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PROFESSOR: Hi, welcome
to Calculus Part 2,

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where our theme for
the entire course

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will essentially be functions
of several variables.

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But a more underlying theme,
a theme which will not only

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permeate this
course, but virtually

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every course in the mathematics
curriculum, and perhaps

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other curricula as
well, is the idea

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of what we mean by a
mathematical structure.

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00:00:58,920 --> 00:01:03,350
And if this sounds a little
bit ominous and frightening,

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00:01:03,350 --> 00:01:05,910
the idea of a
mathematical structure

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00:01:05,910 --> 00:01:10,030
can be compared very
nicely in terms of a game,

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00:01:10,030 --> 00:01:12,180
and for this reason,
I have chosen

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to entitle our first lesson
"The 'Game' of Mathematics".

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00:01:17,960 --> 00:01:19,880
"The 'Game' of Mathematics."

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00:01:19,880 --> 00:01:23,480
And let's emphasize
the word "game" here.

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00:01:23,480 --> 00:01:26,590
We do not mean game
in a trivial sense,

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00:01:26,590 --> 00:01:30,260
where in elementary school, the
first row races the second row

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to see who finishes the
addition problem first.

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00:01:32,960 --> 00:01:36,830
The idea that we want to
talk about is what is a game.

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00:01:36,830 --> 00:01:38,850
And I remember an
old riddle when

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I was in about the fourth or
fifth grade, when somebody said

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to me: "what is it that looks
like a box, smells like cheese,

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and flies?"

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And the answer was
a flying cheese box.

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00:01:49,970 --> 00:01:51,940
And the interesting
point is that this

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00:01:51,940 --> 00:01:55,970
is how, in the scientific world,
we often make up definitions.

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00:01:55,970 --> 00:01:59,590
Namely, to define a
game, we try to think

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00:01:59,590 --> 00:02:04,020
of every single ingredient
that is common to every game,

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00:02:04,020 --> 00:02:06,240
and then we roll all
of these ingredients

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00:02:06,240 --> 00:02:08,550
into one long
definition, and that

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00:02:08,550 --> 00:02:10,949
becomes our final definition.

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00:02:10,949 --> 00:02:16,060
And with that in mind, let
me take the following tack.

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00:02:16,060 --> 00:02:19,890
And again, let me point out that
I am going through this rather

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00:02:19,890 --> 00:02:23,740
hurriedly because the
main aim of the lesson

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00:02:23,740 --> 00:02:26,520
is to give an
overview, with the idea

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00:02:26,520 --> 00:02:31,260
that the supplementary notes
and the exercises in the unit

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00:02:31,260 --> 00:02:34,570
will give you the computational
drill that you need.

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00:02:34,570 --> 00:02:38,930
But for a first approximation,
let us say that, in any game,

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00:02:38,930 --> 00:02:43,950
you must have definitions, so to
know what terminology you have.

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00:02:43,950 --> 00:02:46,650
Then you must have
rules of the game,

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00:02:46,650 --> 00:02:48,630
and notice that the
rules of the game

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00:02:48,630 --> 00:02:51,800
are relationships
between the terms.

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00:02:51,800 --> 00:02:54,460
Oh, as a trivial example,
in playing cards,

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there are many
different card games

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that can be played with
the same deck of cards.

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00:02:59,850 --> 00:03:04,100
What makes the game
different are not

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00:03:04,100 --> 00:03:07,520
the definitions involved,
but the rules of the game.

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00:03:07,520 --> 00:03:10,010
And finally, there
are objectives.

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00:03:10,010 --> 00:03:12,650
Well, obviously, the objective
of any game is to win.

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What we really mean
by an objective

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is the art of carrying
out a winning situation

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by successfully employing the
definitions and the rules.

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00:03:24,550 --> 00:03:28,220
And the way we do that is
usually called strategy.

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Strategy is the art of using
the definitions and rules

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to carry out the objective
in an inescapable manner.

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And the reason I like this
particular little setup

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is it gives me a
way to show you,

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in juxtaposition, the role
of rote verses reason, logic

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verses memory, in any
mathematical situation

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or real-life situation.

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Namely, the strategy
part of a game is logic,

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00:03:55,230 --> 00:03:58,532
and things like the
definitions and the rules

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are things that we memorize.

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00:04:03,160 --> 00:04:05,820
OK, now, the thing
that we're going

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00:04:05,820 --> 00:04:09,770
to do, in our particular
course, is always come back

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to this particular structure.

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In other words, our
definition of a game

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is any system which consists
of definitions, rules,

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and objectives, where the
objective is carried out

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as an inescapable consequence
of the definitions and the rules

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by means of strategy.

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And by the way, don't
take this lightly.

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This is a very serious topic.

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Later in the course,
the computation

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will become
sufficiently difficult

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00:04:37,660 --> 00:04:41,030
that we may lose sight of the
forest because of the trees.

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00:04:41,030 --> 00:04:44,080
Right now, what I want to
do is emphasize this to you

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00:04:44,080 --> 00:04:47,060
in terms of topics that
you're already familiar with,

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00:04:47,060 --> 00:04:49,220
so that you can see what
the overall structure

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00:04:49,220 --> 00:04:51,260
of mathematics is,
so that you will not

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00:04:51,260 --> 00:04:53,490
be preoccupied with
this when we're learning

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00:04:53,490 --> 00:04:55,290
more computational things.

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00:04:55,290 --> 00:04:58,140
Let me just take
a look at, well,

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00:04:58,140 --> 00:05:00,980
a relatively trivial example.

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00:05:00,980 --> 00:05:02,930
I call this a new
look at counting.

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00:05:02,930 --> 00:05:05,130
We all know how to count.

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00:05:05,130 --> 00:05:07,390
We start with a number
called 1, and we

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have various definitions, 1
plus 1 is 2, 2 plus 1 is 3,

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00:05:12,400 --> 00:05:18,550
3 plus 1 is 4, 4 plus 1 is
5, 5 plus 1 is 6, 6 plus 1

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00:05:18,550 --> 00:05:20,090
is 7, et cetera.

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00:05:20,090 --> 00:05:22,480
OK, so far, so good.

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Now, we ask the question:
"how much is 4 plus 3?"

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00:05:26,940 --> 00:05:31,130
Now, obviously, we all
know that 4 plus 3 is 7.

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00:05:31,130 --> 00:05:34,140
We're not saying what are we
looking at this problem for.

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What we're trying to show now
is a very important aspect

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of the game of mathematics.

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00:05:39,620 --> 00:05:42,570
You see, notice that in
our list of definitions,

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00:05:42,570 --> 00:05:46,340
no place do we have the
sum 4 plus 3 defined.

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What we are interested
in, now, is not

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so much the truthful
statement that 4 plus 3 is 7,

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00:05:52,160 --> 00:05:55,370
but whether the result
4 plus 3 equals 7

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00:05:55,370 --> 00:06:00,260
follows inescapably from the
definitions that we've listed.

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00:06:00,260 --> 00:06:03,510
Now, you see, the point is all
we've listed are definitions.

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00:06:03,510 --> 00:06:07,220
We haven't told any particular
rules of the game yet.

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00:06:07,220 --> 00:06:10,620
Well, let's make up some rules
as we go along, and as I say,

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we'll discuss these in
more details in our notes

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and in the exercises.

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00:06:14,960 --> 00:06:17,020
We essentially do
something like this.

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We write down 4 plus
3, then we say, OK, we

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do know how to add
by ones; that's

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00:06:24,940 --> 00:06:26,400
what our definition says.

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00:06:26,400 --> 00:06:29,090
So we rewrite 3 as 2 plus 1.

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00:06:29,090 --> 00:06:32,990
In other words, we substitute
2 plus 1, which is equal to 3

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00:06:32,990 --> 00:06:34,560
by definition.

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00:06:34,560 --> 00:06:39,860
Then we say, OK, 2 plus 1
is the same as 1 plus 2.

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00:06:39,860 --> 00:06:42,100
Now, by the way, notice
the tacit assumption

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that we're making.

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00:06:42,950 --> 00:06:45,970
We're assuming that the order
in which you add two numbers

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00:06:45,970 --> 00:06:47,330
makes no difference.

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00:06:47,330 --> 00:06:50,430
Obviously, this is not a
rule in every game of life.

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00:06:50,430 --> 00:06:53,400
In most things in life,
order does make a difference.

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00:06:53,400 --> 00:06:55,360
Consider, for example,
the statements

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00:06:55,360 --> 00:06:59,065
first I undress, and then I take
a shower; or first I shower,

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00:06:59,065 --> 00:07:00,430
and then I undress.

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00:07:00,430 --> 00:07:04,000
Without meaning to pass
judgment as to which is proper,

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00:07:04,000 --> 00:07:06,899
at least notice there is a
difference between the two.

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00:07:06,899 --> 00:07:08,690
What we're saying is
that, somehow or other

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00:07:08,690 --> 00:07:10,690
in the game of
arithmetic, we assume

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00:07:10,690 --> 00:07:13,820
that addition has the property
that the order in which you add

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00:07:13,820 --> 00:07:15,060
makes no difference.

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00:07:15,060 --> 00:07:18,630
So we say, OK, let's accept
that as a rule of the game.

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00:07:18,630 --> 00:07:21,200
If we accept that as a
rule the game, 2 plus 1

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00:07:21,200 --> 00:07:24,880
could then be substituted
for by 1 plus 2.

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00:07:24,880 --> 00:07:26,900
We make the
additional assumption

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00:07:26,900 --> 00:07:28,780
that, when you
add three numbers,

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the answer does not depend
on voice inflection.

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In other words that
4 plus 1 plus 2

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00:07:34,610 --> 00:07:38,400
is equal to 4 plus 1 plus 2.

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00:07:38,400 --> 00:07:40,450
And the strategy
behind doing that

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is that we know that another
name for 4 plus 1 is 5.

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00:07:44,070 --> 00:07:47,880
In other words, we now arrive
at the fact that 4 plus 3

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is equal to 5 plus 2.

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We now rewrite 2 as
1 plus 1, and we now

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00:07:54,210 --> 00:07:59,340
have the 5 plus 2 is the
same as 5 plus 1 plus 1.

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00:07:59,340 --> 00:08:02,110
We now again use the fact
that voice inflection

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00:08:02,110 --> 00:08:05,230
makes no difference,
and we rewrite this as 5

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00:08:05,230 --> 00:08:08,360
plus 1 plus 1.

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00:08:08,360 --> 00:08:13,230
5 plus 1, we know by
definition, is 6, and 6 plus 1,

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00:08:13,230 --> 00:08:16,190
we know by definition, is seven.

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00:08:16,190 --> 00:08:19,920
In other words, subject to the
rules that we've talked about

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00:08:19,920 --> 00:08:23,400
implicitly here but have not
stated in our game format

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00:08:23,400 --> 00:08:26,560
explicitly, what
we have shown is

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that if we accept certain rules,
it follows from our definitions

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that 4 plus 3 equals 7 is
an inescapable conclusion.

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Notice that the inescapability
of the conclusion

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hinges on the fact that
we've accepted certain rules.

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00:08:44,350 --> 00:08:47,840
If we change the rules, we
can change the conclusions.

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00:08:47,840 --> 00:08:49,800
In other words,
this thing called

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drawing inescapable conclusions
is something called validity,

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00:08:54,000 --> 00:08:58,490
and validity involves the art of
drawing inescapable conclusions

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00:08:58,490 --> 00:09:02,010
using given rules and
given definitions.

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00:09:02,010 --> 00:09:04,500
We'll talk about
that in more detail,

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00:09:04,500 --> 00:09:07,380
but for now, notice
the difference here.

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00:09:07,380 --> 00:09:13,470
Before we did this, we knew as
a conjecture, a past experience

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00:09:13,470 --> 00:09:17,150
thing, that 4 plus 3 equals
7 is a true statement.

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00:09:17,150 --> 00:09:21,310
What we now know is in terms
of certain rules of the game,

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00:09:21,310 --> 00:09:25,020
coupled with the definitions
that we've accepted, 4 plus 3

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00:09:25,020 --> 00:09:29,160
equals 7 is an inescapable
conclusion, henceforth to be

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called a "theorem" in our game.

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00:09:32,250 --> 00:09:35,250
Let's see if we can't get
away from this rather simple

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00:09:35,250 --> 00:09:38,180
example, and again, let me
emphasize that as simple

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00:09:38,180 --> 00:09:40,860
as this example is,
throughout our course,

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00:09:40,860 --> 00:09:44,050
we will be using the
same technique, only

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00:09:44,050 --> 00:09:48,080
at a more sophisticated
level of computational skill.

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00:09:48,080 --> 00:09:52,040
But let's take a look here and
see what we're really saying.

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00:09:52,040 --> 00:09:56,310
What mathematical
structure really involves

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00:09:56,310 --> 00:09:59,200
is a logic machine
type of thing.

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00:09:59,200 --> 00:10:02,280
We have a logic
machine, a machine

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00:10:02,280 --> 00:10:06,150
that, being fed any kinds
of definitions, rules,

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00:10:06,150 --> 00:10:12,340
assumptions, et cetera, grinds
these things through and has,

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00:10:12,340 --> 00:10:16,450
as its output,
inescapable conclusions.

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00:10:16,450 --> 00:10:20,570
In other words, you feed in
definitions, rules, et cetera,

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00:10:20,570 --> 00:10:23,070
which by the way,
don't really have

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00:10:23,070 --> 00:10:26,150
to be true in the real-life
sense of being true.

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00:10:26,150 --> 00:10:28,030
I mean, for example,
in the baseball game,

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00:10:28,030 --> 00:10:30,120
to say the rule is
three strikes is an out,

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00:10:30,120 --> 00:10:32,190
there was certainly
no basic truth

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00:10:32,190 --> 00:10:34,000
the said that had
to be the case.

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00:10:34,000 --> 00:10:37,350
What is true is that,
in the world of science,

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00:10:37,350 --> 00:10:39,600
the scientist is the
interpreter of nature.

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00:10:39,600 --> 00:10:43,770
What happens in real life
happens whether the scientist

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00:10:43,770 --> 00:10:44,740
predicts it or not.

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00:10:44,740 --> 00:10:48,720
What he tries to do is to make
up definitions and rules, which

201
00:10:48,720 --> 00:10:51,710
are compatible with
his experience, things

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00:10:51,710 --> 00:10:53,640
which he calls truth.

203
00:10:53,640 --> 00:10:56,650
He feeds those through
his logic machine,

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00:10:56,650 --> 00:11:02,150
draws inescapable conclusions,
and if the conclusions

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00:11:02,150 --> 00:11:05,750
follow inescapably from the
definitions and the rules,

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00:11:05,750 --> 00:11:09,200
we call the resulting
argument valid.

207
00:11:09,200 --> 00:11:12,430
In other words, truth
is a value judgment

208
00:11:12,430 --> 00:11:14,940
that we make about a
particular statement.

209
00:11:14,940 --> 00:11:17,940
Validity is a more
objective thing

210
00:11:17,940 --> 00:11:20,030
that we attribute
to an argument.

211
00:11:20,030 --> 00:11:23,080
In other words, a statement
is either true or false.

212
00:11:23,080 --> 00:11:26,030
An argument is either
valid or invalid,

213
00:11:26,030 --> 00:11:29,040
meaning that we only judge
whether the conclusion

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00:11:29,040 --> 00:11:31,760
of the argument follows
inescapably from the given

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00:11:31,760 --> 00:11:35,030
assumptions, independently
of whether those assumptions

216
00:11:35,030 --> 00:11:37,090
happen to be true or not.

217
00:11:37,090 --> 00:11:40,030
By the way, as an aside,
one of the reasons

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00:11:40,030 --> 00:11:43,790
that the scientist
prefers to dodge issues

219
00:11:43,790 --> 00:11:48,160
such as "what is truth" and
leaves that to the philosopher,

220
00:11:48,160 --> 00:11:51,210
is that truth, in many
cases, is a relative thing.

221
00:11:51,210 --> 00:11:53,670
It's based on the
available knowledge.

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00:11:53,670 --> 00:11:57,070
It's also based on the situation
that we want to handle.

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00:11:57,070 --> 00:12:00,200
In other words, what is truth?

224
00:12:00,200 --> 00:12:04,130
And my claim is that the answer
depends on the situation.

225
00:12:04,130 --> 00:12:07,970
Well, let me give you again a
trivial arithmetic situation.

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00:12:07,970 --> 00:12:15,100
Does 1/2 plus 1/3 equal 5/6,
or does 1/2 plus 1/3 equal 2/5?

227
00:12:15,100 --> 00:12:19,050
And again, the
answer is it depends

228
00:12:19,050 --> 00:12:22,050
on what real-life problem
you're dealing with.

229
00:12:22,050 --> 00:12:26,150
For example, if a person is
working for you by the hour,

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00:12:26,150 --> 00:12:29,460
and he works for you
for a half hour one day

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00:12:29,460 --> 00:12:33,250
and a third of an hour the next
day, the total time he's worked

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00:12:33,250 --> 00:12:34,740
is 5/6 of an hour.

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00:12:34,740 --> 00:12:35,290
Why?

234
00:12:35,290 --> 00:12:37,960
Because this agrees with
our real-life experience

235
00:12:37,960 --> 00:12:42,830
that 30 minutes plus 20
minutes is 50 minutes.

236
00:12:42,830 --> 00:12:45,420
On the other hand,
if a baseball player

237
00:12:45,420 --> 00:12:48,130
goes one for two in the
first game of a doubleheader

238
00:12:48,130 --> 00:12:51,690
and one for three in the
second game of a doubleheader,

239
00:12:51,690 --> 00:12:55,730
he has batted two hits
in five times at bat.

240
00:12:55,730 --> 00:12:57,830
In fact, if you could
convince the public

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00:12:57,830 --> 00:13:00,300
that he had five hits
and six times at bat,

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00:13:00,300 --> 00:13:04,110
you could become a director
of any economy program

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00:13:04,110 --> 00:13:05,550
in the nation, I guess.

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00:13:05,550 --> 00:13:07,340
The point, however, is this.

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In most arithmetic
questions that we deal with,

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we are used to the
physical interpretation

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00:13:12,930 --> 00:13:17,680
in which 1/2 plus 1/3 equals
5/6 reflects the real life

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00:13:17,680 --> 00:13:18,700
situation.

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00:13:18,700 --> 00:13:22,060
Whereas this example here
may seem trivial, namely,

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00:13:22,060 --> 00:13:25,390
how often are you going to be
involved with batting averages

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00:13:25,390 --> 00:13:27,730
unless you're doing
sixth grade arithmetic.

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The point remains, ironically
or whatever you want to call it,

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that this little batting
average problem is not trivial.

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In the world of engineering,
we know this example

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00:13:38,090 --> 00:13:40,820
as the weighted average problem.

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00:13:40,820 --> 00:13:44,940
In other words, you'll notice
that 2 over 5, as a fraction,

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00:13:44,940 --> 00:13:49,750
is more nearly equal to 1/3
that it is equal to 1/2,

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00:13:49,750 --> 00:13:53,250
and the reason for that
is that the player batted

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00:13:53,250 --> 00:13:56,480
at the low average, 1 over
3, one hit in three times

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at bat, for more times at bat.

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In other words, the three has
a heavier weighting factor

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00:14:01,840 --> 00:14:05,810
than the two, and every time
that you use a weighted average

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00:14:05,810 --> 00:14:09,170
in a scientific engineering
oriented investigation,

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this is the truth, not this.

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00:14:12,460 --> 00:14:16,690
In other words, it is neither
true or false that 1/2 plus 1/3

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equals 2/5 or that 1/2
plus 1/3 equals 5/6.

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Which is true depends on the
particular physical situation.

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You see a rather
interesting point here,

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that we sometimes
allow truth to be

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00:14:28,620 --> 00:14:32,420
based on what particular
problem we're trying to solve.

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Another way that
we manipulate truth

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is that we sometimes have a
rule that we like to be true.

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00:14:38,760 --> 00:14:40,420
In fact, I guess
this is probably

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what happens with most
political theories

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00:14:43,680 --> 00:14:45,740
that one starts
with the objectives,

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00:14:45,740 --> 00:14:48,080
knows what it is that
he wants to be true,

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00:14:48,080 --> 00:14:50,590
and then invents the
definitions and the rules

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00:14:50,590 --> 00:14:52,350
to conform with this.

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In much the same way as
around the fourth grade again,

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00:14:55,160 --> 00:14:58,490
we learn such adages as
"Look before you leap,"

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00:14:58,490 --> 00:15:02,030
and two minutes later you learn
"He who hesitates is lost,"

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00:15:02,030 --> 00:15:04,240
and suddenly you come
to the conclusion

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00:15:04,240 --> 00:15:07,950
that you can make your
assumptions validify

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00:15:07,950 --> 00:15:11,130
any conclusion you want just
by choosing your assumptions

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00:15:11,130 --> 00:15:12,180
appropriately.

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00:15:12,180 --> 00:15:14,140
Now, if that sounds
degrading, let

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me show you how it's used
effectively in mathematics.

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00:15:18,730 --> 00:15:22,110
In other words, let me just say
that predetermined rules may

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00:15:22,110 --> 00:15:23,480
control truth.

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00:15:23,480 --> 00:15:25,460
Let me give you an example.

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00:15:25,460 --> 00:15:30,620
Going back to exponents,
why does b to the 0 equal 1?

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00:15:30,620 --> 00:15:33,570
The answer is very
simple, that when

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00:15:33,570 --> 00:15:37,810
we use positive exponents,
positive whole number

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00:15:37,810 --> 00:15:42,810
exponents, we talked about
that many factors of b.

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00:15:42,810 --> 00:15:45,140
And one of the
interesting rules that we

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00:15:45,140 --> 00:15:48,540
saw that was obeyed by
whole number exponents

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00:15:48,540 --> 00:15:51,490
was that if you multiply
b to the m-th power

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00:15:51,490 --> 00:15:55,000
by b to the n-th power,
you got as an answer b

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00:15:55,000 --> 00:15:59,650
to the m plus n power.

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00:15:59,650 --> 00:16:01,390
b to the m plus n.

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00:16:01,390 --> 00:16:06,360
Now, the interesting thing
is that after a while,

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00:16:06,360 --> 00:16:10,440
you never even paid attention
to why this rule worked.

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00:16:10,440 --> 00:16:13,680
What you did know was that
this was a pretty darn

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00:16:13,680 --> 00:16:15,600
convenient rule to use.

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00:16:15,600 --> 00:16:18,630
Computationally,
this rule simplified

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00:16:18,630 --> 00:16:23,380
many particular computations
that you were doing.

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00:16:23,380 --> 00:16:27,460
In particular, then,
as soon as n is 0,

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00:16:27,460 --> 00:16:30,130
you would still like to
be able to use this rule.

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00:16:30,130 --> 00:16:33,120
In other words, we
want this nice rule

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00:16:33,120 --> 00:16:36,430
to apply even when n equals 0.

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00:16:36,430 --> 00:16:39,810
Now, let's look at this from
a computational point of view.

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00:16:39,810 --> 00:16:45,190
If we want this rule to
apply when n equals 0,

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00:16:45,190 --> 00:16:48,160
let's simply
rewrite this exactly

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00:16:48,160 --> 00:16:53,190
as it appears here
with n equal to 0.

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00:16:53,190 --> 00:16:55,820
We then have, what?

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00:16:55,820 --> 00:16:58,600
We have b-- we'll just
repeat everything here.

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00:16:58,600 --> 00:17:01,860
b to the m times b.

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00:17:01,860 --> 00:17:06,079
Now, we're replacing n by
0, so that's b to the 0,

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00:17:06,079 --> 00:17:15,390
and that must equal b to the
m plus n, that's m plus 0.

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00:17:15,390 --> 00:17:17,990
But the interesting
point is that we

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00:17:17,990 --> 00:17:23,089
know how to add numbers, and
for numbers, m plus 0 is m.

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00:17:23,089 --> 00:17:26,849
In other words, this
is still b to the m.

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00:17:26,849 --> 00:17:30,560
Now, we look at this,
and what we're saying

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00:17:30,560 --> 00:17:36,090
is if we want the rule b
to the m times b to the n

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00:17:36,090 --> 00:17:40,710
to equal b to the m plus n
to be true even when n is 0,

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00:17:40,710 --> 00:17:44,730
this says that b to the 0
must be that number such

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00:17:44,730 --> 00:17:49,340
that when we multiply it by b
to the m, we get b to the m.

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00:17:49,340 --> 00:17:51,770
Now, what number
has the property

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00:17:51,770 --> 00:17:55,940
that when you multiply it by
b to the m you get b to the m?

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00:17:55,940 --> 00:17:58,300
And if you're real
quick, you say 1,

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00:17:58,300 --> 00:17:59,970
and if you're
algebra-oriented, you

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00:17:59,970 --> 00:18:05,110
say b to the 0 is therefore
equal b to the m divided

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00:18:05,110 --> 00:18:08,980
by b to the m, and
again you say 1,

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00:18:08,980 --> 00:18:12,890
except of course that b must
be unequal to zero because,

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00:18:12,890 --> 00:18:15,910
hopefully, by this time we
understand why we must never

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00:18:15,910 --> 00:18:18,100
divide by 0.

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00:18:18,100 --> 00:18:20,900
In other words, what we now
have is the old high school

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00:18:20,900 --> 00:18:30,700
rule that b to the 0 equals
1 provided b is unequal to 0.

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00:18:30,700 --> 00:18:32,600
But here's the important point.

340
00:18:32,600 --> 00:18:35,120
If I have never
defined b to the 0,

341
00:18:35,120 --> 00:18:36,880
and somebody says
to me: "make up

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00:18:36,880 --> 00:18:40,540
a definition for b to the
0," and I say, OK, b to the 0

343
00:18:40,540 --> 00:18:44,350
is going to be 36, I have
every right in the world

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00:18:44,350 --> 00:18:45,770
to make up that definition.

345
00:18:45,770 --> 00:18:48,450
What I don't have
a right to do is,

346
00:18:48,450 --> 00:18:51,420
when I'm ever using an
expression like b to the 0,

347
00:18:51,420 --> 00:18:53,780
is to assume that
I have the right

348
00:18:53,780 --> 00:18:56,030
to use this particular recipe.

349
00:18:56,030 --> 00:19:00,040
In other words, if I want to
be able to use the nice recipe,

350
00:19:00,040 --> 00:19:05,290
even when n is 0, I have no
choice but to define b to the 0

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00:19:05,290 --> 00:19:06,840
to equal 1.

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00:19:06,840 --> 00:19:08,990
Therefore, we make
up that definition

353
00:19:08,990 --> 00:19:11,820
because we want that
recipe to apply,

354
00:19:11,820 --> 00:19:14,050
and that's exactly what
we're going to be doing

355
00:19:14,050 --> 00:19:15,420
through most of this course.

356
00:19:15,420 --> 00:19:18,310
We are going to look at
certain real-life situations,

357
00:19:18,310 --> 00:19:21,610
we are going to look at certain
recipes that we want to apply,

358
00:19:21,610 --> 00:19:23,980
and we are going to make
up definitions this way,

359
00:19:23,980 --> 00:19:28,100
and then see how our
inescapable conclusions follow

360
00:19:28,100 --> 00:19:29,760
from these definitions.

361
00:19:29,760 --> 00:19:33,870
More about that will be said
in the remainder of this unit,

362
00:19:33,870 --> 00:19:35,770
and our next
lecture will pick up

363
00:19:35,770 --> 00:19:38,450
the game of mathematics
in a different context.

364
00:19:38,450 --> 00:19:40,300
But until next time, good bye.

365
00:19:46,710 --> 00:19:49,080
Funding for the
publication of this video

366
00:19:49,080 --> 00:19:53,960
was provided by the Gabriella
and Paul Rosenbaum Foundation.

367
00:19:53,960 --> 00:19:58,130
Help OCW continue to provide
free and open access to MIT

368
00:19:58,130 --> 00:20:02,548
courses by making a donation
at ocw.mit.edu/donate.