lec4b_512kb.mp4.srt

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[MUSIC-- "JESU, JOY OF
MAN'S DESIRING" BY

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JOHANN SEBASTIAN BACH]

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PROFESSOR: So far in this course
we've been talking a

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lot about data abstraction.

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And remember the idea is that
we build systems that have

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these horizontal barriers in
them, these abstraction

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barriers that separate use,
the way you might use some

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data object, from the way
you might represent it.

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Or another way to think of that
is up here you have the

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boss who's going to be using
some sort of data object.

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And down here is George
who's implemented it.

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Now this notion of separating
use from representation so you

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can think about these two
problems separately is a very,

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very powerful programming
methodology, data abstraction.

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On the other hand, it's not
really sufficient for really

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complex systems. And the problem
with this is George.

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Or actually, the problem
is that there

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are a lot of Georges.

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Let's be concrete.

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Let's suppose there is George,
and there's also Martha.

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OK, now George and Martha are
both working on this system,

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both designing representations,
and

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absolutely are incompatible.

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They wouldn't cooperate on a
representation under any

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circumstances.

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And the problem is you would
like to have some system where

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both George and Martha are
designing representations, and

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yet, if you're above this
abstraction barrier you don't

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want to have to worry about
that, whether something is

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done by George or by Martha.

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And you don't want George
and Martha to

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interfere with each other.

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Somehow in designing a system,
you not only want these

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horizontal barriers, but you
also want some kind of

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vertical barrier to keep George
and Martha separate.

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Let me be a little bit
more concrete.

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Imagine that you're thinking
about personnel records for a

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large company with a lot of
loosely linked divisions that

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don't cooperate very
well either.

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And imagine even that this
company is formed by merging a

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whole bunch of companies that
already have their personnel

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record system set up.

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And imagine that once these
divisions are all linked in

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some kind of very sophisticated
satellite

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network, and all these databases
are put together.

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And what you'd like to do is,
from any place in the company,

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to be able to say things like,
oh, what's the name in a

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personnel record?

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Or, what's the job description
in a personnel record?

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And not have to worry about the
fact that each division

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obviously is going to have
completely separate

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conventions for how you might
implement these records.

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From this point you don't
want to know about that.

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Well how could you
possibly do that?

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One way, of course, is to send
down an edict from somewhere

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that everybody has to change
their format to some fixed

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compatible thing.

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That's what people often try,
and of course it never works.

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Another thing that you might
want to do is somehow arrange

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it so you can have these
vertical barriers.

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So that when you ask for the
name of a personnel record,

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somehow, whatever format it
happens to be, name will

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figure out how to do
the right thing.

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We want name to be, so-called,
a generic operator.

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Generic operator means what it
sort of precisely does depends

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on the kind of data that
it's looking at.

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More than that, you'd like to
design the system so that the

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next time a new division comes
into the company they don't

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have to make any big changes in
what they're already doing

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to link into this system, and
the rest of the company

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doesn't have to make any big
changes to admit their stuff

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to the system.

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So that's the problem you should
be thinking about.

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Like it's sort of
just your work.

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You want to be able to
include new things by

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making minimal changes.

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OK, well that's the problem
that we'll be

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talking about today.

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And you should have this sort
of distributed personnel

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record system in your mind.

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But actually the one I'll be
talking about is a problem

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that's a little bit more
self-contained than that.

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that'll bring up the issues,
I think, more clearly.

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That's the problem of doing a
system that does arithmetic on

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complex numbers.

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So let's take a look here.

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Just as a little review,
there are things

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called complex numbers.

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Complex number you can think
of as a point in

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the plane, or z.

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And you can represent a point
either by its real-part and

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its imaginary-part.

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So if this is z and its
real-part is this much, and

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its imaginary-part is that
much, and you write z

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equals x plus iy.

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Or another way to represent a
complex number is by saying,

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what's the distance from the
origin, and what's the angle?

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So that represents a complex
number as its

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radius times an angle.

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This one's called-- the
original one's called

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rectangular form, rectangular
representation, real- and

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imaginary-part, or polar
representation.

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Magnitude and angle--

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and if you know the real- and
imaginary-part, you can figure

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out the magnitude and angle.

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If you know x and y, you can
get r by this formula.

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Square root of sum of the
squares, and you can get the

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angle as an arctangent.

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Or conversely, if you knew
r and A you could

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figure out x and y.

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x is r times the cosine of A,
and y is r times the sine of

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A. All right, so there's
these two.

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They're complex numbers.

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You can think of them
either in polar form

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or rectangular form.

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What we would like to do is
make a system that does

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arithmetic on complex numbers.

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In other words, what
we'd like--

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just like the rational
number example--

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is to have some operations plus
c, which is going to take

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two complex numbers and add
them, subtract them, and

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multiply them, and
divide them.

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OK, well there's little bit
of mathematics behind it.

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What are the actual formulas for
manipulating such things?

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And it's sort of not important
where they come from, but just

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as an implementer let's see--

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if you want to add two complex
numbers it's pretty easy to

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get its real-part and
its imaginary-part.

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The real-part of the sum of
two complex numbers, the

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real-part of the z1 plus z2 is
the real-part of z1 plus the

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real-part of z2.

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And the imaginary-part of z1
plus z2 is the imaginary part

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of z1 plus the imaginary
part of z2.

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So it's pretty easy to
add complex numbers.

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You just add the corresponding
parts and make a new complex

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number with those parts.

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If you want to multiply them,
it's kind of nice to do it in

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polar form.

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Because if you have two complex
numbers, the magnitude

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of their product is here, the
product of the magnitudes.

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And the angle of the product
is the sum of the angles.

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So that's sort of mathematics
that allows you to do

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arithmetic on complex numbers.

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Let's actually think about
the implementation.

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Well we do it just like
rational numbers.

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We come down, we assume
we have some

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constructors and selectors.

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What would we like?

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Well let's assume that we make
a data object cloud, which is

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a complex number that has some
stuff in it, and that we can

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get out from a complex number
the real-part, or the

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imaginary-part, or the
magnitude, or the angle.

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We want some ways of making
complex numbers-- not only

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selectors, but constructors.

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So we'll assume we have a thing
called make-rectangular.

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What make-rectangular is going
to do is take a real-part and

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an imaginary-part and construct
a complex number

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with those parts.

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Similarly, we can have
make-polar which will take a

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magnitude and an angle, and
construct a complex number

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which has that magnitude
and angle.

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So here's a system.

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We'll have two constructors
and four selectors.

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And now, just like before, in
terms of that abstract data

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we'll go ahead and implement our
complex number operations.

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And here you can see translated
into Lisp code just

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the arithmetic formulas
I put down before.

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If I want to add two complex
numbers I will make a complex

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number out of its real-
and imaginary-parts.

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The real part of the complex
number I'm going to make is

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the sum of the real-parts.

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The imaginary part of the
complex number I'm going to

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make is the sum of the
imaginary-parts.

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I put those together, make
a complex number.

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That's how I implement complex
number addition.

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Subtraction is essentially
the same.

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All I do is subtract the parts
rather than add them.

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To multiply two complex
numbers, I

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use the other formula.

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I'll make a complex number out
of a magnitude and angle.

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The magnitude is going to be the
product of the magnitudes

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of the two complex numbers
I'm multiplying.

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And the angle is going to be the
sum of the angles of the

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two complex numbers
I'm multiplying.

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So there's multiplication.

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And then division, division
is almost the same.

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Here I divide the magnitudes
and subtract the angles.

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Now I've implemented
the operations.

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And what do we do?

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We call on George.

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We've done the use, let's
worry about the

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representation.

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We'll call on George and say to
George, go ahead and build

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us a complex number
representation.

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Well that's fine.

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George can say, we'll implement
a complex number

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simply as a pair that has
the real-part and the

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imaginary-part.

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So if I want to make a complex
number with a certain

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real-part and an imaginary-part,
I'll just use

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cons to form a pair, and that
will-- that's George's

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representation of a
complex number.

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So if I want to get out the
real-part of something, I just

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extract the car,
the first part.

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If I want to get the
imaginary-part, I extract the

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cdr. How do I deal with the
magnitude and angle?

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Well if I want to extract the
magnitude of one of these

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things, I get the square root
of the sum of the square of

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the car plus the square of the
cdr. If I want to get the

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angle, I compute the
arctangent of

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the cdr in the car.

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This is a list procedure for
computing arctangent.

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And if somebody hands me a
magnitude and an angle and

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says, make me a complex number,
well I compute the

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real-part and the
imaginary-part, or our cosine

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of a and our sine of
a, and stick them

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together into a pair.

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OK so we're done.

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In fact, what I just did,
conceptually, is absolutely no

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different from the rational
number representation that we

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looked at last time.

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It's the same sort of idea.

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You implement the operators,
you pick a representation.

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Nothing different.

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Now let's worry about Martha.

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See, Martha has a
different idea.

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She doesn't want to represent a
complex number as a pair of

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a real-part and an
imaginary-part.

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What she would like to do is
represent a complex number as

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a pair of a magnitude
and an angle.

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So if instead of calling up
George we ask Martha to design

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our representation, we get
something like this.

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We get make-polar.

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Sure, if I give you a magnitude
and an angle we're

249
00:13:50,220 --> 00:13:55,430
just going to form a pair that
has magnitude and angle.

250
00:13:55,430 --> 00:13:57,680
If you want to extract the
magnitude, that's easy.

251
00:13:57,680 --> 00:13:59,780
You just pull out the
car or the pair.

252
00:13:59,780 --> 00:14:02,670
If you want to extract the
angle, sure, that's easy.

253
00:14:02,670 --> 00:14:05,480
You just pull out the cdr.
If you want to look for

254
00:14:05,480 --> 00:14:07,660
real-parts and imaginary-parts,
well then you

255
00:14:07,660 --> 00:14:08,590
have to do some work.

256
00:14:08,590 --> 00:14:14,580
If you want the real-part, you
have to get r cosine a.

257
00:14:14,580 --> 00:14:19,990
In other words, r, the car of
the pair, times the cosine of

258
00:14:19,990 --> 00:14:20,910
the cdr of the pair.

259
00:14:20,910 --> 00:14:26,230
So this is r times
the cosine of a,

260
00:14:26,230 --> 00:14:28,330
and that's the real-part.

261
00:14:28,330 --> 00:14:30,810
If you want to get the
imaginary-part, it's r times

262
00:14:30,810 --> 00:14:32,660
the sine of a.

263
00:14:32,660 --> 00:14:37,930
And if I hand you a real-part
and an imaginary-part and say,

264
00:14:37,930 --> 00:14:42,030
make me a complex number
with that real-part and

265
00:14:42,030 --> 00:14:44,170
imaginary-part, well I figure
out what the magnitude and

266
00:14:44,170 --> 00:14:45,540
angle should be.

267
00:14:45,540 --> 00:14:48,090
The magnitude's the square root
of the sum of the squares

268
00:14:48,090 --> 00:14:49,230
and the angle's the
arctangent.

269
00:14:49,230 --> 00:14:52,090
I put those together
to make a pair.

270
00:14:52,090 --> 00:14:54,170
So there's Martha's idea.

271
00:14:54,170 --> 00:14:56,690


272
00:14:56,690 --> 00:14:59,680
Well which is better?

273
00:14:59,680 --> 00:15:02,850
Well if you're doing a lot of
additions, probably George's

274
00:15:02,850 --> 00:15:04,810
is better, because you're doing
a lot of real-parts and

275
00:15:04,810 --> 00:15:05,850
imaginary-parts.

276
00:15:05,850 --> 00:15:07,920
If mostly you're going to be
doing multiplications and

277
00:15:07,920 --> 00:15:11,140
divisions, then maybe Martha's
idea is better.

278
00:15:11,140 --> 00:15:16,590
Or maybe, and this is the real
point, you can't decide.

279
00:15:16,590 --> 00:15:21,170
Or maybe you just have to let
them both hang around, for

280
00:15:21,170 --> 00:15:23,480
personality reasons.

281
00:15:23,480 --> 00:15:25,870
Maybe you just really
can't ever decide

282
00:15:25,870 --> 00:15:28,560
what you would like.

283
00:15:28,560 --> 00:15:31,520
And again, what we would really
like is a system that

284
00:15:31,520 --> 00:15:32,320
looks like this.

285
00:15:32,320 --> 00:15:37,090
That somehow there's George
over here, who has built

286
00:15:37,090 --> 00:15:41,470
rectangular complex numbers.

287
00:15:41,470 --> 00:15:46,120
And Martha, who has polar
complex numbers.

288
00:15:46,120 --> 00:15:54,200
And somehow we have operations
that can add, and subtract,

289
00:15:54,200 --> 00:15:59,710
and multiply, and divide, and it
shouldn't matter that there

290
00:15:59,710 --> 00:16:02,790
are two incompatible
representations of complex

291
00:16:02,790 --> 00:16:04,410
numbers floating around
this system.

292
00:16:04,410 --> 00:16:09,640
In other words, not only like
an abstraction barrier here

293
00:16:09,640 --> 00:16:15,770
that has things in it like
a real-part, and an

294
00:16:15,770 --> 00:16:23,830
imaginary-part, and magnitude,
and angle.

295
00:16:23,830 --> 00:16:26,850
So not only is there an
abstraction barrier that hides

296
00:16:26,850 --> 00:16:30,310
the actual representation from
us, but also there's some kind

297
00:16:30,310 --> 00:16:33,620
of vertical barrier here that
allows both of these

298
00:16:33,620 --> 00:16:36,270
representations to
exist without

299
00:16:36,270 --> 00:16:38,570
interfering with each other.

300
00:16:38,570 --> 00:16:41,900
The idea is that the
things in here--

301
00:16:41,900 --> 00:16:44,120
real-part, imaginary-part,
magnitude, and angle--

302
00:16:44,120 --> 00:16:47,310
will be generic operators.

303
00:16:47,310 --> 00:16:50,190
If you ask for the real-part,
it will worry about what

304
00:16:50,190 --> 00:16:53,880
representation it's
looking at.

305
00:16:53,880 --> 00:16:56,840
OK, well how can we do that?

306
00:16:56,840 --> 00:17:00,290
There's actually a really
obvious idea, if you're used

307
00:17:00,290 --> 00:17:02,770
to thinking about
complex numbers.

308
00:17:02,770 --> 00:17:06,390
If you're used to thinking
about compound data.

309
00:17:06,390 --> 00:17:10,690
See, suppose you could just tell
by looking at a complex

310
00:17:10,690 --> 00:17:13,190
number whether it
was constructed

311
00:17:13,190 --> 00:17:15,790
by George or Martha.

312
00:17:15,790 --> 00:17:18,900
In other words, so it's not that
what's floating around

313
00:17:18,900 --> 00:17:20,910
here are ordinary, just complex
numbers, right?

314
00:17:20,910 --> 00:17:24,390
They're fancy, designer
complex numbers.

315
00:17:24,390 --> 00:17:27,260
So you look at a complex numbers
as it's not just a

316
00:17:27,260 --> 00:17:29,190
complex number, it's got a label
on it that says, this

317
00:17:29,190 --> 00:17:31,450
one is by Martha.

318
00:17:31,450 --> 00:17:34,480
Or this is a complex
number by George.

319
00:17:34,480 --> 00:17:34,700
Right?

320
00:17:34,700 --> 00:17:36,860
They're signed.

321
00:17:36,860 --> 00:17:40,155
See, and then whenever we looked
at a complex number we

322
00:17:40,155 --> 00:17:45,800
could just read the label, and
then we'd know how you expect

323
00:17:45,800 --> 00:17:48,030
to operate on that.

324
00:17:48,030 --> 00:17:49,850
In other words, what
we want is not just

325
00:17:49,850 --> 00:17:51,190
ordinary data objects.

326
00:17:51,190 --> 00:17:53,120
We want to introduce the
notion of what's

327
00:17:53,120 --> 00:17:54,370
called typed data.

328
00:17:54,370 --> 00:17:59,760


329
00:17:59,760 --> 00:18:03,940
Typed data means, again, there's
some sort of cloud.

330
00:18:03,940 --> 00:18:08,930
And what it's got in it is an
ordinary data object like

331
00:18:08,930 --> 00:18:10,180
we've been thinking about.

332
00:18:10,180 --> 00:18:13,180


333
00:18:13,180 --> 00:18:16,540
Pulled out the contents, sort
of the actual data.

334
00:18:16,540 --> 00:18:19,320


335
00:18:19,320 --> 00:18:24,220
But also a thing called a
type, but it's signed by

336
00:18:24,220 --> 00:18:25,850
either George or Martha.

337
00:18:25,850 --> 00:18:28,340
So we're going to go from
regular data to type data.

338
00:18:28,340 --> 00:18:31,950


339
00:18:31,950 --> 00:18:32,710
How do we build that?

340
00:18:32,710 --> 00:18:33,990
Well that's easy.

341
00:18:33,990 --> 00:18:34,980
We know how to build clouds.

342
00:18:34,980 --> 00:18:37,920
We build them out of pairs.

343
00:18:37,920 --> 00:18:41,050
So here's a little
representation that supports

344
00:18:41,050 --> 00:18:43,510
typed data.

345
00:18:43,510 --> 00:18:49,020
There's a thing called take a
type and attach it to a piece

346
00:18:49,020 --> 00:18:51,530
of contents, and we
just use cons.

347
00:18:51,530 --> 00:18:53,770
And if we have a piece of typed
data, we can look at the

348
00:18:53,770 --> 00:18:56,290
type, which is the car.

349
00:18:56,290 --> 00:19:00,460
We can look at the contents,
which is the cdr. Now along

350
00:19:00,460 --> 00:19:05,420
with that, the way we use our
type data will test, when

351
00:19:05,420 --> 00:19:07,520
we're given a piece of data,
what type it is.

352
00:19:07,520 --> 00:19:10,510
So we have some type
predicates with us.

353
00:19:10,510 --> 00:19:13,730
For example, to see whether
a complex number is one of

354
00:19:13,730 --> 00:19:16,860
George's, whether it's
rectangular, we just check to

355
00:19:16,860 --> 00:19:23,850
see if the type of that is the
symbol rectangular, right?

356
00:19:23,850 --> 00:19:25,100
The symbol rectangular.

357
00:19:25,100 --> 00:19:27,200


358
00:19:27,200 --> 00:19:30,650
And to check whether a complex
number is one of Martha's, we

359
00:19:30,650 --> 00:19:33,430
check to see whether the type
is the symbol polar.

360
00:19:33,430 --> 00:19:36,460


361
00:19:36,460 --> 00:19:38,710
So that's a way to test
what kind of number

362
00:19:38,710 --> 00:19:40,350
we're looking at.

363
00:19:40,350 --> 00:19:42,070
Now let's think about
how we can use that

364
00:19:42,070 --> 00:19:43,870
to build the system.

365
00:19:43,870 --> 00:19:46,170
So let's suppose that George
and Martha were off working

366
00:19:46,170 --> 00:19:50,710
separately, and each of them
had designed their complex

367
00:19:50,710 --> 00:19:52,640
number representation
packages.

368
00:19:52,640 --> 00:19:58,980
What do they have to do to
become part of the system, to

369
00:19:58,980 --> 00:20:00,140
exist compatibly?

370
00:20:00,140 --> 00:20:02,860
Well it's really pretty easy.

371
00:20:02,860 --> 00:20:05,970
Remember, George had
this package.

372
00:20:05,970 --> 00:20:08,980
Here's George's original
package, or half of it.

373
00:20:08,980 --> 00:20:12,090
And underlined in red are the
changes he has to make.

374
00:20:12,090 --> 00:20:16,010
So before, when George made a
complex number out of an x and

375
00:20:16,010 --> 00:20:20,930
y, he just put them together
to make a pair.

376
00:20:20,930 --> 00:20:24,090
And the only difference is
that now he signs them.

377
00:20:24,090 --> 00:20:26,920
He attaches the type,
which is the symbol

378
00:20:26,920 --> 00:20:30,600
rectangular to that pair.

379
00:20:30,600 --> 00:20:33,920
Everything else George does
is the same, except that--

380
00:20:33,920 --> 00:20:35,970
see, George and Martha both
have procedures named

381
00:20:35,970 --> 00:20:38,700
real-part and imaginary-part.

382
00:20:38,700 --> 00:20:44,220
So to allow them both to exist
in the same Lisp environment,

383
00:20:44,220 --> 00:20:45,920
George had changed the names
of his procedures.

384
00:20:45,920 --> 00:20:49,045
So we'll say, this is George's
real-part procedure.

385
00:20:49,045 --> 00:20:52,710
It's the real-part rectangular
procedure, the imaginary-part

386
00:20:52,710 --> 00:20:55,170
rectangular procedure.

387
00:20:55,170 --> 00:20:59,130
And then here's the rest
of George's package.

388
00:20:59,130 --> 00:21:02,060
He'd had magnitude and angle,
just renames them magnitude

389
00:21:02,060 --> 00:21:05,702
rectangular and angle
rectangular.

390
00:21:05,702 --> 00:21:09,860
And Martha has to do basically
the same thing.

391
00:21:09,860 --> 00:21:15,200
Martha previously, when she made
a complex number out of a

392
00:21:15,200 --> 00:21:19,270
magnitude and angle,
she just cons them.

393
00:21:19,270 --> 00:21:25,330
Now she attaches the type polar,
and she changes the

394
00:21:25,330 --> 00:21:28,100
name so her real-part procedure
won't conflict in

395
00:21:28,100 --> 00:21:30,710
name with George's.

396
00:21:30,710 --> 00:21:34,540
It's a real-part-polar,
imaginary-part-polar,

397
00:21:34,540 --> 00:21:38,060
magnitude polar, and
angle polar.

398
00:21:38,060 --> 00:21:45,000


399
00:21:45,000 --> 00:21:46,130
Now we have the system.

400
00:21:46,130 --> 00:21:49,160
Right there's George
and Martha.

401
00:21:49,160 --> 00:21:51,050
And now we've got to get some
kind of manager to look at

402
00:21:51,050 --> 00:21:52,300
these types.

403
00:21:52,300 --> 00:21:55,050


404
00:21:55,050 --> 00:21:57,530
How are these things actually
going to work now that George

405
00:21:57,530 --> 00:22:00,530
and Martha have supplied
us with typed data?

406
00:22:00,530 --> 00:22:05,260
Well what we have are a bunch
of generic selectors.

407
00:22:05,260 --> 00:22:07,800
Generic selectors for complex
numbers real-part,

408
00:22:07,800 --> 00:22:10,630
imaginary-part, magnitude,
and angle.

409
00:22:10,630 --> 00:22:14,140


410
00:22:14,140 --> 00:22:15,410
Let's look at them
more closely.

411
00:22:15,410 --> 00:22:17,930


412
00:22:17,930 --> 00:22:19,310
What does a real-part do?

413
00:22:19,310 --> 00:22:24,070
If I ask for the real part
of a complex number,

414
00:22:24,070 --> 00:22:25,800
well I look at it.

415
00:22:25,800 --> 00:22:26,690
I look at its type.

416
00:22:26,690 --> 00:22:27,940
I say, is it rectangular?

417
00:22:27,940 --> 00:22:31,020


418
00:22:31,020 --> 00:22:36,970
If so, I apply George's real
part procedure to the contents

419
00:22:36,970 --> 00:22:38,220
of that complex number.

420
00:22:38,220 --> 00:22:41,230


421
00:22:41,230 --> 00:22:43,720
This is a number that
has a type on it.

422
00:22:43,720 --> 00:22:46,340
I strip off the type
using contents and

423
00:22:46,340 --> 00:22:47,590
apply George's procedure.

424
00:22:47,590 --> 00:22:50,700


425
00:22:50,700 --> 00:22:53,950
Or is this a polar
complex number?

426
00:22:53,950 --> 00:22:56,890
If I want the real part, I
apply Martha's real part

427
00:22:56,890 --> 00:22:59,850
procedure to the contents
of that number.

428
00:22:59,850 --> 00:23:02,260
So that's how real part works.

429
00:23:02,260 --> 00:23:04,670
And then similarly there's
imaginary-part, which is

430
00:23:04,670 --> 00:23:06,770
almost the same.

431
00:23:06,770 --> 00:23:09,600
It looks at the number and
if it's rectangular, uses

432
00:23:09,600 --> 00:23:11,130
George's imaginary-part
procedure.

433
00:23:11,130 --> 00:23:13,380
If it's polar, uses Martha's.

434
00:23:13,380 --> 00:23:17,240
And then there's a magnitude
and an angle.

435
00:23:17,240 --> 00:23:19,880


436
00:23:19,880 --> 00:23:21,130
So there's a system.

437
00:23:21,130 --> 00:23:23,460


438
00:23:23,460 --> 00:23:24,260
Has three parts.

439
00:23:24,260 --> 00:23:26,760
There's sort of George, and
Martha, and the manager.

440
00:23:26,760 --> 00:23:28,970
And that's how you get generic
operators implemented.

441
00:23:28,970 --> 00:23:33,500
Let's look at just a simple
example, just to pin it down.

442
00:23:33,500 --> 00:23:40,240
But exactly how this is going to
work, suppose you're going

443
00:23:40,240 --> 00:23:44,460
to be looking at the complex
number who's real-part is one,

444
00:23:44,460 --> 00:23:46,090
and who's imaginary-part
is two.

445
00:23:46,090 --> 00:23:50,310
So that would be one plus 2i.

446
00:23:50,310 --> 00:23:56,350
What would happen is up here,
up here above where the

447
00:23:56,350 --> 00:23:58,530
operations have to happen,
that number would be

448
00:23:58,530 --> 00:24:10,320
represented as a pair of 1 and
2 together with typed data.

449
00:24:10,320 --> 00:24:11,870
That would be the contents.

450
00:24:11,870 --> 00:24:16,300
And the whole data would be
that thing with the symbol

451
00:24:16,300 --> 00:24:17,960
rectangular added onto that.

452
00:24:17,960 --> 00:24:20,980
And that's the way that complex
number would exist in

453
00:24:20,980 --> 00:24:22,330
the system.

454
00:24:22,330 --> 00:24:26,560
When you went to take the
real-part, the manager would

455
00:24:26,560 --> 00:24:30,270
look at this and say, oh
it's one of George's.

456
00:24:30,270 --> 00:24:34,440
He'll strip off the type
and hand down to

457
00:24:34,440 --> 00:24:37,532
George the pair 1, 2.

458
00:24:37,532 --> 00:24:41,420
And that's the kind of data
that George developed his

459
00:24:41,420 --> 00:24:42,670
system to use.

460
00:24:42,670 --> 00:24:44,950


461
00:24:44,950 --> 00:24:46,680
So it gets stripped down.

462
00:24:46,680 --> 00:24:51,240
Later on, if you ask George to
construct a complex number,

463
00:24:51,240 --> 00:24:55,370
George would construct some
complex number as a pair, and

464
00:24:55,370 --> 00:24:59,630
before he passes it back up
through the manager would

465
00:24:59,630 --> 00:25:00,880
attach the type rectangular.

466
00:25:00,880 --> 00:25:03,920


467
00:25:03,920 --> 00:25:04,650
So you see what happens.

468
00:25:04,650 --> 00:25:05,850
There's no confusion
in this system.

469
00:25:05,850 --> 00:25:13,780
It doesn't matter in the least
that the pair 1, 2 means

470
00:25:13,780 --> 00:25:15,750
something completely different
in Martha's world.

471
00:25:15,750 --> 00:25:18,440
In Martha's world this pair
means the complex number whose

472
00:25:18,440 --> 00:25:21,190
magnitude is 1 and
whose angle is 2.

473
00:25:21,190 --> 00:25:23,930
And there's no confusion,
because by the time any pair

474
00:25:23,930 --> 00:25:27,250
like this gets handed back
through the manager to the

475
00:25:27,250 --> 00:25:31,210
main system it's going to have
the type polar attached.

476
00:25:31,210 --> 00:25:33,670
Whereas this one would have the
type rectangular attached.

477
00:25:33,670 --> 00:25:36,930


478
00:25:36,930 --> 00:25:38,180
OK, let's take a break.

479
00:25:38,180 --> 00:25:40,770


480
00:25:40,770 --> 00:25:41,057
[MUSIC-- "JESU, JOY OF
MAN'S DESIRING" BY

481
00:25:41,057 --> 00:25:42,307
JOHANN SEBASTIAN BACH]

482
00:25:42,307 --> 00:26:20,210


483
00:26:20,210 --> 00:26:22,080
We just looked at
a strategy for

484
00:26:22,080 --> 00:26:24,150
implementing generic operators.

485
00:26:24,150 --> 00:26:31,400
That strategy has a name: it's
called dispatch type.

486
00:26:31,400 --> 00:26:34,310


487
00:26:34,310 --> 00:26:38,480
And the idea is that you
break your system

488
00:26:38,480 --> 00:26:39,360
into a bunch of pieces.

489
00:26:39,360 --> 00:26:43,250
There's George and Martha, who
are making representations,

490
00:26:43,250 --> 00:26:46,320
and then there's the manager.

491
00:26:46,320 --> 00:26:49,880
Looks at the types on the data
and then dispatches them to

492
00:26:49,880 --> 00:26:51,990
the right person.

493
00:26:51,990 --> 00:26:55,320
Well what criticisms can we
make of that as a system

494
00:26:55,320 --> 00:26:56,570
organization?

495
00:26:56,570 --> 00:26:58,150


496
00:26:58,150 --> 00:27:00,400
Well first of all there was this
little, annoying problem

497
00:27:00,400 --> 00:27:02,350
that George and Martha had to
change the names of their

498
00:27:02,350 --> 00:27:04,220
procedures.

499
00:27:04,220 --> 00:27:06,160
George originally had a
real-part procedure, and he

500
00:27:06,160 --> 00:27:09,110
had to go name it real-part
rectangular so it wouldn't

501
00:27:09,110 --> 00:27:11,170
interfere with Martha's
real-part procedure, which is

502
00:27:11,170 --> 00:27:14,410
now named real-part-polar, so it
wouldn't interfere with the

503
00:27:14,410 --> 00:27:17,310
manager's real-part procedure,
who's now named real-part.

504
00:27:17,310 --> 00:27:19,460
That's kind of an annoying
problem.

505
00:27:19,460 --> 00:27:21,270
But I'm not going to talk
about that one now.

506
00:27:21,270 --> 00:27:24,450
We'll see later on when we think
about the structure of

507
00:27:24,450 --> 00:27:27,480
Lisp names and environments that
there really are ways to

508
00:27:27,480 --> 00:27:30,390
package all those so-called name
spaces separately so they

509
00:27:30,390 --> 00:27:32,500
don't interfere with
each other.

510
00:27:32,500 --> 00:27:35,720
Not going to think about
that problem now.

511
00:27:35,720 --> 00:27:38,740
The problem that I actually
want to focus on is what

512
00:27:38,740 --> 00:27:44,510
happens when you bring somebody
new into the system.

513
00:27:44,510 --> 00:27:45,320
What has to happen?

514
00:27:45,320 --> 00:27:47,690
Well George and Martha
don't care.

515
00:27:47,690 --> 00:27:52,830
George is sitting there in his
rectangular world, has his

516
00:27:52,830 --> 00:27:54,090
procedures and his types.

517
00:27:54,090 --> 00:27:56,260
Martha sits in her
polar world.

518
00:27:56,260 --> 00:27:59,380
She doesn't care.

519
00:27:59,380 --> 00:28:01,540
But let's look at the manager.

520
00:28:01,540 --> 00:28:03,180
What's the manager have to do?

521
00:28:03,180 --> 00:28:07,360
The manager comes through and
had these operations.

522
00:28:07,360 --> 00:28:09,040
There was a test
for rectangular

523
00:28:09,040 --> 00:28:10,140
and a test for polar.

524
00:28:10,140 --> 00:28:17,210
If Harry comes in with some new
kind of complex number,

525
00:28:17,210 --> 00:28:20,430
and Harry has a new type, Harry
type complex number, the

526
00:28:20,430 --> 00:28:25,240
manager has to go in and change
all those procedures.

527
00:28:25,240 --> 00:28:28,940
So the inflexibility in the
system, the place where work

528
00:28:28,940 --> 00:28:34,890
has to happen to accommodate
change, is in the manager.

529
00:28:34,890 --> 00:28:35,990
That's pretty annoying.

530
00:28:35,990 --> 00:28:40,300
It's even more annoying when you
realize the manager's not

531
00:28:40,300 --> 00:28:42,590
doing anything.

532
00:28:42,590 --> 00:28:46,690
The manager is just being
a paper pusher.

533
00:28:46,690 --> 00:28:51,760
Let's look again at these
programs. What are they doing?

534
00:28:51,760 --> 00:28:52,880
What does real-part do?

535
00:28:52,880 --> 00:28:56,170
Real-part says, oh, is it the
kind of complex number that

536
00:28:56,170 --> 00:28:57,000
George can handle?

537
00:28:57,000 --> 00:28:59,410
If so, send it off to George.

538
00:28:59,410 --> 00:29:01,910
Is it the kind of complex number
that Martha can handle?

539
00:29:01,910 --> 00:29:05,040
If so, send it off to Martha.

540
00:29:05,040 --> 00:29:08,720
So it's really annoying that the
bottleneck in this system,

541
00:29:08,720 --> 00:29:13,040
the thing that's preventing
flexibility and change, is

542
00:29:13,040 --> 00:29:15,000
completely in the bureaucracy.

543
00:29:15,000 --> 00:29:19,700
It's not in anybody who's
doing any of the work.

544
00:29:19,700 --> 00:29:23,300
Not an uncommon situation,
unfortunately.

545
00:29:23,300 --> 00:29:24,570
See, what's really going on--

546
00:29:24,570 --> 00:29:28,100
abstractly in the system,
there's a table.

547
00:29:28,100 --> 00:29:30,150
So what's really happening is
somewhere there's a table.

548
00:29:30,150 --> 00:29:32,780


549
00:29:32,780 --> 00:29:34,400
There're types.

550
00:29:34,400 --> 00:29:38,565
There's polar and rectangular.

551
00:29:38,565 --> 00:29:41,550


552
00:29:41,550 --> 00:29:44,380
And Harry's may be over here.

553
00:29:44,380 --> 00:29:48,050
And there are operators.

554
00:29:48,050 --> 00:29:50,340
There's an operator
like real-part.

555
00:29:50,340 --> 00:29:55,600


556
00:29:55,600 --> 00:30:00,010
Or imaginary-part.

557
00:30:00,010 --> 00:30:05,830
Or a magnitude and angle.

558
00:30:05,830 --> 00:30:19,280
And sitting in this table are
the right procedures.

559
00:30:19,280 --> 00:30:21,990
So sitting here for the type
polar and real-part is

560
00:30:21,990 --> 00:30:24,730
Martha's procedure
real-part-polar.

561
00:30:24,730 --> 00:30:30,570


562
00:30:30,570 --> 00:30:33,740
And over here in the table
is George's procedure

563
00:30:33,740 --> 00:30:34,990
real-part-rectangular.

564
00:30:34,990 --> 00:30:37,740


565
00:30:37,740 --> 00:30:40,680
And over here would be, say,
Martha's procedure

566
00:30:40,680 --> 00:30:46,780
magnitude-polar, and
George's procedure

567
00:30:46,780 --> 00:30:49,760
magnitude-rectangular,
right, and so on.

568
00:30:49,760 --> 00:30:52,390
The rest of this table's
filled in.

569
00:30:52,390 --> 00:30:54,260
And that's really
what's going on.

570
00:30:54,260 --> 00:30:57,630


571
00:30:57,630 --> 00:31:03,380
So in some sense, all the
manager is doing is acting as

572
00:31:03,380 --> 00:31:04,630
this table.

573
00:31:04,630 --> 00:31:06,860


574
00:31:06,860 --> 00:31:08,610
Well how do we fix our system?

575
00:31:08,610 --> 00:31:12,110


576
00:31:12,110 --> 00:31:13,770
How do you fix bureaucracies
a lot of the time?

577
00:31:13,770 --> 00:31:16,240
What you do is you get
rid of the manager.

578
00:31:16,240 --> 00:31:20,170
We just take the manager and
replace him by a computer.

579
00:31:20,170 --> 00:31:23,320
We're going to automate
him out of existence.

580
00:31:23,320 --> 00:31:25,970
Namely, instead of having the
manager who basically consults

581
00:31:25,970 --> 00:31:31,020
this table, we'll have our
system use the table directly.

582
00:31:31,020 --> 00:31:32,110
What do I mean by that?

583
00:31:32,110 --> 00:31:38,730
Let's assume, again using data
abstraction, that we have some

584
00:31:38,730 --> 00:31:40,880
kind of data structure
that's a table.

585
00:31:40,880 --> 00:31:43,080
And we have ways of sticking
things in and ways of getting

586
00:31:43,080 --> 00:31:44,356
things out.

587
00:31:44,356 --> 00:31:47,000
And to be explicit, let me
assume that there's an

588
00:31:47,000 --> 00:31:52,710
operation called "put." And put
is going to take, in this

589
00:31:52,710 --> 00:32:00,130
case two things I'll call
"keys." Key1 and key2.

590
00:32:00,130 --> 00:32:01,380
And a value.

591
00:32:01,380 --> 00:32:06,200


592
00:32:06,200 --> 00:32:11,490
And that stores the value in the
table under key1 and key2.

593
00:32:11,490 --> 00:32:15,530
And then we'll assume there's
a thing called "get," such

594
00:32:15,530 --> 00:32:19,680
that if later on I say, get me
what's in the table stored

595
00:32:19,680 --> 00:32:25,010
under key1 and key2, it'll
retrieve whatever value was

596
00:32:25,010 --> 00:32:26,730
stored there.

597
00:32:26,730 --> 00:32:30,000
And let's not worry about how
tables are implemented.

598
00:32:30,000 --> 00:32:33,060
That's yet another data
abstraction, George's problem.

599
00:32:33,060 --> 00:32:34,700
And maybe we'll see later--

600
00:32:34,700 --> 00:32:36,970
talk about how you might
actually build tables in Lisp.

601
00:32:36,970 --> 00:32:40,710


602
00:32:40,710 --> 00:32:44,850
Well given this organization,
what did George and Martha

603
00:32:44,850 --> 00:32:47,380
have to do?

604
00:32:47,380 --> 00:32:50,010
Well when they build their
system, they each have the

605
00:32:50,010 --> 00:32:52,750
responsibility to set
up their appropriate

606
00:32:52,750 --> 00:32:55,210
column in the table.

607
00:32:55,210 --> 00:33:00,620
So what George does, for
example, when he defines his

608
00:33:00,620 --> 00:33:04,020
procedures, all he has to do
is go off and put into the

609
00:33:04,020 --> 00:33:06,990
table under the
type-rectangular.

610
00:33:06,990 --> 00:33:09,820


611
00:33:09,820 --> 00:33:14,100
And the name of the operation
is real-part, his procedure

612
00:33:14,100 --> 00:33:16,250
real-part-rectangular.

613
00:33:16,250 --> 00:33:17,780
So notice what's going
into this table.

614
00:33:17,780 --> 00:33:22,100
The two keys here are symbols,
rectangular and real-part.

615
00:33:22,100 --> 00:33:24,400
That's the quote.

616
00:33:24,400 --> 00:33:27,410
And what's going into the table
is the actual procedure

617
00:33:27,410 --> 00:33:28,870
that he wrote, real-part
rectangular.

618
00:33:28,870 --> 00:33:32,040


619
00:33:32,040 --> 00:33:35,000
And then puts an imaginary part
into the table, filed

620
00:33:35,000 --> 00:33:39,370
under the keys rectangular-
and imaginary-part, and

621
00:33:39,370 --> 00:33:44,020
magnitude under the keys
rectangular magnitude, angle

622
00:33:44,020 --> 00:33:45,270
under rectangular-angle.

623
00:33:45,270 --> 00:33:47,350


624
00:33:47,350 --> 00:33:50,840
So that's what George has to do
to be part of this system.

625
00:33:50,840 --> 00:33:54,420


626
00:33:54,420 --> 00:33:57,740
Martha similarly sets
up the column and

627
00:33:57,740 --> 00:33:59,430
the table under polar.

628
00:33:59,430 --> 00:34:02,160
Polar and real-part.

629
00:34:02,160 --> 00:34:04,340
Is the procedure
real-part-polar?

630
00:34:04,340 --> 00:34:09,030
And imaginary-part, and
magnitude, and angle.

631
00:34:09,030 --> 00:34:11,409
So this is what Martha has to
do to be part of the system.

632
00:34:11,409 --> 00:34:13,550
Everyone who makes a
representation has the

633
00:34:13,550 --> 00:34:17,840
responsibility for setting
up a column in the table.

634
00:34:17,840 --> 00:34:19,900
And what does Harry do when
Harry comes in with his

635
00:34:19,900 --> 00:34:21,800
brilliant idea for implementing
complex numbers?

636
00:34:21,800 --> 00:34:25,170
Well he makes whatever procedure
he wants and builds

637
00:34:25,170 --> 00:34:28,550
a new column in this table.

638
00:34:28,550 --> 00:34:31,330
OK, well what happened
to the manager?

639
00:34:31,330 --> 00:34:34,610
The manager has been automated
out of existence and is

640
00:34:34,610 --> 00:34:37,110
replaced by a procedure
called operate.

641
00:34:37,110 --> 00:34:40,380
And this is the key procedure
in the whole system.

642
00:34:40,380 --> 00:34:45,920
Let's say define operate.

643
00:34:45,920 --> 00:34:51,060


644
00:34:51,060 --> 00:34:57,750
Operate is going to take an
operation that you want to do,

645
00:34:57,750 --> 00:35:01,840
the name of an operation, and an
object that you would like

646
00:35:01,840 --> 00:35:04,210
to apply that operation to.

647
00:35:04,210 --> 00:35:07,400
So for example, the real-part
of some particular complex

648
00:35:07,400 --> 00:35:09,890
number, what does it do?

649
00:35:09,890 --> 00:35:12,650
Well the first thing it does,
it looks in the table.

650
00:35:12,650 --> 00:35:20,710
Goes into the table and tries
to find a procedure that's

651
00:35:20,710 --> 00:35:23,320
stored in the table.

652
00:35:23,320 --> 00:35:29,830
So it gets from the table, using
as keys the type of the

653
00:35:29,830 --> 00:35:40,450
object and the operator, but
looks on the table and sees

654
00:35:40,450 --> 00:35:42,300
what's stored under the type
of the object and the

655
00:35:42,300 --> 00:35:44,440
operator, sees if anything's
stored.

656
00:35:44,440 --> 00:35:45,930
Let's assume that get
is implemented.

657
00:35:45,930 --> 00:35:52,560
So if nothing is stored there,
it'll return the empty list.

658
00:35:52,560 --> 00:35:55,130
So it says, if there's actually
something stored

659
00:35:55,130 --> 00:36:04,920
there, if the procedure here is
not no, then it'll take the

660
00:36:04,920 --> 00:36:11,240
procedure that it found in the
table and apply it to the

661
00:36:11,240 --> 00:36:15,120
contents of the object.

662
00:36:15,120 --> 00:36:18,042


663
00:36:18,042 --> 00:36:21,445
And otherwise if there was
nothing stored there, it'll--

664
00:36:21,445 --> 00:36:22,435
well we can decide.

665
00:36:22,435 --> 00:36:25,920
In this case let's have it put
out an error message saying,

666
00:36:25,920 --> 00:36:28,650
undefined operator.

667
00:36:28,650 --> 00:36:30,230
No operator for this type.

668
00:36:30,230 --> 00:36:32,770


669
00:36:32,770 --> 00:36:34,285
Or some appropriate
error message.

670
00:36:34,285 --> 00:36:39,150


671
00:36:39,150 --> 00:36:39,300
OK?

672
00:36:39,300 --> 00:36:41,890
And that replaces the manager.

673
00:36:41,890 --> 00:36:43,960
How do we really use it?

674
00:36:43,960 --> 00:36:48,580
Well what we say is we'll go
off and define our generic

675
00:36:48,580 --> 00:36:50,040
selectors using operate.

676
00:36:50,040 --> 00:36:57,140
We'll say that the real-part
of an object is found by

677
00:36:57,140 --> 00:37:05,010
operating on the object with
the name of the operation

678
00:37:05,010 --> 00:37:06,260
being real-part.

679
00:37:06,260 --> 00:37:08,070


680
00:37:08,070 --> 00:37:10,870
And then similarly,
imaginary-part is operate

681
00:37:10,870 --> 00:37:16,080
using the name imaginary-part
and magnitude and angle.

682
00:37:16,080 --> 00:37:17,430
That's our implementation.

683
00:37:17,430 --> 00:37:21,330
That plus the tape plus
the operate procedure.

684
00:37:21,330 --> 00:37:23,100
And the table effectively
replaces what the

685
00:37:23,100 --> 00:37:24,150
manager used to do.

686
00:37:24,150 --> 00:37:27,040
Let's just go through that
slowly to show you

687
00:37:27,040 --> 00:37:27,900
what's going on.

688
00:37:27,900 --> 00:37:33,000
Suppose I have one of Martha's
complex numbers.

689
00:37:33,000 --> 00:37:35,520


690
00:37:35,520 --> 00:37:39,100
It's got magnitude
1 and angle 2.

691
00:37:39,100 --> 00:37:40,220
And it's one of Martha's.

692
00:37:40,220 --> 00:37:47,120
So it's labeled here, polar.

693
00:37:47,120 --> 00:37:48,000
Let's call that z.

694
00:37:48,000 --> 00:37:49,250
Suppose that's z.

695
00:37:49,250 --> 00:37:51,770


696
00:37:51,770 --> 00:37:54,320
And suppose with this
implementation someone comes

697
00:37:54,320 --> 00:37:57,110
up and asks for the
real-part of z.

698
00:37:57,110 --> 00:38:04,870


699
00:38:04,870 --> 00:38:08,920
Well real-part now is defined
in terms of operate.

700
00:38:08,920 --> 00:38:18,470
So that's equivalent to saying
operate with the name of the

701
00:38:18,470 --> 00:38:27,060
operator being real-part, the
symbol real-part on z.

702
00:38:27,060 --> 00:38:28,090
And now operate comes.

703
00:38:28,090 --> 00:38:31,720
It's going to look in the table,
and it's going to try

704
00:38:31,720 --> 00:38:34,005
and find something
stored under--

705
00:38:34,005 --> 00:38:38,830


706
00:38:38,830 --> 00:38:42,160
the operation is going to apply
by looking in the table

707
00:38:42,160 --> 00:38:46,225
under the type of the object.

708
00:38:46,225 --> 00:38:48,790
And the type of z is polar.

709
00:38:48,790 --> 00:38:52,990
So it's going to look and say,
can I get using polar?

710
00:38:52,990 --> 00:38:58,250
And the operation name,
which was real-part.

711
00:38:58,250 --> 00:39:05,960


712
00:39:05,960 --> 00:39:09,490
It's going to look in there
and apply that to

713
00:39:09,490 --> 00:39:14,930
the contents of z.

714
00:39:14,930 --> 00:39:15,650
And that?

715
00:39:15,650 --> 00:39:20,350
If everything was set up
correctly, this thing is the

716
00:39:20,350 --> 00:39:21,700
procedure that Martha
put there.

717
00:39:21,700 --> 00:39:22,950
This is real-part-polar.

718
00:39:22,950 --> 00:39:30,790


719
00:39:30,790 --> 00:39:35,130
And this is z without
its type.

720
00:39:35,130 --> 00:39:37,860
The thing that Martha originally
designed those

721
00:39:37,860 --> 00:39:40,340
procedures to work on,
which is 1, 2.

722
00:39:40,340 --> 00:39:43,790


723
00:39:43,790 --> 00:39:47,210
And so operate sort of does
uniformly what the manager

724
00:39:47,210 --> 00:39:49,450
used to do sort of all
over the system.

725
00:39:49,450 --> 00:39:52,170
It finds the right thing, looks
in the table, strips off

726
00:39:52,170 --> 00:39:56,600
the type, and passes
it down into the

727
00:39:56,600 --> 00:39:59,160
person who handles it.

728
00:39:59,160 --> 00:40:04,980
This is another, and, you can
see, more flexible for most

729
00:40:04,980 --> 00:40:07,990
purposes, way of implementing
generic operators.

730
00:40:07,990 --> 00:40:15,505
And it's called data-directed
programming.

731
00:40:15,505 --> 00:40:20,350


732
00:40:20,350 --> 00:40:24,920
And the idea of that is in some
sense the data objects

733
00:40:24,920 --> 00:40:27,260
themselves, those little complex
numbers that are

734
00:40:27,260 --> 00:40:30,340
floating around the system,
are carrying with them the

735
00:40:30,340 --> 00:40:35,390
information about how you
should operate on them.

736
00:40:35,390 --> 00:40:36,640
Let's break for questions.

737
00:40:36,640 --> 00:40:41,000


738
00:40:41,000 --> 00:40:41,240
Yes.

739
00:40:41,240 --> 00:40:43,390
AUDIENCE: What do you have
stored in that data object?

740
00:40:43,390 --> 00:40:47,850
You have the data itself, you
have its type, and you have

741
00:40:47,850 --> 00:40:49,690
the operations for that type?

742
00:40:49,690 --> 00:40:53,600
Or where are the operations
that you found?

743
00:40:53,600 --> 00:40:54,980
PROFESSOR: OK, let me--

744
00:40:54,980 --> 00:40:56,500
yeah, that's a good question.

745
00:40:56,500 --> 00:40:59,700
Because it raises other
possibilities of how

746
00:40:59,700 --> 00:41:00,750
you might do it.

747
00:41:00,750 --> 00:41:04,200
And of course there are a
lot of possibilities.

748
00:41:04,200 --> 00:41:06,820
In this particular
implementation, what's sitting

749
00:41:06,820 --> 00:41:11,630
in this data object, for
example, is the data itself--

750
00:41:11,630 --> 00:41:14,980
which in this case is
a pair of 1 and 2--

751
00:41:14,980 --> 00:41:16,550
and also a symbol.

752
00:41:16,550 --> 00:41:21,140
This is the symbol, the word
P-O-L-A-R, and that's what's

753
00:41:21,140 --> 00:41:22,390
sitting in this data object.

754
00:41:22,390 --> 00:41:24,870


755
00:41:24,870 --> 00:41:26,690
Where are the operations
themselves?

756
00:41:26,690 --> 00:41:29,850
The operations are sitting
in the table.

757
00:41:29,850 --> 00:41:35,450
So in this table, the rows and
columns of the table are

758
00:41:35,450 --> 00:41:38,230
labeled by symbols.

759
00:41:38,230 --> 00:41:40,810
So when I store something in
this table, the key might be

760
00:41:40,810 --> 00:41:48,240
the symbol polar and the
symbol magnitude.

761
00:41:48,240 --> 00:41:51,310
And I think by writing it this
way I've been very confusing.

762
00:41:51,310 --> 00:41:53,160
Because what's really
sitting here isn't--

763
00:41:53,160 --> 00:41:58,360
when I wrote magnitude polar,
what I mean is the procedure

764
00:41:58,360 --> 00:41:59,850
magnitude polar.

765
00:41:59,850 --> 00:42:02,580
And probably what I really
should have written--

766
00:42:02,580 --> 00:42:04,200
except it's too small
for me to write

767
00:42:04,200 --> 00:42:05,580
in this little space--

768
00:42:05,580 --> 00:42:11,250
is something like lambda
of z, the thing that

769
00:42:11,250 --> 00:42:14,710
Martha wrote to implement.

770
00:42:14,710 --> 00:42:16,620
And then you can see from that,
there's another way that

771
00:42:16,620 --> 00:42:20,250
I alluded to of solving this
name conflict problem, which

772
00:42:20,250 --> 00:42:22,380
is that George and Martha
never have to name their

773
00:42:22,380 --> 00:42:23,150
procedures at all.

774
00:42:23,150 --> 00:42:26,710
They can just stick the
anonymous things generated by

775
00:42:26,710 --> 00:42:28,660
lambda directly into
the table.

776
00:42:28,660 --> 00:42:32,540
There's also another thing that
your question raises, is

777
00:42:32,540 --> 00:42:36,045
the possibility that maybe what
I would like somehow is

778
00:42:36,045 --> 00:42:40,120
to store in this data object not
the symbol P-O-L-A-R but

779
00:42:40,120 --> 00:42:43,520
maybe actually all the
operations themselves.

780
00:42:43,520 --> 00:42:45,860
And that's another way to
organize the system, called

781
00:42:45,860 --> 00:42:48,650
message passing.

782
00:42:48,650 --> 00:42:49,970
So there are a lot of
ways you can do it.

783
00:42:49,970 --> 00:42:54,640


784
00:42:54,640 --> 00:42:58,040
AUDIENCE: Therefore if Martha
and George had used the same

785
00:42:58,040 --> 00:43:01,230
procedure names, it would be
OK because it wouldn't look

786
00:43:01,230 --> 00:43:02,560
[UNINTELLIGIBLE].

787
00:43:02,560 --> 00:43:03,010
PROFESSOR: That's right.

788
00:43:03,010 --> 00:43:04,890
That's right.

789
00:43:04,890 --> 00:43:07,060
See, they wouldn't even
have to name their

790
00:43:07,060 --> 00:43:09,470
procedures at all.

791
00:43:09,470 --> 00:43:12,440
What George could have written
instead of saying put in the

792
00:43:12,440 --> 00:43:16,890
table under rectangular- and
real-part, the procedure

793
00:43:16,890 --> 00:43:19,660
real-part rectangular, George
could have written put under

794
00:43:19,660 --> 00:43:23,080
rectangular real-part, lambda
of z, such and such,

795
00:43:23,080 --> 00:43:24,540
and such and such.

796
00:43:24,540 --> 00:43:27,330
And the system would work
completely the same.

797
00:43:27,330 --> 00:43:31,750
AUDIENCE: My question is, Martha
could have put key1

798
00:43:31,750 --> 00:43:37,120
key2 real-part, and George
could have put key1 key2

799
00:43:37,120 --> 00:43:40,060
real-part, and as long as they
defined them differently they

800
00:43:40,060 --> 00:43:41,290
wouldn't have had any
conflicts, right?

801
00:43:41,290 --> 00:43:45,130
PROFESSOR: Yes, that would all
be OK except for the fact that

802
00:43:45,130 --> 00:43:47,130
if you imagine George and Martha
typing at the same

803
00:43:47,130 --> 00:43:50,090
console with the same meanings
for all their names, and it

804
00:43:50,090 --> 00:43:51,720
would get confused by real-part,
but there are ways

805
00:43:51,720 --> 00:43:52,800
to arrange that, too.

806
00:43:52,800 --> 00:43:54,980
And in principle you're
absolutely right.

807
00:43:54,980 --> 00:43:56,290
If their names didn't
conflict--

808
00:43:56,290 --> 00:43:58,190
it's the objects that go in
the table, not the names.

809
00:43:58,190 --> 00:44:08,200


810
00:44:08,200 --> 00:44:09,450
OK, let's take a break.

811
00:44:09,450 --> 00:44:12,493


812
00:44:12,493 --> 00:44:12,836
[MUSIC-- "JESU, JOY OF
MAN'S DESIRING" BY

813
00:44:12,836 --> 00:44:14,086
JOHANN SEBASTIAN BACH]

814
00:44:14,086 --> 00:45:12,880


815
00:45:12,880 --> 00:45:17,680
All right, well we just looked
at data-directed programming

816
00:45:17,680 --> 00:45:21,590
as a way of implementing a
system that does arithmetic on

817
00:45:21,590 --> 00:45:22,840
complex numbers.

818
00:45:22,840 --> 00:45:27,420


819
00:45:27,420 --> 00:45:32,880
So I had these operations in it
called plus C and minus C,

820
00:45:32,880 --> 00:45:38,230
and multiply, and divide,
and maybe some others.

821
00:45:38,230 --> 00:45:46,030
And that sat on top of-- and
this is the key point-- sat on

822
00:45:46,030 --> 00:45:50,340
top of two different
representations.

823
00:45:50,340 --> 00:45:55,110
A rectangular package here,
and a polar package.

824
00:45:55,110 --> 00:45:58,240


825
00:45:58,240 --> 00:45:59,150
And maybe some more.

826
00:45:59,150 --> 00:46:01,640
And we saw that the whole idea
is that maybe some more are

827
00:46:01,640 --> 00:46:04,670
now very easy to add.

828
00:46:04,670 --> 00:46:08,900
But that doesn't really show the
power of this methodology.

829
00:46:08,900 --> 00:46:10,150
Shows you what's going on.

830
00:46:10,150 --> 00:46:13,260
The power of the methodology
only becomes apparent when you

831
00:46:13,260 --> 00:46:17,080
start embedding this in some
more complex system.

832
00:46:17,080 --> 00:46:19,180
What I'm going to do now is
embed this in some more

833
00:46:19,180 --> 00:46:20,250
complex system.

834
00:46:20,250 --> 00:46:23,960
Let's assume that what we really
have is a general kind

835
00:46:23,960 --> 00:46:25,280
of arithmetic system.

836
00:46:25,280 --> 00:46:27,240
So called generic arithmetic
system.

837
00:46:27,240 --> 00:46:32,060
And at the top level here,
somebody can say add two

838
00:46:32,060 --> 00:46:38,450
things, or subtract two things,
or multiply two

839
00:46:38,450 --> 00:46:41,180
things, or divide two things.

840
00:46:41,180 --> 00:46:44,140


841
00:46:44,140 --> 00:46:47,930
And underneath that there's
an abstraction barrier.

842
00:46:47,930 --> 00:46:50,510
And underneath this barrier,
is, say, a

843
00:46:50,510 --> 00:46:52,850
complex arithmetic package.

844
00:46:52,850 --> 00:46:55,110
And you can say, add two
complex numbers.

845
00:46:55,110 --> 00:46:57,540
Or you might also have--
remember we did a rational

846
00:46:57,540 --> 00:47:00,190
number package-- you might
have that sitting there.

847
00:47:00,190 --> 00:47:03,950
And there might be
a rational thing.

848
00:47:03,950 --> 00:47:07,760
And the rational number package,
well, has the things

849
00:47:07,760 --> 00:47:08,320
we implemented.

850
00:47:08,320 --> 00:47:15,490
Plus rat, and times
rat, and so on.

851
00:47:15,490 --> 00:47:17,010
Or you might have ordinary
Lisp numbers.

852
00:47:17,010 --> 00:47:19,310
You might say add
three and four.

853
00:47:19,310 --> 00:47:29,030
So we might have ordinary
numbers, in which case we have

854
00:47:29,030 --> 00:47:36,670
the Lisp supplied plus, and
minus, and times, and slash.

855
00:47:36,670 --> 00:47:39,840
OK, so we might imagine this
complex number system sitting

856
00:47:39,840 --> 00:47:43,660
in a more complicated generic
operator structure at

857
00:47:43,660 --> 00:47:44,910
the next level up.

858
00:47:44,910 --> 00:47:47,730


859
00:47:47,730 --> 00:47:49,050
Well how can we make that?

860
00:47:49,050 --> 00:47:50,240
We already have the
idea, we're just

861
00:47:50,240 --> 00:47:52,780
going to do it again.

862
00:47:52,780 --> 00:47:54,720
We've implemented a rational
number package.

863
00:47:54,720 --> 00:47:56,650
Let's look at how it
has to be changed.

864
00:47:56,650 --> 00:48:01,590


865
00:48:01,590 --> 00:48:02,660
In fact, at this level
it doesn't have to

866
00:48:02,660 --> 00:48:03,730
be changed at all.

867
00:48:03,730 --> 00:48:07,180
This is exactly the code that
we wrote last time.

868
00:48:07,180 --> 00:48:10,140
To add two rational
numbers, remember

869
00:48:10,140 --> 00:48:11,140
there was this formula.

870
00:48:11,140 --> 00:48:14,980
You make a rational number
whose numerator--

871
00:48:14,980 --> 00:48:17,330
the numerator of the first times
the denominator of the

872
00:48:17,330 --> 00:48:20,486
second, plus the denominator
of the first times the

873
00:48:20,486 --> 00:48:21,520
numerator of the second.

874
00:48:21,520 --> 00:48:25,760
And who's denominator is the
product of the denominators.

875
00:48:25,760 --> 00:48:30,580
And minus rat, and star
rat, and slash rat.

876
00:48:30,580 --> 00:48:34,420
And this is exactly the rational
number package that

877
00:48:34,420 --> 00:48:36,310
we made before.

878
00:48:36,310 --> 00:48:38,390
We're ignoring the GCD problem,
but let's not worry

879
00:48:38,390 --> 00:48:40,240
about that.

880
00:48:40,240 --> 00:48:42,980
As implementers of this rational
number package, how

881
00:48:42,980 --> 00:48:45,570
do we install it in the generic
arithmetic system?

882
00:48:45,570 --> 00:48:46,820
Well that's easy.

883
00:48:46,820 --> 00:48:48,980


884
00:48:48,980 --> 00:48:51,840
There's only one thing we
have to do differently.

885
00:48:51,840 --> 00:48:56,270
Whereas previously we said that
to make a rational number

886
00:48:56,270 --> 00:49:00,960
you built a pair of the
numerator and denominator,

887
00:49:00,960 --> 00:49:03,300
here we'll not only build the
pair, but we'll sign it.

888
00:49:03,300 --> 00:49:06,120
We'll attach the
type rational.

889
00:49:06,120 --> 00:49:08,940
That's the only thing we have
to do different, make it a

890
00:49:08,940 --> 00:49:12,380
typed data object.

891
00:49:12,380 --> 00:49:14,500
And now we'll stick our
operations in the table.

892
00:49:14,500 --> 00:49:18,920
We'll put under the symbol
rational and the operation add

893
00:49:18,920 --> 00:49:21,820
our procedure, plus rat.

894
00:49:21,820 --> 00:49:23,580
And, again, note this
is a symbol.

895
00:49:23,580 --> 00:49:23,930
Right?

896
00:49:23,930 --> 00:49:26,830
Quote, unquote, but the actual
thing we're putting in the

897
00:49:26,830 --> 00:49:30,060
table is the procedure.

898
00:49:30,060 --> 00:49:33,700
And for how to subtract,
well you subtract

899
00:49:33,700 --> 00:49:38,270
rationals with minus rat.

900
00:49:38,270 --> 00:49:41,090
And multiply, and divide.

901
00:49:41,090 --> 00:49:43,640
And that is exactly and
precisely what we have to do

902
00:49:43,640 --> 00:49:48,510
to fit inside this generic
arithmetic system.

903
00:49:48,510 --> 00:49:51,560
Well how does the whole
thing work?

904
00:49:51,560 --> 00:50:00,170
See, what we want to do is have
some generic operators.

905
00:50:00,170 --> 00:50:01,720
Have add and sub and
[UNINTELLIGIBLE]

906
00:50:01,720 --> 00:50:03,990
be generic operators.

907
00:50:03,990 --> 00:50:18,930
So we're going to define add and
say, to add x and y, that

908
00:50:18,930 --> 00:50:21,840
will be operate--

909
00:50:21,840 --> 00:50:26,080


910
00:50:26,080 --> 00:50:27,490
we were going to call
it operate-2.

911
00:50:27,490 --> 00:50:30,350
This is our operator procedure,
but set up for two

912
00:50:30,350 --> 00:50:37,261
arguments using add
on x and y.

913
00:50:37,261 --> 00:50:40,420
And so this is the analog
to operate.

914
00:50:40,420 --> 00:50:41,680
Let's look at the
code for second.

915
00:50:41,680 --> 00:50:42,930
It's almost like operate.

916
00:50:42,930 --> 00:50:46,040


917
00:50:46,040 --> 00:50:51,550
To operate with some operator
on an argument 1 and an

918
00:50:51,550 --> 00:50:56,370
argument 2, well the first thing
we're going to do is

919
00:50:56,370 --> 00:51:01,900
check and see if the two
arguments have the same type.

920
00:51:01,900 --> 00:51:06,610
So we'll say, is the type of the
first argument the same as

921
00:51:06,610 --> 00:51:07,860
the type of the second
argument?

922
00:51:07,860 --> 00:51:10,350


923
00:51:10,350 --> 00:51:15,070
And if they're not, we'll go
off and complain, and say,

924
00:51:15,070 --> 00:51:15,670
that's an error.

925
00:51:15,670 --> 00:51:19,140
We don't know how to do that.

926
00:51:19,140 --> 00:51:20,920
If they do have the same
type, we'll do

927
00:51:20,920 --> 00:51:22,080
exactly what we did before.

928
00:51:22,080 --> 00:51:26,460
We'll go look and filed under
the type of the argument--

929
00:51:26,460 --> 00:51:30,420
arg 1 and arg 2 have the same
type, so it doesn't matter.

930
00:51:30,420 --> 00:51:33,640
So we'll look in the table,
find the procedure.

931
00:51:33,640 --> 00:51:38,870
If there is a procedure there,
then we'll apply it to the

932
00:51:38,870 --> 00:51:43,030
contents of the argument 1 and
the contents of arg 2.

933
00:51:43,030 --> 00:51:44,760
And otherwise we'll
say, error.

934
00:51:44,760 --> 00:51:46,890
Undefined operator.

935
00:51:46,890 --> 00:51:48,140
And so there's operate-2.

936
00:51:48,140 --> 00:51:51,326


937
00:51:51,326 --> 00:51:55,160
And that's all we have to do.

938
00:51:55,160 --> 00:51:57,640
We just built the complex
number package before.

939
00:51:57,640 --> 00:52:00,140
How do we embed that complex
number package in

940
00:52:00,140 --> 00:52:02,140
this generic system?

941
00:52:02,140 --> 00:52:03,390
Almost the same.

942
00:52:03,390 --> 00:52:06,410


943
00:52:06,410 --> 00:52:11,060
We make a procedure called
make-complex that takes

944
00:52:11,060 --> 00:52:14,100
whatever George and Martha
hand to us and add the

945
00:52:14,100 --> 00:52:15,350
type-complex.

946
00:52:15,350 --> 00:52:18,170


947
00:52:18,170 --> 00:52:25,840
And then we say, to add complex
numbers, plus complex,

948
00:52:25,840 --> 00:52:32,240
we use our internal procedure,
plus c, and attach a type,

949
00:52:32,240 --> 00:52:33,490
make that a complex number.

950
00:52:33,490 --> 00:52:37,560


951
00:52:37,560 --> 00:52:42,840
So our original package had
names plus c and minus c that

952
00:52:42,840 --> 00:52:45,250
we're using to communicate
with George and Martha.

953
00:52:45,250 --> 00:52:47,730
And then to communicate with the
outside world, we have a

954
00:52:47,730 --> 00:52:52,380
thing called plus-complex
and minus-complex.

955
00:52:52,380 --> 00:52:55,920


956
00:52:55,920 --> 00:52:56,530
And so on.

957
00:52:56,530 --> 00:52:59,000
And the only difference
is that these return

958
00:52:59,000 --> 00:53:01,120
values that are tight.

959
00:53:01,120 --> 00:53:02,850
So they can be looked
at up here.

960
00:53:02,850 --> 00:53:04,690
And these are internal
operations.

961
00:53:04,690 --> 00:53:09,250


962
00:53:09,250 --> 00:53:10,680
Let's go look at that
slide again.

963
00:53:10,680 --> 00:53:13,740
There's one more thing we do.

964
00:53:13,740 --> 00:53:19,280
After defining plus-complex, we
put under the type complex

965
00:53:19,280 --> 00:53:23,200
and the symbol add, that
procedure plus complex.

966
00:53:23,200 --> 00:53:27,130
And then similarly for
subtracting complex numbers,

967
00:53:27,130 --> 00:53:29,130
and multiplying them,
and dividing them.

968
00:53:29,130 --> 00:53:31,700


969
00:53:31,700 --> 00:53:35,250
OK, how do we install
ordinary numbers?

970
00:53:35,250 --> 00:53:38,160
Exactly the same way.

971
00:53:38,160 --> 00:53:40,500
Come off and say, well we'll
make a thing called

972
00:53:40,500 --> 00:53:41,750
make-number.

973
00:53:41,750 --> 00:53:44,340


974
00:53:44,340 --> 00:53:48,500
Make-number takes a number and
attaches a type, which is the

975
00:53:48,500 --> 00:53:50,260
symbol number.

976
00:53:50,260 --> 00:53:55,300
We build a procedure called
plus-number, which is simply,

977
00:53:55,300 --> 00:53:59,220
add the two things using the
ordinary addition, because in

978
00:53:59,220 --> 00:54:01,850
this case we're talking about
ordinary numbers, and attach a

979
00:54:01,850 --> 00:54:04,510
type to it and make
that a number.

980
00:54:04,510 --> 00:54:08,700
And then we put into the table
under the symbol number and

981
00:54:08,700 --> 00:54:12,550
the operation add, this
procedure plus-number, and

982
00:54:12,550 --> 00:54:15,360
then the same thing for
subtracting, and multiplying,

983
00:54:15,360 --> 00:54:16,610
and dividing.

984
00:54:16,610 --> 00:54:22,750


985
00:54:22,750 --> 00:54:26,060
Let's look at an example,
just to make it clear.

986
00:54:26,060 --> 00:54:32,600
Suppose, for instance,
I'm going

987
00:54:32,600 --> 00:54:34,150
to perform the operation.

988
00:54:34,150 --> 00:54:38,220
So I sit up here and I'm going
to perform the operation,

989
00:54:38,220 --> 00:54:40,930
which looks like multiplying
two complex numbers.

990
00:54:40,930 --> 00:54:49,786
So I would multiply, say,
3 plus 4i and 2 plus 6i.

991
00:54:49,786 --> 00:54:51,740
And that's something that
I might want to take

992
00:54:51,740 --> 00:54:52,840
hand that to mul.

993
00:54:52,840 --> 00:54:57,170
I'll write mul as my generic
operator here.

994
00:54:57,170 --> 00:54:58,280
How's that going to work?

995
00:54:58,280 --> 00:55:05,020
Well 3 plus 4i, say, sits in
the system at this level as

996
00:55:05,020 --> 00:55:06,250
something that looks
like this.

997
00:55:06,250 --> 00:55:08,280
Let's say it was one
of George's.

998
00:55:08,280 --> 00:55:14,695
So it would have a 3 and a 4.

999
00:55:14,695 --> 00:55:18,490


1000
00:55:18,490 --> 00:55:25,330
And attached to that would be
George's type, which would say

1001
00:55:25,330 --> 00:55:29,510
rectangular, it came
from George.

1002
00:55:29,510 --> 00:55:31,230
And attached to that--

1003
00:55:31,230 --> 00:55:35,630
and this itself would be the
data view from the next level

1004
00:55:35,630 --> 00:55:37,700
up, which it is--

1005
00:55:37,700 --> 00:55:41,030
so that itself would be a
type-data object which would

1006
00:55:41,030 --> 00:55:42,280
say complex.

1007
00:55:42,280 --> 00:55:44,820


1008
00:55:44,820 --> 00:55:49,240
So that's what this object would
look like up here at the

1009
00:55:49,240 --> 00:55:52,300
very highest level, where
the really super-generic

1010
00:55:52,300 --> 00:55:55,560
operations are looking at it.

1011
00:55:55,560 --> 00:55:58,220
Now what happens, mul
eventually's going to come

1012
00:55:58,220 --> 00:56:00,400
along and say, oh,
what's it's type?

1013
00:56:00,400 --> 00:56:01,650
It's type is complex.

1014
00:56:01,650 --> 00:56:04,270


1015
00:56:04,270 --> 00:56:08,460
Go through to operate-2 and say,
oh, what I want to do is

1016
00:56:08,460 --> 00:56:10,440
apply what's in the table,
which is going to be the

1017
00:56:10,440 --> 00:56:17,150
procedure star complex, on
this thing with the type

1018
00:56:17,150 --> 00:56:17,950
stripped off.

1019
00:56:17,950 --> 00:56:22,400
So it's going to strip off the
type, take that much, and send

1020
00:56:22,400 --> 00:56:26,288
that down into the
complex world.

1021
00:56:26,288 --> 00:56:28,950
The complex world looks at its
operations and says, oh, I

1022
00:56:28,950 --> 00:56:31,280
have to apply star c.

1023
00:56:31,280 --> 00:56:34,490
Star c might say, oh, at some
point I want to look at the

1024
00:56:34,490 --> 00:56:39,420
magnitude of this object that
it's in, that it's got.

1025
00:56:39,420 --> 00:56:40,160
And they'll say, oh, it's

1026
00:56:40,160 --> 00:56:41,870
rectangular, it's one of George's.

1027
00:56:41,870 --> 00:56:47,340
So it'll then strip off the next
version of type, and hand

1028
00:56:47,340 --> 00:56:52,160
that down to George to take
the magnitude of.

1029
00:56:52,160 --> 00:56:55,290
So you see what's going
on is that there are

1030
00:56:55,290 --> 00:56:59,320
these chains of types.

1031
00:56:59,320 --> 00:57:01,530
And the length of the chain is
sort of the number of levels

1032
00:57:01,530 --> 00:57:05,090
that you're going to be going
up in this table.

1033
00:57:05,090 --> 00:57:09,590
And what a type tells you, every
time you have a vertical

1034
00:57:09,590 --> 00:57:12,350
barrier in this table, where
there's some ambiguity about

1035
00:57:12,350 --> 00:57:15,010
where you should go down to the
next level, the type is

1036
00:57:15,010 --> 00:57:17,440
telling you where to go.

1037
00:57:17,440 --> 00:57:19,950
And then everybody at the
bottom, as they construct data

1038
00:57:19,950 --> 00:57:22,810
and filter it up, they stick
their type back on.

1039
00:57:22,810 --> 00:57:25,350


1040
00:57:25,350 --> 00:57:30,750
So that's the general structure
of the system.

1041
00:57:30,750 --> 00:57:33,410


1042
00:57:33,410 --> 00:57:34,820
OK.

1043
00:57:34,820 --> 00:57:38,660
Now that we've got this, let's
go and make this thing even

1044
00:57:38,660 --> 00:57:39,910
more complex.

1045
00:57:39,910 --> 00:57:41,890


1046
00:57:41,890 --> 00:57:46,150
Let's talk about adding to the
system not only these kinds of

1047
00:57:46,150 --> 00:57:49,680
numbers, but it's also
meaningful to start talking

1048
00:57:49,680 --> 00:57:51,510
about adding polynomials.

1049
00:57:51,510 --> 00:57:53,360
Might do arithmetic
on polynomials.

1050
00:57:53,360 --> 00:57:57,570
Like we could have x to the
fifteenth plus 2x to the

1051
00:57:57,570 --> 00:58:04,480
seventh plus 5.

1052
00:58:04,480 --> 00:58:06,380
That might be some polynomial.

1053
00:58:06,380 --> 00:58:08,720
And if we have two such gadgets
we can add them or

1054
00:58:08,720 --> 00:58:10,530
multiply them.

1055
00:58:10,530 --> 00:58:12,140
Let's not worry about
dividing them.

1056
00:58:12,140 --> 00:58:15,870
Just add them, multiply them,
then we'll subtract them.

1057
00:58:15,870 --> 00:58:16,660
What do we have to do?

1058
00:58:16,660 --> 00:58:21,830
Well let's think about how we
might represent a polynomial.

1059
00:58:21,830 --> 00:58:24,950
It's going to be some
typed data object.

1060
00:58:24,950 --> 00:58:29,690
So let's say a polynomial to
this system might look like a

1061
00:58:29,690 --> 00:58:32,000
thing that starts with
the type polynomial.

1062
00:58:32,000 --> 00:58:33,710
And then maybe it says the
next thing is what

1063
00:58:33,710 --> 00:58:34,550
variable its in.

1064
00:58:34,550 --> 00:58:38,960
So I might say I'm a polynomial
in the variable x.

1065
00:58:38,960 --> 00:58:40,500
And then it'll have some
information about

1066
00:58:40,500 --> 00:58:42,250
what the terms are.

1067
00:58:42,250 --> 00:58:45,620
And there're just tons of ways
to do this, but one way is to

1068
00:58:45,620 --> 00:58:51,520
say we're going to have a thing
called a term-list. And

1069
00:58:51,520 --> 00:58:53,700
a term-list--

1070
00:58:53,700 --> 00:58:54,830
well, in our case we'll
use something

1071
00:58:54,830 --> 00:58:56,360
that looks like this.

1072
00:58:56,360 --> 00:58:59,010
We'll make it a bunch of pairs
which have an order in a

1073
00:58:59,010 --> 00:58:59,690
coefficient.

1074
00:58:59,690 --> 00:59:09,070
So this polynomial would be
represented by this term-list.

1075
00:59:09,070 --> 00:59:12,910
And what that means is that
this polynomial starts off

1076
00:59:12,910 --> 00:59:19,710
with a term of order 15
and coefficient 1.

1077
00:59:19,710 --> 00:59:23,820


1078
00:59:23,820 --> 00:59:26,780
And the next thing in it is
a term of order 7 and

1079
00:59:26,780 --> 00:59:29,680
coefficient 2, a term of order
0, which is constant in

1080
00:59:29,680 --> 00:59:31,450
coefficient 5.

1081
00:59:31,450 --> 00:59:35,600
And there are lots and lots of
ways, and lots and lots of

1082
00:59:35,600 --> 00:59:37,890
trade-offs when you really think
about making algebraic

1083
00:59:37,890 --> 00:59:40,570
manipulation packages about
exactly how you should

1084
00:59:40,570 --> 00:59:41,730
represent these things.

1085
00:59:41,730 --> 00:59:44,180
But this is a fairly
standard one.

1086
00:59:44,180 --> 00:59:47,770
It's useful in a lot
of contexts.

1087
00:59:47,770 --> 00:59:50,815
OK, well how do we implement
our polynomial arithmetic?

1088
00:59:50,815 --> 00:59:54,270


1089
00:59:54,270 --> 00:59:55,520
Let's start out.

1090
00:59:55,520 --> 00:59:57,950


1091
00:59:57,950 --> 01:00:00,760
What we'll do to make
a polynomial--

1092
01:00:00,760 --> 01:00:05,690
we'll first have a way
to make polynomials.

1093
01:00:05,690 --> 01:00:08,560
We're going to make a polynomial
out of variable

1094
01:00:08,560 --> 01:00:13,180
like x and term-list. And all
that does is we'll package

1095
01:00:13,180 --> 01:00:14,290
them together someway.

1096
01:00:14,290 --> 01:00:18,740
We'll put the variable together
with the term list

1097
01:00:18,740 --> 01:00:21,380
using cons, and then attached
to that the type polynomial.

1098
01:00:21,380 --> 01:00:26,270


1099
01:00:26,270 --> 01:00:29,280
OK, how do we add
two polynomials?

1100
01:00:29,280 --> 01:00:33,330
To add a polynomial, p1 and
p2, and then just for

1101
01:00:33,330 --> 01:00:36,060
simplicity let's say
we will only add

1102
01:00:36,060 --> 01:00:37,380
things in the same variable.

1103
01:00:37,380 --> 01:00:40,740
So if they have the same
variable, and same variable

1104
01:00:40,740 --> 01:00:43,160
here is going to be some
selector we write, whose

1105
01:00:43,160 --> 01:00:45,150
details we don't care about.

1106
01:00:45,150 --> 01:00:48,280
If the two polynomials have the
same variable, then we'll

1107
01:00:48,280 --> 01:00:48,810
do something.

1108
01:00:48,810 --> 01:00:52,350
If they don't have the same
variable, we'll give an error,

1109
01:00:52,350 --> 01:00:55,480
polynomials not in the
same variable.

1110
01:00:55,480 --> 01:00:58,120
And if they do have the same
variable, what we'll do is

1111
01:00:58,120 --> 01:01:01,130
we'll make a polynomial whose
variable is whatever that

1112
01:01:01,130 --> 01:01:05,570
variable is, and whose term-list
is something we'll

1113
01:01:05,570 --> 01:01:10,170
call sum-terms. Plus terms will
add the two term lists.

1114
01:01:10,170 --> 01:01:13,500
So we'll add the two term
lists to the polynomial.

1115
01:01:13,500 --> 01:01:16,755
That'll give us a term-list.
We'll add on, we'll say it's a

1116
01:01:16,755 --> 01:01:19,500
polynomial in the variable
with that

1117
01:01:19,500 --> 01:01:22,550
term-list. That's plus poly.

1118
01:01:22,550 --> 01:01:26,360
And then we're going to put in
our table under the type

1119
01:01:26,360 --> 01:01:30,520
polynomial, add them
using plus poly.

1120
01:01:30,520 --> 01:01:31,750
And of course we really
haven't done much.

1121
01:01:31,750 --> 01:01:34,360
What we've really done is pushed
all the work onto this

1122
01:01:34,360 --> 01:01:38,480
thing, plus-terms, which is
supposed to add term-lists.

1123
01:01:38,480 --> 01:01:40,920
Let's look at that.

1124
01:01:40,920 --> 01:01:48,900
Here's an overview of how we
might add two term-lists.

1125
01:01:48,900 --> 01:01:51,860
So L1 and L2 were going
to be two term-lists.

1126
01:01:51,860 --> 01:01:55,700
And a term-list is a bunch of
pairs, coefficient in order.

1127
01:01:55,700 --> 01:01:56,950
And it's a big case analysis.

1128
01:01:56,950 --> 01:01:59,860


1129
01:01:59,860 --> 01:02:03,470
And the first thing we'll check
for and see if there are

1130
01:02:03,470 --> 01:02:07,020
any terms. We're going to
recursively work down these

1131
01:02:07,020 --> 01:02:09,980
term-lists, so eventually we'll
get to a place where

1132
01:02:09,980 --> 01:02:12,270
either L1 or L2 might
be empty.

1133
01:02:12,270 --> 01:02:15,160
And if either one is empty,
our answer will

1134
01:02:15,160 --> 01:02:15,850
be the other one.

1135
01:02:15,850 --> 01:02:20,720
So if L1 is empty we'll return
L2, and if L2 is empty

1136
01:02:20,720 --> 01:02:23,470
we'll return L1.

1137
01:02:23,470 --> 01:02:27,220
Otherwise there are sort of
three interesting cases.

1138
01:02:27,220 --> 01:02:30,560
What we're going to do is grab
the first term in each of

1139
01:02:30,560 --> 01:02:37,660
those lists, called t1 and t2.

1140
01:02:37,660 --> 01:02:43,090
And we're going to look at
three cases, depending on

1141
01:02:43,090 --> 01:02:47,230
whether the order of t1 is
greater than the order of t2,

1142
01:02:47,230 --> 01:02:50,470
or less than t2, or the same.

1143
01:02:50,470 --> 01:02:53,290


1144
01:02:53,290 --> 01:02:54,910
Those are the three cases
we're going to look at.

1145
01:02:54,910 --> 01:02:56,160
Let's look at this case.

1146
01:02:56,160 --> 01:02:58,640


1147
01:02:58,640 --> 01:03:03,550
If the order of t1 is greater
than the order of t2, then

1148
01:03:03,550 --> 01:03:08,280
what that means is that our
answer is going to start with

1149
01:03:08,280 --> 01:03:11,480
this term of the order of t1.

1150
01:03:11,480 --> 01:03:14,455
Because it won't combine with
any lower order terms. So what

1151
01:03:14,455 --> 01:03:19,720
we do is add the lower order
terms. We recursively add

1152
01:03:19,720 --> 01:03:21,900
together all the terms
in the rest of the

1153
01:03:21,900 --> 01:03:26,880
term-list in L1 and L2.

1154
01:03:26,880 --> 01:03:30,120
That's going to be the lower
order terms of the answer.

1155
01:03:30,120 --> 01:03:31,490
And then we're going to
adjoin to that the

1156
01:03:31,490 --> 01:03:33,180
highest order term.

1157
01:03:33,180 --> 01:03:35,120
And I'm using here a whole bunch
of procedures I haven't

1158
01:03:35,120 --> 01:03:39,360
defined, like a adjoin-term, and
rest-terms, and selectors

1159
01:03:39,360 --> 01:03:41,410
that get order.

1160
01:03:41,410 --> 01:03:44,730
But you can imagine
what those are.

1161
01:03:44,730 --> 01:03:48,550
So if the first term-list has
a higher order than the

1162
01:03:48,550 --> 01:03:51,830
second, we recursively add all
the lower terms and then stick

1163
01:03:51,830 --> 01:03:55,540
on that last term.

1164
01:03:55,540 --> 01:03:56,890
The other case, the same way.

1165
01:03:56,890 --> 01:04:05,400
If the first term has a smaller
order, well then we

1166
01:04:05,400 --> 01:04:07,740
add the first term-list and the
rest of the terms in the

1167
01:04:07,740 --> 01:04:11,430
second one, and adjoin on
this highest order term.

1168
01:04:11,430 --> 01:04:14,570


1169
01:04:14,570 --> 01:04:16,660
So so far nothing's much
happened, we've just sort of

1170
01:04:16,660 --> 01:04:19,700
pushed this thing off into
adding lower order terms. The

1171
01:04:19,700 --> 01:04:22,870
last case where you actually get
to a coefficients that you

1172
01:04:22,870 --> 01:04:24,240
have to add, this will
be the case where

1173
01:04:24,240 --> 01:04:27,240
the orders are equal.

1174
01:04:27,240 --> 01:04:30,340
What we do is, well again
recursively add the lower

1175
01:04:30,340 --> 01:04:33,460
order terms. But now we have to
really combine something.

1176
01:04:33,460 --> 01:04:38,960
What we do is we make a term
whose order is the order of

1177
01:04:38,960 --> 01:04:40,820
the term we're looking at.

1178
01:04:40,820 --> 01:04:44,320
By now t1 and t2 have
the same order.

1179
01:04:44,320 --> 01:04:45,090
That's its order.

1180
01:04:45,090 --> 01:04:50,400
And its coefficient is gotten
by adding the coefficient of

1181
01:04:50,400 --> 01:04:52,230
t1 and the coefficient of t2.

1182
01:04:52,230 --> 01:04:56,360


1183
01:04:56,360 --> 01:04:59,800
This is a big recursive working
down of terms, but

1184
01:04:59,800 --> 01:05:03,070
really there's only one
interesting symbol in this

1185
01:05:03,070 --> 01:05:05,900
procedure, only one
interesting idea.

1186
01:05:05,900 --> 01:05:08,500
The interesting idea
is this add.

1187
01:05:08,500 --> 01:05:12,390


1188
01:05:12,390 --> 01:05:15,330
And the reason that's
interesting is because

1189
01:05:15,330 --> 01:05:18,220
something completely wonderful
just happened.

1190
01:05:18,220 --> 01:05:25,440
We reduced adding polynomials,
not to sort of plus, but to

1191
01:05:25,440 --> 01:05:28,820
the generic add.

1192
01:05:28,820 --> 01:05:33,270
In other words, by implementing
it that way, not

1193
01:05:33,270 --> 01:05:37,530
only do we have our system where
we can have rational

1194
01:05:37,530 --> 01:05:42,090
numbers, or complex numbers,
or ordinary numbers, we've

1195
01:05:42,090 --> 01:05:43,340
just added on polynomials.

1196
01:05:43,340 --> 01:05:48,520


1197
01:05:48,520 --> 01:05:51,820
But the coefficients of the
polynomials can be anything

1198
01:05:51,820 --> 01:05:53,590
that the system can add.

1199
01:05:53,590 --> 01:05:57,450
So these could be polynomials
whose coefficients are

1200
01:05:57,450 --> 01:06:04,110
rational numbers or complex
numbers, which in turn could

1201
01:06:04,110 --> 01:06:11,250
be either rectangular, or polar,
or ordinary numbers.

1202
01:06:11,250 --> 01:06:19,860


1203
01:06:19,860 --> 01:06:23,460
So what I mean precisely
is our system right now

1204
01:06:23,460 --> 01:06:30,200
automatically can handle things
like adding together

1205
01:06:30,200 --> 01:06:35,830
polynomials that have this one:
2/3 of x squared plus

1206
01:06:35,830 --> 01:06:40,940
5/17 x plus 11/4.

1207
01:06:40,940 --> 01:06:44,210
Or automatically handle
polynomials that look like 3

1208
01:06:44,210 --> 01:06:54,160
plus 2i times x to the fifth
plus 4 plus 7i, or something.

1209
01:06:54,160 --> 01:06:56,210
You can automatically
handle those things.

1210
01:06:56,210 --> 01:06:57,820
Why is that?

1211
01:06:57,820 --> 01:07:03,280
That's merely because, or
profoundly because we reduced

1212
01:07:03,280 --> 01:07:06,790
adding polynomials to adding
their coefficients.

1213
01:07:06,790 --> 01:07:09,670
And adding coefficients was
done by the generic add

1214
01:07:09,670 --> 01:07:12,970
operator, which said, I don't
care what your types are as

1215
01:07:12,970 --> 01:07:15,170
long as I know how to add you.

1216
01:07:15,170 --> 01:07:17,800
So automatically for
free we get the

1217
01:07:17,800 --> 01:07:20,880
ability to handle that.

1218
01:07:20,880 --> 01:07:24,920
What's even better than that,
because remember one of the

1219
01:07:24,920 --> 01:07:29,870
things we did is we put into the
table that the way you add

1220
01:07:29,870 --> 01:07:34,660
polynomials is using
plus poly.

1221
01:07:34,660 --> 01:07:37,480
That means that polynomials
themselves are

1222
01:07:37,480 --> 01:07:39,370
things that can be added.

1223
01:07:39,370 --> 01:07:42,110
So for instance let
me write one here.

1224
01:07:42,110 --> 01:07:45,260


1225
01:07:45,260 --> 01:07:46,510
Here's a polynomial.

1226
01:07:46,510 --> 01:07:50,560


1227
01:07:50,560 --> 01:07:55,080
So this gadget here I'm
writing up, this is a

1228
01:07:55,080 --> 01:08:02,710
polynomial in y whose
coefficients are

1229
01:08:02,710 --> 01:08:04,690
polynomials in x.

1230
01:08:04,690 --> 01:08:08,610


1231
01:08:08,610 --> 01:08:13,110
So you see, simply by saying,
polynomials are themselves

1232
01:08:13,110 --> 01:08:15,590
things that can be added, we can
go off and say, well not

1233
01:08:15,590 --> 01:08:19,560
only can we deal with rationals,
or complex, or

1234
01:08:19,560 --> 01:08:22,330
ordinary numbers, but we can
deal with polynomials whose

1235
01:08:22,330 --> 01:08:25,420
coefficients are rationals, or
complex, or ordinary numbers,

1236
01:08:25,420 --> 01:08:31,979
or polynomials whose
coefficients are rationals, or

1237
01:08:31,979 --> 01:08:37,569
complex, rectangular, polar,
or ordinary numbers, or

1238
01:08:37,569 --> 01:08:42,609
polynomials whose coefficients
are rationals, complex, or

1239
01:08:42,609 --> 01:08:43,670
ordinary numbers.

1240
01:08:43,670 --> 01:08:45,950
And so on, and so
on, and so on.

1241
01:08:45,950 --> 01:08:50,830
So this is sort of an infinite
or maybe a recursive tower of

1242
01:08:50,830 --> 01:08:53,880
types that we've built up.

1243
01:08:53,880 --> 01:08:56,420
And it's all exactly from
that one little symbol.

1244
01:08:56,420 --> 01:08:59,615
A-D-D. Writing "add" instead
of "plus" in

1245
01:08:59,615 --> 01:09:02,270
the polynomial thing.

1246
01:09:02,270 --> 01:09:04,620
Slightly different way to
think about it is that

1247
01:09:04,620 --> 01:09:08,740
polynomials are a constructor
for types.

1248
01:09:08,740 --> 01:09:12,149
Namely you give it a type, like
integer, and it returns

1249
01:09:12,149 --> 01:09:16,279
for you polynomials in x whose
coefficients are integers.

1250
01:09:16,279 --> 01:09:20,010
And the important thing about
that is that the operations on

1251
01:09:20,010 --> 01:09:22,729
polynomials reduce to the
operations on the

1252
01:09:22,729 --> 01:09:23,500
coefficients.

1253
01:09:23,500 --> 01:09:25,840
And there are a lot of
things like that.

1254
01:09:25,840 --> 01:09:28,870
So for example, let's go back
and rational numbers.

1255
01:09:28,870 --> 01:09:32,410
We thought about rational
numbers as an integer over an

1256
01:09:32,410 --> 01:09:34,229
integer, but there's
the general notion

1257
01:09:34,229 --> 01:09:36,240
of a rational object.

1258
01:09:36,240 --> 01:09:43,010
Like we might think about 3x
plus 7 over x squared plus 1.

1259
01:09:43,010 --> 01:09:47,430
That's general rational object
whose numerator and

1260
01:09:47,430 --> 01:09:50,310
denominator are polynomials.

1261
01:09:50,310 --> 01:09:52,990
And to add two of them we use
the same formula, numerator

1262
01:09:52,990 --> 01:09:55,720
times denominator plus
denominator times numerator

1263
01:09:55,720 --> 01:09:57,290
over product of denominators.

1264
01:09:57,290 --> 01:09:59,430
How could we install
that in our system?

1265
01:09:59,430 --> 01:10:01,820
Well here's our original
rational

1266
01:10:01,820 --> 01:10:04,250
number arithmetic package.

1267
01:10:04,250 --> 01:10:08,660
And all we have to do in order
to make the entire system

1268
01:10:08,660 --> 01:10:12,530
continue working with general
rational objects, is replace

1269
01:10:12,530 --> 01:10:16,480
these particular pluses and
stars by the generic operator.

1270
01:10:16,480 --> 01:10:19,870
So if we simply change that
procedure to this one, here

1271
01:10:19,870 --> 01:10:23,100
we've changed plus and star
to add a mul, those are

1272
01:10:23,100 --> 01:10:28,170
absolutely the only change,
then suddenly our entire

1273
01:10:28,170 --> 01:10:34,000
system can start talking about
objects that look like this.

1274
01:10:34,000 --> 01:10:40,350
So for example, here is a
rational object whose

1275
01:10:40,350 --> 01:10:44,030
numerator is a polynomial in
x whose coefficients are

1276
01:10:44,030 --> 01:10:47,350
rational numbers.

1277
01:10:47,350 --> 01:10:53,740
Or here is a rational object
whose numerator is polynomials

1278
01:10:53,740 --> 01:11:00,480
in x whose coefficients are
rational objects constructed

1279
01:11:00,480 --> 01:11:03,390
out of complex numbers.

1280
01:11:03,390 --> 01:11:04,850
And then there are a lot of
other things like that.

1281
01:11:04,850 --> 01:11:07,500
See, whenever you have a thing
where the operations reduce to

1282
01:11:07,500 --> 01:11:10,450
operations on the pieces,
another example would be two

1283
01:11:10,450 --> 01:11:12,310
by two matrices.

1284
01:11:12,310 --> 01:11:17,030
I have the idea, there might
be a matrix here of general

1285
01:11:17,030 --> 01:11:18,650
things that I don't
care about.

1286
01:11:18,650 --> 01:11:25,180
But if I add two of them, the
answer over here is gotten by

1287
01:11:25,180 --> 01:11:29,030
adding this one and that one,
however they like to add.

1288
01:11:29,030 --> 01:11:31,110
So I can implement that
the same way.

1289
01:11:31,110 --> 01:11:33,520
And if I do that, then again
suddenly my system can start

1290
01:11:33,520 --> 01:11:35,480
handling things like this.

1291
01:11:35,480 --> 01:11:39,460
So here's a matrix whose
elements happen to be--

1292
01:11:39,460 --> 01:11:43,330
we'll say this element here
is a rational object whose

1293
01:11:43,330 --> 01:11:47,230
numerator and denominators
are polynomials.

1294
01:11:47,230 --> 01:11:49,510
And all that comes for free.

1295
01:11:49,510 --> 01:11:52,580


1296
01:11:52,580 --> 01:11:53,920
What's really going on here?

1297
01:11:53,920 --> 01:11:58,910
What's really going on is
getting rid of this manager

1298
01:11:58,910 --> 01:12:02,060
who's sitting there poking his
nose into who everybody's

1299
01:12:02,060 --> 01:12:03,120
business is.

1300
01:12:03,120 --> 01:12:05,900
We built a system that has
decentralized control.

1301
01:12:05,900 --> 01:12:14,350


1302
01:12:14,350 --> 01:12:18,340
So when you come into and no
one's poking around saying,

1303
01:12:18,340 --> 01:12:21,080
gee, are you in the
official list of

1304
01:12:21,080 --> 01:12:22,440
people who can be added?

1305
01:12:22,440 --> 01:12:24,850
Rather you say, well go off
and add yourself how your

1306
01:12:24,850 --> 01:12:27,810
parts like to be added.

1307
01:12:27,810 --> 01:12:31,030
And the result of that is you
can get this very, very, very

1308
01:12:31,030 --> 01:12:33,870
complex hierarchy where a lot
of things just get done and

1309
01:12:33,870 --> 01:12:36,482
rooted to the right place
automatically.

1310
01:12:36,482 --> 01:12:37,732
Let's stop for questions.

1311
01:12:37,732 --> 01:12:40,380


1312
01:12:40,380 --> 01:12:43,020
AUDIENCE: You say you
get this for free.

1313
01:12:43,020 --> 01:12:46,920
One thing that strikes me is
that now you've lost kind of

1314
01:12:46,920 --> 01:12:50,150
the cleanness of the break
between what's on top and

1315
01:12:50,150 --> 01:12:50,910
what's underneath.

1316
01:12:50,910 --> 01:12:52,770
In other words, now you're
defining some of the

1317
01:12:52,770 --> 01:12:54,850
lower-level procedures
in terms of things

1318
01:12:54,850 --> 01:12:56,610
above their own line.

1319
01:12:56,610 --> 01:13:00,350
Isn't that dangerous?

1320
01:13:00,350 --> 01:13:05,440
Or, if nothing more, a little
less structured?

1321
01:13:05,440 --> 01:13:06,125
PROFESSOR: No, I--

1322
01:13:06,125 --> 01:13:07,770
the question is whether that's
less structured.

1323
01:13:07,770 --> 01:13:08,690
Depends on what you
mean by structure.

1324
01:13:08,690 --> 01:13:11,050
All this is doing
is recursion.

1325
01:13:11,050 --> 01:13:15,780
See, it's saying that the
way you add these

1326
01:13:15,780 --> 01:13:18,640
guys is to use that.

1327
01:13:18,640 --> 01:13:20,520
And that's not less structured,
it's just a

1328
01:13:20,520 --> 01:13:22,610
recursive structure.

1329
01:13:22,610 --> 01:13:24,730
So I don't think it's
particularly any less clean.

1330
01:13:24,730 --> 01:13:27,250
AUDIENCE: Now when you want to
change the multiplier or the

1331
01:13:27,250 --> 01:13:31,380
add operator, suddenly you've
got tremendous consequences

1332
01:13:31,380 --> 01:13:34,480
underneath that you're not
even sure the extent of.

1333
01:13:34,480 --> 01:13:37,080
PROFESSOR: That's right, but
it depends what you mean.

1334
01:13:37,080 --> 01:13:38,470
See, this goes both ways.

1335
01:13:38,470 --> 01:13:41,790


1336
01:13:41,790 --> 01:13:44,690
What would be a good example?

1337
01:13:44,690 --> 01:13:47,500
I ignored greatest common
divisor, for instance.

1338
01:13:47,500 --> 01:13:50,280
I ignored that problem just to
keep the example simple.

1339
01:13:50,280 --> 01:13:59,820
But if I suddenly decided that
plus rat here should do a GCD

1340
01:13:59,820 --> 01:14:04,750
computation and install that,
then that immediately becomes

1341
01:14:04,750 --> 01:14:08,280
available to all of these, to
that guy, and that guy, and

1342
01:14:08,280 --> 01:14:11,560
that guy, and all
the way down.

1343
01:14:11,560 --> 01:14:13,890
So it depends what you mean by
the coherence of your system.

1344
01:14:13,890 --> 01:14:17,030
It's certainly true that you
might want to have a special

1345
01:14:17,030 --> 01:14:18,950
different one that didn't
filter down through the

1346
01:14:18,950 --> 01:14:21,400
coefficients, but the nice thing
about this particular

1347
01:14:21,400 --> 01:14:25,440
example is that mostly you do.

1348
01:14:25,440 --> 01:14:27,630
AUDIENCE: Isn't that the
problem, I think, that you're

1349
01:14:27,630 --> 01:14:32,950
getting to tied in with the fact
that the structuring, the

1350
01:14:32,950 --> 01:14:36,330
recursiveness of that
structuring there is actually

1351
01:14:36,330 --> 01:14:40,340
in execution as opposed to just
definition of the actual

1352
01:14:40,340 --> 01:14:41,590
types themselves?

1353
01:14:41,590 --> 01:14:44,680


1354
01:14:44,680 --> 01:14:46,120
PROFESSOR: I think I understand
the question.

1355
01:14:46,120 --> 01:14:48,650
The point is that these types
evolve and get more and more

1356
01:14:48,650 --> 01:14:50,400
complex as the thing's
actually running.

1357
01:14:50,400 --> 01:14:50,730
Is that what--

1358
01:14:50,730 --> 01:14:50,990
AUDIENCE: Yes.

1359
01:14:50,990 --> 01:14:51,790
As it's running.

1360
01:14:51,790 --> 01:14:51,956
PROFESSOR: --what
you're saying?

1361
01:14:51,956 --> 01:14:52,090
Yes, the point is--

1362
01:14:52,090 --> 01:14:54,180
AUDIENCE: As opposed to
the basic definitions.

1363
01:14:54,180 --> 01:14:54,830
PROFESSOR: Right.

1364
01:14:54,830 --> 01:14:57,210
The type structure is
sort of recursive.

1365
01:14:57,210 --> 01:15:02,770
It's not that you can make this
finite list of the actual

1366
01:15:02,770 --> 01:15:04,850
things they might look like
before the system runs.

1367
01:15:04,850 --> 01:15:06,780
It's something that evolves.

1368
01:15:06,780 --> 01:15:09,610
So if you want to specify that
system, you have to do in some

1369
01:15:09,610 --> 01:15:12,275
other way than by this finite
list. You have to do it by a

1370
01:15:12,275 --> 01:15:13,670
recursive structure.

1371
01:15:13,670 --> 01:15:16,960
AUDIENCE: Because the basic
structure of the types is

1372
01:15:16,960 --> 01:15:17,900
pretty clean and simple.

1373
01:15:17,900 --> 01:15:20,125
PROFESSOR: Right.

1374
01:15:20,125 --> 01:15:21,460
Yes?

1375
01:15:21,460 --> 01:15:22,870
AUDIENCE: I have a question.

1376
01:15:22,870 --> 01:15:25,980
I understand once you have your
data structure set up,

1377
01:15:25,980 --> 01:15:29,230
how it pulls off complex and
passes that down, and then

1378
01:15:29,230 --> 01:15:30,640
pulls off rect, passes
that down.

1379
01:15:30,640 --> 01:15:32,790
But if you're just a user and
you don't know anything about

1380
01:15:32,790 --> 01:15:35,610
rect or polar or whatever, how
do you initially set up that

1381
01:15:35,610 --> 01:15:37,330
data structure so that
everything goes

1382
01:15:37,330 --> 01:15:38,390
to the right spot?

1383
01:15:38,390 --> 01:15:41,210
If I just have the equation over
there on the left and I

1384
01:15:41,210 --> 01:15:42,500
just want to add, multiply
complex numbers--

1385
01:15:42,500 --> 01:15:43,640
PROFESSOR: Well that's
the wonderful thing.

1386
01:15:43,640 --> 01:15:47,730
If you're just a user
you say "mul."

1387
01:15:47,730 --> 01:15:50,280
AUDIENCE: And it figures out
that I mean complex numbers?

1388
01:15:50,280 --> 01:15:51,420
Or how do I tell it
that I want--

1389
01:15:51,420 --> 01:15:51,950
PROFESSOR: Well you're
going to have in your

1390
01:15:51,950 --> 01:15:53,050
hands complex numbers.

1391
01:15:53,050 --> 01:15:56,490
See what you would have at some
level, as a real user, is

1392
01:15:56,490 --> 01:15:58,370
a constructor for
complex numbers.

1393
01:15:58,370 --> 01:15:59,470
AUDIENCE: So then I have to
make complex numbers?

1394
01:15:59,470 --> 01:16:00,350
PROFESSOR: So you have
to make them.

1395
01:16:00,350 --> 01:16:03,180
What you would probably have as
a user is some little thing

1396
01:16:03,180 --> 01:16:07,390
in the reader loop, which would
give you some plausible

1397
01:16:07,390 --> 01:16:09,850
way to type in a complex
number, in

1398
01:16:09,850 --> 01:16:11,590
whatever format you like.

1399
01:16:11,590 --> 01:16:14,360
Or it might be that you're
never typing them in.

1400
01:16:14,360 --> 01:16:16,170
Someone's just handing
you a complex number.

1401
01:16:16,170 --> 01:16:19,500
AUDIENCE: OK, so if I had a
complex number that had a

1402
01:16:19,500 --> 01:16:21,505
polynomial in it, I'd have to
make my polynomial and then

1403
01:16:21,505 --> 01:16:21,960
make my complex number.

1404
01:16:21,960 --> 01:16:24,220
PROFESSOR: Right if you wanted
it constructed from scratch.

1405
01:16:24,220 --> 01:16:25,710
At some point you construct
them from scratch.

1406
01:16:25,710 --> 01:16:27,880
But what you don't have to know
of that is when you have

1407
01:16:27,880 --> 01:16:32,345
the object you can just say
"mul." And it'll multiply.

1408
01:16:32,345 --> 01:16:33,279
Yeah?

1409
01:16:33,279 --> 01:16:36,450
AUDIENCE: I think the question
that was being posed here is,

1410
01:16:36,450 --> 01:16:40,220
say if I want to change my
presentation of complexes, or

1411
01:16:40,220 --> 01:16:46,330
some operation of complex, how
much real code I will have to

1412
01:16:46,330 --> 01:16:49,860
gets around with, or change
to change it in

1413
01:16:49,860 --> 01:16:52,270
one specific operation?

1414
01:16:52,270 --> 01:16:53,490
PROFESSOR: [UNINTELLIGIBLE]
what you have to change.

1415
01:16:53,490 --> 01:16:54,690
And the point is that
you only have to

1416
01:16:54,690 --> 01:16:56,070
change what you're changing.

1417
01:16:56,070 --> 01:17:00,320
See if Martha decides that
she would rather--

1418
01:17:00,320 --> 01:17:01,440
let's see something silly--

1419
01:17:01,440 --> 01:17:04,040
like change the order
in the pair.

1420
01:17:04,040 --> 01:17:09,700
Like angle and magnitude in
the other order, she just

1421
01:17:09,700 --> 01:17:10,970
makes that change locally.

1422
01:17:10,970 --> 01:17:12,750
And the whole thing will
propagate through the system

1423
01:17:12,750 --> 01:17:14,790
in the right way.

1424
01:17:14,790 --> 01:17:18,040
Or if suddenly you said, gee, I
have another representation

1425
01:17:18,040 --> 01:17:19,700
for rationals.

1426
01:17:19,700 --> 01:17:22,740
And I'm going to stick it
here, by filing those

1427
01:17:22,740 --> 01:17:24,820
operations in the table.

1428
01:17:24,820 --> 01:17:27,220
Then suddenly all of these
polynomials whose coefficients

1429
01:17:27,220 --> 01:17:29,240
are coefficients of
coefficients, or whatever,

1430
01:17:29,240 --> 01:17:32,970
also can automatically have
available that representation.

1431
01:17:32,970 --> 01:17:35,952
That's the power of this
particular one.

1432
01:17:35,952 --> 01:17:37,625
AUDIENCE: I'm not sure if I can
even pose an intelligent

1433
01:17:37,625 --> 01:17:38,700
sounding question.

1434
01:17:38,700 --> 01:17:42,920
But somehow this whole thing
went really nicely to this

1435
01:17:42,920 --> 01:17:44,910
beautiful finish where
all the things seemed

1436
01:17:44,910 --> 01:17:47,280
to fall into place.

1437
01:17:47,280 --> 01:17:48,530
Sort of seemed a little
contrived.

1438
01:17:48,530 --> 01:17:50,930


1439
01:17:50,930 --> 01:17:52,670
That's all for the sake,
I'm sure, of teaching.

1440
01:17:52,670 --> 01:17:55,100
I doubt that the guys
who first did this--

1441
01:17:55,100 --> 01:17:56,510
and I could be wrong--

1442
01:17:56,510 --> 01:17:59,200
figured it all out so that when
they just all put it all

1443
01:17:59,200 --> 01:18:02,410
together, you could all of the
sudden, blam, do any kind of

1444
01:18:02,410 --> 01:18:04,860
arithmetic on any
kind of object.

1445
01:18:04,860 --> 01:18:07,930
It seems like maybe they had
to play with it for a while

1446
01:18:07,930 --> 01:18:11,800
and had to bash it
and rework it.

1447
01:18:11,800 --> 01:18:14,120
And it seems like that's the
kind of problem we're really

1448
01:18:14,120 --> 01:18:16,540
faced with we start trying to
design a really complex

1449
01:18:16,540 --> 01:18:19,390
system, is having lots of
different kinds of parts and

1450
01:18:19,390 --> 01:18:22,730
not even knowing what kinds of
operations we're going to want

1451
01:18:22,730 --> 01:18:24,620
to do on those parts.

1452
01:18:24,620 --> 01:18:27,580
How to organize the operations
in this nice way so that no

1453
01:18:27,580 --> 01:18:29,630
matter what you do, when you
start putting them together

1454
01:18:29,630 --> 01:18:31,700
everything starts falling
out for free.

1455
01:18:31,700 --> 01:18:33,090
PROFESSOR: OK, well
that's certainly a

1456
01:18:33,090 --> 01:18:34,340
very intelligent question.

1457
01:18:34,340 --> 01:18:37,020


1458
01:18:37,020 --> 01:18:40,560
One part is this is a very good
methodology that people

1459
01:18:40,560 --> 01:18:44,590
have discovered a lot coming
from symbolic algebra.

1460
01:18:44,590 --> 01:18:47,590
Because there are a lot
of complications.

1461
01:18:47,590 --> 01:18:50,710
To allow you to implement these
things before you decide

1462
01:18:50,710 --> 01:18:52,130
what you want all the
operations to

1463
01:18:52,130 --> 01:18:53,310
be, and all of that.

1464
01:18:53,310 --> 01:18:55,580
So in some sense it's an
answer that people have

1465
01:18:55,580 --> 01:18:58,560
discovered by wading
through this stuff.

1466
01:18:58,560 --> 01:19:02,160
In another sense, it is a
very contrived example.

1467
01:19:02,160 --> 01:19:06,240
AUDIENCE: It seems like to be
able to do this you do have to

1468
01:19:06,240 --> 01:19:08,320
wade through it for a certain
amount of time before you can

1469
01:19:08,320 --> 01:19:09,010
become good at it.

1470
01:19:09,010 --> 01:19:12,220
PROFESSOR: Let me show you how
terribly contrived this is.

1471
01:19:12,220 --> 01:19:14,130
So you can write all these
wonderful things.

1472
01:19:14,130 --> 01:19:17,600
But the system that I wrote
here, and if we had another

1473
01:19:17,600 --> 01:19:19,820
half an hour to give this
lecture I would have given

1474
01:19:19,820 --> 01:19:23,470
this part of it, which says,
notice that it breaks down if

1475
01:19:23,470 --> 01:19:30,880
I tell it to do something as
foolish as add 3 plus 7/2.

1476
01:19:30,880 --> 01:19:33,980
Because what will happen is
you'll get to operate-2, and

1477
01:19:33,980 --> 01:19:36,180
operate-2 will say, oh
this is type number,

1478
01:19:36,180 --> 01:19:37,560
and that's type rational.

1479
01:19:37,560 --> 01:19:38,810
I don't know how to add them.

1480
01:19:38,810 --> 01:19:41,530


1481
01:19:41,530 --> 01:19:43,600
So you'd like the system at
least to be able to say

1482
01:19:43,600 --> 01:19:48,660
something like, gee,
before you do that

1483
01:19:48,660 --> 01:19:50,480
change that to 3/1.

1484
01:19:50,480 --> 01:19:52,250
Turn it into a rational number,
hand that to the

1485
01:19:52,250 --> 01:19:53,500
rational package.

1486
01:19:53,500 --> 01:19:55,510


1487
01:19:55,510 --> 01:19:58,860
That's the thing I didn't talk
about in this lecture.

1488
01:19:58,860 --> 01:20:00,880
It's a little bit in the book,
which talks about the problem

1489
01:20:00,880 --> 01:20:03,390
of what's called coercion.

1490
01:20:03,390 --> 01:20:05,310
Where you wanted--

1491
01:20:05,310 --> 01:20:08,280
see, having so carefully set
up all of these types as

1492
01:20:08,280 --> 01:20:11,720
distinct objects, a lot of times
you want to also put in

1493
01:20:11,720 --> 01:20:16,650
knowledge about how to view
an ordinary number

1494
01:20:16,650 --> 01:20:19,110
as a kind of rational.

1495
01:20:19,110 --> 01:20:21,620
Or view an ordinary number
as a kind of complex.

1496
01:20:21,620 --> 01:20:24,580
That's where the complexity in
the system really starts

1497
01:20:24,580 --> 01:20:27,110
happening, where you talk
about, see where

1498
01:20:27,110 --> 01:20:28,420
do I put that knowledge?

1499
01:20:28,420 --> 01:20:30,810
Is it rational to know that
ordinary numbers might be

1500
01:20:30,810 --> 01:20:33,130
pieces of [UNINTELLIGIBLE]
of them?

1501
01:20:33,130 --> 01:20:38,790
Or they're terrible, terrible
examples, like if I might want

1502
01:20:38,790 --> 01:20:47,510
to add a complex number
to a rational number.

1503
01:20:47,510 --> 01:20:50,080


1504
01:20:50,080 --> 01:20:50,760
Bad example.

1505
01:20:50,760 --> 01:20:52,010
5/7.

1506
01:20:52,010 --> 01:20:53,860


1507
01:20:53,860 --> 01:20:57,300
Then somebody's got to know that
I have to convert these

1508
01:20:57,300 --> 01:20:59,790
to another type, which is
complex numbers whose parts

1509
01:20:59,790 --> 01:21:01,540
might be rationals.

1510
01:21:01,540 --> 01:21:02,680
And who worries about that?

1511
01:21:02,680 --> 01:21:03,950
Does complex worry about that?

1512
01:21:03,950 --> 01:21:05,030
Does rational worry
about that?

1513
01:21:05,030 --> 01:21:06,900
Does plus worry about that?

1514
01:21:06,900 --> 01:21:08,520
That's where the real
complexity comes in.

1515
01:21:08,520 --> 01:21:11,380
And that's where it's pretty
well sorted out.

1516
01:21:11,380 --> 01:21:14,810
And a lot of, in fact, all of
this message passing stuff was

1517
01:21:14,810 --> 01:21:18,460
motivated by problems
like this.

1518
01:21:18,460 --> 01:21:21,630
And when you really push it,
people are-- somehow the

1519
01:21:21,630 --> 01:21:25,330
algebraic manipulation problem
seems to be so complex that

1520
01:21:25,330 --> 01:21:27,410
the people who are always at the
edge of it are exactly in

1521
01:21:27,410 --> 01:21:28,050
the state you said.

1522
01:21:28,050 --> 01:21:29,940
They're wading through this
thing, mucking around, seeing

1523
01:21:29,940 --> 01:21:33,470
what they use, trying
to distill stuff.

1524
01:21:33,470 --> 01:21:36,030
AUDIENCE: I just want to come
back to this issue of

1525
01:21:36,030 --> 01:21:39,250
complexity once more.

1526
01:21:39,250 --> 01:21:44,550
It certainly seems to be true
that you have a great deal of

1527
01:21:44,550 --> 01:21:49,580
flexibility in altering the
lower level kinds of things.

1528
01:21:49,580 --> 01:21:54,320
But it is true that you are,
in a sense, freezing higher

1529
01:21:54,320 --> 01:21:55,450
level operations.

1530
01:21:55,450 --> 01:21:58,510
Or at least if you change them
you don't know where all of

1531
01:21:58,510 --> 01:22:02,060
the changes are going to show
up, or how they are.

1532
01:22:02,060 --> 01:22:04,840
PROFESSOR: OK, that's an
extremely good question.

1533
01:22:04,840 --> 01:22:10,130
What I have to do is, if I
decide there's a new general

1534
01:22:10,130 --> 01:22:16,300
operation called equality test,
then all of these people

1535
01:22:16,300 --> 01:22:19,835
have to decide whether or not
they would like to have an

1536
01:22:19,835 --> 01:22:24,650
equality test by looking
in the table.

1537
01:22:24,650 --> 01:22:27,870
There're ways to decentralize
it even more.

1538
01:22:27,870 --> 01:22:31,430
That's what I sort of hinted at
last time, where I said you

1539
01:22:31,430 --> 01:22:34,240
could not only have this type as
a symbol, but you actually

1540
01:22:34,240 --> 01:22:37,850
might store in each object
the operations

1541
01:22:37,850 --> 01:22:40,450
that it knows of that.

1542
01:22:40,450 --> 01:22:44,670
So you might have things like
greatest common divisor, which

1543
01:22:44,670 --> 01:22:47,540
is a thing here which is defined
only for integers, and

1544
01:22:47,540 --> 01:22:51,030
not in general for
rational numbers.

1545
01:22:51,030 --> 01:22:53,110
So it might be a very, very
fragmented system.

1546
01:22:53,110 --> 01:22:56,570
And then depending on where
you want your flexibility,

1547
01:22:56,570 --> 01:22:58,190
there's a whole spectrum
of places that you

1548
01:22:58,190 --> 01:22:59,960
can build that in.

1549
01:22:59,960 --> 01:23:02,320
But you're pointing at the place
where this starts being

1550
01:23:02,320 --> 01:23:04,540
weak, that there has to be some
agreement on top here

1551
01:23:04,540 --> 01:23:06,370
about these general
operations.

1552
01:23:06,370 --> 01:23:08,390
Or at least people have
to think about them.

1553
01:23:08,390 --> 01:23:10,340
Or you might decide, you might
have a table that's very

1554
01:23:10,340 --> 01:23:14,010
sparse, that only has
a few things in it.

1555
01:23:14,010 --> 01:23:15,490
But there are lot of ways
to play that game.

1556
01:23:15,490 --> 01:23:19,780


1557
01:23:19,780 --> 01:23:21,030
OK, thank you.

1558
01:23:21,030 --> 01:23:23,534


1559
01:23:23,534 --> 01:23:23,849
[MUSIC: "JESU, JOY OF
MAN'S DESIRING" BY

1560
01:23:23,849 --> 01:23:25,099
JOHANN SEBASTIAN BACH]

1561
01:23:25,099 --> 01:23:36,682