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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.