:36:25.000
which involve complex numbers.
There's a lot of this.
00:36:22.000 --> 00:36:28.000
So, now is the time to learn to
do it, and to feel skillful at
00:36:26.000 --> 00:36:32.000
it.
So, it's this times e to the
00:36:28.000 --> 00:36:34.000
negative x times e to the ix,
00:36:31.000 --> 00:36:37.000
which is cosine x plus
i sine x.
00:36:36.000 --> 00:36:42.000
Now, I'm not ready,
yet, to do the calculation to
00:36:39.000 --> 00:36:45.000
find the real part because I
don't like the way this looks.
00:36:43.000 --> 00:36:49.000
I want to go back to the thing
I did right at the very
00:36:46.000 --> 00:36:52.000
beginning of the hour,
and turn it into an a plus bi
00:36:50.000 --> 00:36:56.000
type of complex
number.
00:36:52.000 --> 00:36:58.000
In other words,
what we have to do is the
00:36:55.000 --> 00:37:01.000
division.
So, the division is going to
00:36:57.000 --> 00:37:03.000
be, now, I'm going to ask you to
do it in your head.
00:37:02.000 --> 00:37:08.000
I multiply the top and bottom
by negative one minus I.
00:37:06.000 --> 00:37:12.000
What does that put in the
denominator?
00:37:09.000 --> 00:37:15.000
One squared plus one squared:
Two.
00:37:13.000 --> 00:37:19.000
And in the numerator,
negative one minus i.
00:37:17.000 --> 00:37:23.000
This is the same as that.
00:37:20.000 --> 00:37:26.000
But now, it looks at the form a
+ bi.
00:37:24.000 --> 00:37:30.000
It's negative one over two
minus i times one half.
00:37:28.000 --> 00:37:34.000
So, this is multiplied by e to
00:37:33.000 --> 00:37:39.000
the minus x and cosine x.
00:37:36.000 --> 00:37:42.000
So, if you are doing it,
and practice a little bit,
00:37:40.000 --> 00:37:46.000
please don't put in all these
steps.
00:37:42.000 --> 00:37:48.000
Go from here;
well, I would go from here to
00:37:46.000 --> 00:37:52.000
here by myself.
Maybe you shouldn't.
00:37:48.000 --> 00:37:54.000
Practice a little before you do
that.
00:37:51.000 --> 00:37:57.000
And now, what do we do with
this?
00:37:53.000 --> 00:37:59.000
Now, this is in a form to pick
out the real part.
00:37:57.000 --> 00:38:03.000
We want the real part of this.
So, you don't have to write the
00:38:03.000 --> 00:38:09.000
whole thing out as a complex
number.
00:38:05.000 --> 00:38:11.000
In other words,
you don't have to do all the
00:38:08.000 --> 00:38:14.000
multiplications.
You only have to find the real
00:38:11.000 --> 00:38:17.000
part of it, which is what?
Well, e to the negative x
00:38:14.000 --> 00:38:20.000
will be simply a factor.
00:38:16.000 --> 00:38:22.000
That's a real factor,
which I don't have to worry
00:38:20.000 --> 00:38:26.000
about.
And, in that category,
00:38:21.000 --> 00:38:27.000
I can include the two also.
So, I only have to pick out the
00:38:25.000 --> 00:38:31.000
real part of this times that.
And, what's that?
00:38:30.000 --> 00:38:36.000
It's negative cosine x.
00:38:32.000 --> 00:38:38.000
And, the other real part comes
from the product of these two
00:38:37.000 --> 00:38:43.000
things.
I times negative i is one.
00:38:40.000 --> 00:38:46.000
And, what's left is sine x.
00:38:42.000 --> 00:38:48.000
So, that's the answer to the
00:38:45.000 --> 00:38:51.000
question.
That's the integral of e to the
00:38:48.000 --> 00:38:54.000
negative x * cosine x.
00:38:52.000 --> 00:38:58.000
Notice, it's a completely
straightforward process.
00:38:56.000 --> 00:39:02.000
It doesn't involve any tricks,
unless you call going to the
00:39:00.000 --> 00:39:06.000
complex domain a trick.
But, I don't.
00:39:04.000 --> 00:39:10.000
As soon as you see in this
course the combination of e to
00:39:08.000 --> 00:39:14.000
ax times cosine bx or sine bx,
00:39:11.000 --> 00:39:17.000
you should immediately think,
00:39:14.000 --> 00:39:20.000
and you're going to get plenty
of it in the couple of weeks
00:39:18.000 --> 00:39:24.000
after the exam,
you are going to get plenty of
00:39:21.000 --> 00:39:27.000
it, and you should immediately
think of passing to the complex
00:39:25.000 --> 00:39:31.000
domain.
That will be the standard way
00:39:27.000 --> 00:39:33.000
we solve such problems.
So, you're going to get lots of
00:39:32.000 --> 00:39:38.000
practice doing this.
But, this was the first time.
00:39:37.000 --> 00:39:43.000
Now, I guess in the time
remaining, I'm not going to talk
00:39:42.000 --> 00:39:48.000
about in the notes,
i, R, at all,
00:39:44.000 --> 00:39:50.000
but I would like to talk a
little bit about the extraction
00:39:49.000 --> 00:39:55.000
of the complex roots,
since you have a problem about
00:39:54.000 --> 00:40:00.000
that and because it's another
beautiful application of this
00:39:59.000 --> 00:40:05.000
polar way of writing complex
numbers.
00:40:04.000 --> 00:40:10.000
Suppose I want to calculate.
So, the basic problem is to
00:40:09.000 --> 00:40:15.000
calculate the nth roots of one.
Now, in the real domain,
00:40:15.000 --> 00:40:21.000
of course, the answer is,
sometimes there's only one of
00:40:21.000 --> 00:40:27.000
these, one itself,
and sometimes there are two,
00:40:26.000 --> 00:40:32.000
depending on whether n is an
even number or an odd number.
00:40:34.000 --> 00:40:40.000
But, in the complex domain,
there are always n answers as
00:40:40.000 --> 00:40:46.000
complex numbers.
One always has n nth roots.
00:40:45.000 --> 00:40:51.000
Now, where are they?
Well, geometrically,
00:40:50.000 --> 00:40:56.000
it's easy to see where they
are.
00:40:54.000 --> 00:41:00.000
Here's the unit circle.
Here's the unit circle.
00:41:01.000 --> 00:41:07.000
One of the roots is right here
at one.
00:41:04.000 --> 00:41:10.000
Now, where are the others?
Well, do you see that if I
00:41:09.000 --> 00:41:15.000
place, let's take n equal five
because that's a nice,
00:41:14.000 --> 00:41:20.000
dramatic number.
If I place these peptides
00:41:18.000 --> 00:41:24.000
equally spaced points around the
unit circle, so,
00:41:23.000 --> 00:41:29.000
in other words,
this angle is alpha.
00:41:26.000 --> 00:41:32.000
Alpha should be the angle.
What would be the expression
00:41:32.000 --> 00:41:38.000
for that?
If there were five such equally
00:41:37.000 --> 00:41:43.000
spaced, it would be one fifth of
all the way around the circle.
00:41:43.000 --> 00:41:49.000
All the way around the circle
is two pi.
00:41:47.000 --> 00:41:53.000
So, it will be one fifth of two
pi in radians.
00:41:52.000 --> 00:41:58.000
Now, it's geometrically clear
that those are the five fifth
00:41:58.000 --> 00:42:04.000
roots because,
how do I multiply complex
00:42:02.000 --> 00:42:08.000
numbers?
I multiply the moduli.
00:42:06.000 --> 00:42:12.000
Well, they all have moduli one.
So, if I take this guy,
00:42:11.000 --> 00:42:17.000
let's call that complex number,
oh, I hate to give you,
00:42:16.000 --> 00:42:22.000
they are always giving you
Greek notation.
00:42:20.000 --> 00:42:26.000
All right, why not torture you?
Zeta.
00:42:23.000 --> 00:42:29.000
At least you will learn how to
make a zeta in this period,
00:42:28.000 --> 00:42:34.000
small zeta, so that's zeta.
There's our fifth root of
00:42:34.000 --> 00:42:40.000
unity.
It's the first one that occurs
00:42:36.000 --> 00:42:42.000
on the circle that isn't the
trivial one, one.
00:42:40.000 --> 00:42:46.000
Now, do you see that,
how would I calculate zeta to
00:42:44.000 --> 00:42:50.000
the fifth?
Well, if I write zeta in polar
00:42:47.000 --> 00:42:53.000
notation, what would it be?
The modulus would be one,
00:42:51.000 --> 00:42:57.000
and therefore it will be
simply, the r will be one
00:42:56.000 --> 00:43:02.000
for it because its length is
one.
00:42:59.000 --> 00:43:05.000
Its modulus is one.
What's up here?
00:43:03.000 --> 00:43:09.000
I times that angle,
and that angle is two pi over
00:43:06.000 --> 00:43:12.000
five. So, there's just,
00:43:09.000 --> 00:43:15.000
geometrically I see where zeta
is.
00:43:11.000 --> 00:43:17.000
And, if I translate that
geometry into the e to the i
00:43:15.000 --> 00:43:21.000
theta form for
the formula, I see that it must
00:43:20.000 --> 00:43:26.000
be that number.
Now, let's say somebody gives
00:43:23.000 --> 00:43:29.000
you that number and says,
hey, is this the fifth root of
00:43:27.000 --> 00:43:33.000
one?
I forbid you to draw any
00:43:30.000 --> 00:43:36.000
pictures.
What would you do?
00:43:33.000 --> 00:43:39.000
You say, well,
I'll raise it to the fifth
00:43:36.000 --> 00:43:42.000
power.
What's zeta to the fifth power?
00:43:39.000 --> 00:43:45.000
Well, it's e to the i two pi /
five,
00:43:43.000 --> 00:43:49.000
and now, if I think of raising
that to the fifth power,
00:43:48.000 --> 00:43:54.000
by the exponential law,
that's the same thing as
00:43:51.000 --> 00:43:57.000
putting a five in front of the
exponent.
00:43:54.000 --> 00:44:00.000
So, this times five,
and what's that?
00:43:57.000 --> 00:44:03.000
That's e to the i times two pi.
00:44:01.000 --> 00:44:07.000
And, what is that?
Well, it's the angle.
00:44:06.000 --> 00:44:12.000
If the angle is two pi,
I've gone all the way around
00:44:13.000 --> 00:44:19.000
the circle and come back here
again.
00:44:17.000 --> 00:44:23.000
I've got the number one.
So, this is one.
00:44:22.000 --> 00:44:28.000
Since the argument,
two pi, is the same as an
00:44:28.000 --> 00:44:34.000
angle, it's the same as,
well, let's not write it that
00:44:35.000 --> 00:44:41.000
way. It's wrong.
00:44:39.000 --> 00:44:45.000
It's just wrong since two pi
and zero are the same angle.
00:44:52.000 --> 00:44:58.000
So, I could replace this by
zero.
00:45:01.000 --> 00:45:07.000
Oh dear.
Well, I guess I have to stop
00:45:09.000 --> 00:45:15.000
right in the middle of things.
So, you're going to have to
00:45:22.000 --> 00:45:28.000
read a little bit about how to
find roots in order to do that
00:45:36.000 --> 00:45:42.000
problem.
And, we will go on from that
00:45:44.000 --> 00:45:50.000
point Friday.