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PROFESSOR: Hi.
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00:00:33,700 --> 00:00:37,190
Our lecture today concerns
mathematical induction, which,
10
00:00:37,190 --> 00:00:40,490
roughly speaking, is a technique
that one uses to
11
00:00:40,490 --> 00:00:43,520
prove something when one already
has a pretty good
12
00:00:43,520 --> 00:00:46,460
suspicion as to what the
right answer is.
13
00:00:46,460 --> 00:00:48,710
Now, rather than to philosophize
about this too
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00:00:48,710 --> 00:00:52,800
long, let's tear right into a
problem and see, in action,
15
00:00:52,800 --> 00:00:54,700
just what the concept means.
16
00:00:54,700 --> 00:00:57,800
Recall that, in an earlier
lecture, we have proven that
17
00:00:57,800 --> 00:01:01,460
the limit of a sum is equal
to the sum of the limits
18
00:01:01,460 --> 00:01:03,800
provided, of course, that
there are only two
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00:01:03,800 --> 00:01:05,600
terms in the sum.
20
00:01:05,600 --> 00:01:09,470
That is, if we have two
functions, 'f 1' and 'f 2',
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00:01:09,470 --> 00:01:13,180
the limit of 'f1 of x' plus 'f2
of x' as 'x' approaches
22
00:01:13,180 --> 00:01:16,940
'a' is the limit of 'f1 of x' as
'x' approaches 'a' plus the
23
00:01:16,940 --> 00:01:19,710
limit of 'f2 of x' as
'x' approaches 'a'.
24
00:01:19,710 --> 00:01:22,650
The question, now, is how
about the limit of 'x'
25
00:01:22,650 --> 00:01:25,870
approaches 'a' of 'f1
of x', plus 'f2 of
26
00:01:25,870 --> 00:01:28,410
x', plus 'f3 of x'?
27
00:01:28,410 --> 00:01:32,060
Now, to tackle a problem like
this, we do what is so often
28
00:01:32,060 --> 00:01:35,010
done in any mathematical
logical procedure.
29
00:01:35,010 --> 00:01:39,960
We try to reduce an unfamiliar
problem to a familiar problem
30
00:01:39,960 --> 00:01:41,800
which has already been solved.
31
00:01:41,800 --> 00:01:44,220
Let's see what I mean by that.
32
00:01:44,220 --> 00:01:46,780
If this had been only two
terms in here that we're
33
00:01:46,780 --> 00:01:50,180
adding, we would have known how
to handle this problem.
34
00:01:50,180 --> 00:01:53,720
So what we observe is that since
the sum of two functions
35
00:01:53,720 --> 00:01:57,750
is, again, a function, we can
assume that our expression is
36
00:01:57,750 --> 00:01:59,530
written this way.
37
00:01:59,530 --> 00:02:02,710
Now, you see, we've reduced our
problem to the sum of two
38
00:02:02,710 --> 00:02:07,240
functions, namely, 'f1 of x'
plus 'f2 of x' being one of
39
00:02:07,240 --> 00:02:11,520
our functions, 'f3 of x' being
another of our functions.
40
00:02:11,520 --> 00:02:12,860
We know, now, what?
41
00:02:12,860 --> 00:02:15,350
That the limit of a sum is
the sum of the limits.
42
00:02:15,350 --> 00:02:19,760
If we have two functions,
you see this is, what?
43
00:02:19,760 --> 00:02:30,100
The limit 'f1 of x' plus 'f2 of
x', you see, plus the limit
44
00:02:30,100 --> 00:02:33,750
as 'x' approaches
'a', 'f3 of x'.
45
00:02:33,750 --> 00:02:36,290
But now, look what's in this
expression over here.
46
00:02:36,290 --> 00:02:37,160
This is now, what?
47
00:02:37,160 --> 00:02:39,650
The limit of the sum
of two functions.
48
00:02:39,650 --> 00:02:42,070
And, you see, we know that the
limit of a sum is the sum of
49
00:02:42,070 --> 00:02:45,520
the limits is true if we have
only two functions, so, from
50
00:02:45,520 --> 00:02:46,840
here, we get to here.
51
00:02:46,840 --> 00:02:49,170
And, from here, we
can say, what?
52
00:02:49,170 --> 00:02:54,310
This is the limit 'x' approaches
'a', 'f1 of x',
53
00:02:54,310 --> 00:03:02,740
plus the limit as 'x' approaches
'a', 'f2 of x',
54
00:03:02,740 --> 00:03:06,940
plus the limit as 'x' approaches
'a', 'f3 of x'.
55
00:03:06,940 --> 00:03:09,970
In other words, what we have
shown is that the limit of a
56
00:03:09,970 --> 00:03:13,160
sum equals the sum of the limits
is true not just the
57
00:03:13,160 --> 00:03:17,290
sum of two functions, but for
the sum of 3 as well.
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00:03:17,290 --> 00:03:22,100
And, more importantly, the truth
for 3 hinged directly on
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00:03:22,100 --> 00:03:23,410
the truth for 2.
60
00:03:23,410 --> 00:03:26,290
In other words, it wasn't just
that we proved that the
61
00:03:26,290 --> 00:03:29,430
formula was true for the sum of
three functions, we proved
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00:03:29,430 --> 00:03:33,060
it on the assumption that it was
already true for the sum
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00:03:33,060 --> 00:03:34,080
of two functions.
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00:03:34,080 --> 00:03:37,840
And, by the way, notice how we
may now begin to suspect that
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00:03:37,840 --> 00:03:39,610
this idea generalizes.
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00:03:39,610 --> 00:03:42,640
For example, let's take
a look over here.
67
00:03:42,640 --> 00:03:45,530
Suppose I now said how
about the limit as
68
00:03:45,530 --> 00:03:47,980
'x' approaches 'a'?
69
00:03:47,980 --> 00:03:52,340
And we'll now take the sum of
four functions: 'f1 of x',
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00:03:52,340 --> 00:03:59,391
plus 'f2 of x', plus 'f3
of x', plus 'f4 of x'.
71
00:03:59,391 --> 00:04:01,670
See, what about something
like that?
72
00:04:01,670 --> 00:04:04,730
And, again, we argue the same
way as we did before.
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00:04:04,730 --> 00:04:07,400
We say, you know, if
we had only had two
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00:04:07,400 --> 00:04:09,530
functions in here--
75
00:04:09,530 --> 00:04:11,960
and this gives us the
hint to do this--
76
00:04:11,960 --> 00:04:14,860
see, we do know that the limit
of the sum is the sum of the
77
00:04:14,860 --> 00:04:17,110
limits if we have only
two functions.
78
00:04:17,110 --> 00:04:20,570
You see, now, I could
write this as, what?
79
00:04:20,570 --> 00:04:24,860
It's the limit of the first one,
as 'x' approaches 'a',
80
00:04:24,860 --> 00:04:26,540
but what is the first one?
81
00:04:26,540 --> 00:04:32,060
The first one is the function
'f1 of x', plus 'f2 of x',
82
00:04:32,060 --> 00:04:37,010
plus 'f3 of x', plus the
limit of the second.
83
00:04:37,010 --> 00:04:41,730
The second now you see, is
'f sub four of x' as 'x'
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00:04:41,730 --> 00:04:43,590
approaches 'a'.
85
00:04:43,590 --> 00:04:46,330
Now, you see our previous
case told us, what?
86
00:04:46,330 --> 00:04:48,890
That the limit of a sum is equal
to the sum of the limits
87
00:04:48,890 --> 00:04:51,210
if you have the sum of
three functions.
88
00:04:51,210 --> 00:04:54,250
That's exactly what we have over
here, and now you see we
89
00:04:54,250 --> 00:04:55,130
can say, what?
90
00:04:55,130 --> 00:05:01,020
Ah, this is the limit of the
first as 'x' approaches 'a',
91
00:05:01,020 --> 00:05:07,170
plus the limit of the second
as 'x' approaches 'a', plus
92
00:05:07,170 --> 00:05:14,300
the limit of the third as 'x'
approaches 'a', plus the limit
93
00:05:14,300 --> 00:05:17,230
the fourth as 'x'
approaches 'a'.
94
00:05:17,230 --> 00:05:19,520
In other words, what
have we done now?
95
00:05:19,520 --> 00:05:22,980
We've shown that knowing that
the limit of a sum is the sum
96
00:05:22,980 --> 00:05:26,260
of the limits was true for a
sum of two functions and of
97
00:05:26,260 --> 00:05:27,300
three functions.
98
00:05:27,300 --> 00:05:31,700
We've shown, inescapably, that
the same result holds for the
99
00:05:31,700 --> 00:05:34,710
sum of four functions.
100
00:05:34,710 --> 00:05:37,870
Let's take a little breather
here and make a few asides.
101
00:05:37,870 --> 00:05:40,840
I think, sometimes, when one
starts to work too much with
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00:05:40,840 --> 00:05:44,340
mathematical symbolism, we lose
track of the fact that
103
00:05:44,340 --> 00:05:46,640
things are not quite as
difficult as they might
104
00:05:46,640 --> 00:05:49,340
otherwise have seemed.
105
00:05:49,340 --> 00:05:54,150
You see, for one thing, my claim
is that we have already
106
00:05:54,150 --> 00:05:57,900
tackled this problem as recently
as first grade
107
00:05:57,900 --> 00:05:59,160
arithmetic.
108
00:05:59,160 --> 00:06:00,860
Namely, we learned, what?
109
00:06:00,860 --> 00:06:02,890
We learned tables.
110
00:06:02,890 --> 00:06:04,280
Remember the addition tables?
111
00:06:04,280 --> 00:06:06,510
You learned how to
add two numbers.
112
00:06:06,510 --> 00:06:09,670
All of a sudden somebody
says what is the sum of
113
00:06:09,670 --> 00:06:11,810
1, 2, 3, and 4?
114
00:06:11,810 --> 00:06:13,610
What is this number?
115
00:06:13,610 --> 00:06:15,780
And what we said was, look
at, we'll just add
116
00:06:15,780 --> 00:06:17,240
these two at a time.
117
00:06:17,240 --> 00:06:19,950
1 plus 2 is a number,
which is 3.
118
00:06:19,950 --> 00:06:24,240
3 plus 3 is a number,
namely 6.
119
00:06:24,240 --> 00:06:28,330
And 6 plus 4 is a number: 10.
120
00:06:28,330 --> 00:06:29,860
In other words, we essentially
did, what?
121
00:06:29,860 --> 00:06:32,160
We added the first
to the second.
122
00:06:32,160 --> 00:06:35,860
Then we added the sum of the
first two to the third, the
123
00:06:35,860 --> 00:06:38,320
sum of the first three to
the fourth, and that
124
00:06:38,320 --> 00:06:39,290
was how we did this.
125
00:06:39,290 --> 00:06:43,320
Of course, what we assumed in
doing this was that the sum of
126
00:06:43,320 --> 00:06:45,375
two numbers was, again,
a number.
127
00:06:45,375 --> 00:06:47,470
Up above, we assumed
that the sum of two
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00:06:47,470 --> 00:06:49,040
functions was a function.
129
00:06:49,040 --> 00:06:51,280
And this is not quite as trivial
as it might otherwise
130
00:06:51,280 --> 00:06:52,060
have seemed.
131
00:06:52,060 --> 00:06:56,170
Namely, look at, if you add two
odd numbers, do you get
132
00:06:56,170 --> 00:06:58,570
like things when you combine
like things?
133
00:06:58,570 --> 00:07:03,860
The sum of two odd numbers
is always even.
134
00:07:03,860 --> 00:07:05,940
You see, you can't say let's
replace the sum of two odd
135
00:07:05,940 --> 00:07:07,550
numbers by another odd number.
136
00:07:07,550 --> 00:07:10,080
You can say it, but
it would be wrong.
137
00:07:10,080 --> 00:07:11,830
On the other hand,
another example.
138
00:07:11,830 --> 00:07:12,910
How about subtraction?
139
00:07:12,910 --> 00:07:14,360
That's a nice operation.
140
00:07:14,360 --> 00:07:17,760
If you subtract a positive
number from a positive number,
141
00:07:17,760 --> 00:07:20,910
are you guaranteed that the
result will be positive?
142
00:07:20,910 --> 00:07:25,640
Well, for example, what
about 3 less 5?
143
00:07:25,640 --> 00:07:27,800
The answer would
be negative 2.
144
00:07:27,800 --> 00:07:31,770
Positive minus positive can very
well be negative, so we
145
00:07:31,770 --> 00:07:33,040
must be sure, what?
146
00:07:33,040 --> 00:07:36,160
That, when we combine like
objects, we get like objects.
147
00:07:36,160 --> 00:07:38,770
And another thing that we
assumed was that our answer
148
00:07:38,770 --> 00:07:41,450
did not depend upon
voice inflection.
149
00:07:41,450 --> 00:07:42,600
Now, what does that mean?
150
00:07:42,600 --> 00:07:44,650
Let me show you something
over here.
151
00:07:44,650 --> 00:07:49,340
Look at the expression 12
divided by 6 divided by 2.
152
00:07:49,340 --> 00:07:55,910
If you read this as if it said
12 divided by 6 divided by 2,
153
00:07:55,910 --> 00:07:58,250
the answer is 1.
154
00:07:58,250 --> 00:08:04,010
On the other hand, if you read
the same thing as if it said
155
00:08:04,010 --> 00:08:09,380
12 divided by 6 divided by
2, the answer is, what?
156
00:08:09,380 --> 00:08:12,760
12 divided by 3, which is 4.
157
00:08:12,760 --> 00:08:17,680
In other words, division depends
on voice inflection,
158
00:08:17,680 --> 00:08:19,580
whereas, addition doesn't.
159
00:08:19,580 --> 00:08:22,730
And I simply point out these
asides to show you that, as we
160
00:08:22,730 --> 00:08:25,770
go through advanced mathematical
analysis, we are
161
00:08:25,770 --> 00:08:29,500
always making use of the same
assumptions that we were
162
00:08:29,500 --> 00:08:32,539
making when we dealt with
more simple things.
163
00:08:32,539 --> 00:08:35,000
And I think this is the
healthiest way of seeing how
164
00:08:35,000 --> 00:08:36,070
our subject develops.
165
00:08:36,070 --> 00:08:39,130
We will leap from things that
we already know into
166
00:08:39,130 --> 00:08:41,970
generalizations that are
less familiar to us.
167
00:08:41,970 --> 00:08:45,220
At any rate, let's now return
to our main theme of
168
00:08:45,220 --> 00:08:48,750
mathematical induction
and try to summarize
169
00:08:48,750 --> 00:08:50,420
what's happened so far.
170
00:08:50,420 --> 00:08:53,740
We are working with a certain
conjecture, let's call it.
171
00:08:53,740 --> 00:08:56,910
The conjecture is that the limit
of the sum is the sum of
172
00:08:56,910 --> 00:08:57,660
the limits.
173
00:08:57,660 --> 00:09:01,590
We know that the conjecture
was true for two terms.
174
00:09:01,590 --> 00:09:04,870
We proved that, once it was
true for two terms, it was
175
00:09:04,870 --> 00:09:06,220
true for three terms.
176
00:09:06,220 --> 00:09:09,540
We then prove that, if it was
true for three terms, it was
177
00:09:09,540 --> 00:09:11,120
true for four terms.
178
00:09:11,120 --> 00:09:14,090
And now, if we have any
imagination at all, we might
179
00:09:14,090 --> 00:09:17,590
become suspicious and say, you
know, I think, if it's true,
180
00:09:17,590 --> 00:09:20,340
in general, for 'n' terms,
it's going to be
181
00:09:20,340 --> 00:09:22,810
true for 'n + 1' terms.
182
00:09:22,810 --> 00:09:25,660
And that brings us to our next
stage in our mathematical
183
00:09:25,660 --> 00:09:30,130
induction, namely, suppose the
limit of the sum is equal to
184
00:09:30,130 --> 00:09:33,770
the sum of the limits in the
case that there are 'n' terms
185
00:09:33,770 --> 00:09:35,000
in our sum.
186
00:09:35,000 --> 00:09:40,690
What can we conclude about the
limit of a sum in the case of
187
00:09:40,690 --> 00:09:42,380
'n + 1' terms?
188
00:09:42,380 --> 00:09:45,040
And, without going through the
proof here, we'll do these
189
00:09:45,040 --> 00:09:47,990
things in our supplementary
notes in our learning
190
00:09:47,990 --> 00:09:51,270
exercises, but, here, I just
want to focus our attention on
191
00:09:51,270 --> 00:09:52,630
what the main theme is.
192
00:09:52,630 --> 00:09:56,770
What we say is, since we can
add without worrying about
193
00:09:56,770 --> 00:10:00,060
voice inflection, why don't we
throw in a pair of braces
194
00:10:00,060 --> 00:10:05,230
here, thus reducing our problem
to the limit of a sum
195
00:10:05,230 --> 00:10:08,390
when we're adding but
two functions,
196
00:10:08,390 --> 00:10:09,990
use the theorem there.
197
00:10:09,990 --> 00:10:13,990
And, now, given the limit of the
sum of 'n' functions, we
198
00:10:13,990 --> 00:10:15,810
know that that's the
limit of a sum.
199
00:10:15,810 --> 00:10:20,330
And now, it appears that the
truth for 'n' is going to
200
00:10:20,330 --> 00:10:23,490
imply the truth for 'n + 1'.
201
00:10:23,490 --> 00:10:26,320
Now, of course, there may be
other problems that work
202
00:10:26,320 --> 00:10:30,270
structurally this way other than
the limit of a sum equals
203
00:10:30,270 --> 00:10:31,340
the sum of the limits.
204
00:10:31,340 --> 00:10:33,670
So let's generalize
that result.
205
00:10:33,670 --> 00:10:36,100
And the generalization
is what is known as
206
00:10:36,100 --> 00:10:37,610
mathematical induction.
207
00:10:37,610 --> 00:10:40,900
What mathematical induction says
is this, let's suppose we
208
00:10:40,900 --> 00:10:43,120
have a conjecture.
209
00:10:43,120 --> 00:10:45,420
Now, how we get the conjecture
is something we'll talk about
210
00:10:45,420 --> 00:10:48,040
in a while, but let's suppose
we have the conjecture.
211
00:10:48,040 --> 00:10:50,670
Well, to try to show that the
conjecture is true all the
212
00:10:50,670 --> 00:10:54,170
time, we had better be sure it's
true at least sometimes.
213
00:10:54,170 --> 00:10:58,450
So we say, OK, let's show that
the conjecture is true
214
00:10:58,450 --> 00:11:00,130
for 'n' equals 1.
215
00:11:00,130 --> 00:11:01,530
That's just a simple
verification.
216
00:11:01,530 --> 00:11:04,120
We show that it's true
for 'n' equals 1.
217
00:11:04,120 --> 00:11:09,630
Then we say OK, next, prove that
the truth for 'n' equals
218
00:11:09,630 --> 00:11:14,950
'k' implies the truth for
'n' equals 'k + 1'.
219
00:11:14,950 --> 00:11:17,720
In other words, we're not saying
that it's true for 'k',
220
00:11:17,720 --> 00:11:21,650
all we're saying is that, if
it's true for 'k', if it's
221
00:11:21,650 --> 00:11:25,690
true for 'k', the truth for
'n' equals 'k' implies the
222
00:11:25,690 --> 00:11:28,720
truth for 'n' equals 'k + 1'.
223
00:11:28,720 --> 00:11:32,650
Then, if that's true, our
conjecture is true for all
224
00:11:32,650 --> 00:11:33,490
whole numbers.
225
00:11:33,490 --> 00:11:34,990
Why is that?
226
00:11:34,990 --> 00:11:38,300
Well, let's take a look,
informally, here.
227
00:11:38,300 --> 00:11:40,720
Let's suppose that both of these
conditions are obeyed.
228
00:11:40,720 --> 00:11:43,780
We know the conjecture is
true when 'n' equals 1.
229
00:11:43,780 --> 00:11:45,560
Now, take 'k' to be 1.
230
00:11:45,560 --> 00:11:49,510
Since it's true for 1, this part
tells us it's going to be
231
00:11:49,510 --> 00:11:52,940
true for one more than
1, which is 2.
232
00:11:52,940 --> 00:11:57,120
Now that the conjecture is true
for 2, this says, what?
233
00:11:57,120 --> 00:11:59,170
It's going to be true for 3.
234
00:11:59,170 --> 00:12:01,460
And knowing that it's true
for 3, this will say
235
00:12:01,460 --> 00:12:02,820
it's true for 4.
236
00:12:02,820 --> 00:12:05,690
And now I'll loosely use the
word et cetera, and come back
237
00:12:05,690 --> 00:12:08,260
and reinforce that
as we go along.
238
00:12:08,260 --> 00:12:11,370
I think now, perhaps, the best
thing to do is to look at a
239
00:12:11,370 --> 00:12:12,520
second example.
240
00:12:12,520 --> 00:12:15,910
You see, what we did first of
all was we used an example to
241
00:12:15,910 --> 00:12:18,170
lead in to what the definition
would be.
242
00:12:18,170 --> 00:12:20,720
Now that we have the definition,
let's proceed
243
00:12:20,720 --> 00:12:22,450
directly to use it.
244
00:12:22,450 --> 00:12:24,460
Let me give you a conjecture.
245
00:12:24,460 --> 00:12:29,530
The conjecture which I have in
mind is that the sum of the
246
00:12:29,530 --> 00:12:31,770
first n positive numbers--
247
00:12:31,770 --> 00:12:33,120
and this is an interesting
formula--
248
00:12:33,120 --> 00:12:36,650
it's the last number multiplied
by one more than
249
00:12:36,650 --> 00:12:39,860
the last number divided by 2.
250
00:12:39,860 --> 00:12:42,220
Well, let's just see if that's
true at all for a while.
251
00:12:42,220 --> 00:12:50,400
Look at, if 'n' is 1, the
left-hand side here is 1.
252
00:12:50,400 --> 00:12:55,300
And 1 times 2 divided
by 2 is also 1.
253
00:12:55,300 --> 00:12:57,400
If 'n' is 2, the sum
of the first two
254
00:12:57,400 --> 00:12:58,310
numbers here is, what?
255
00:12:58,310 --> 00:12:59,940
One plus 2 is three.
256
00:12:59,940 --> 00:13:04,710
On the other hand, 2 times
3 divided by 2 is also 3.
257
00:13:04,710 --> 00:13:08,470
So, at least, our conjecture is
true for 'n' equals 1 and
258
00:13:08,470 --> 00:13:09,640
'n' equals 2.
259
00:13:09,640 --> 00:13:12,040
What does mathematical
induction say?
260
00:13:12,040 --> 00:13:13,890
Let's take a look again, now.
261
00:13:13,890 --> 00:13:16,050
You see, we showed that the
conjecture was true
262
00:13:16,050 --> 00:13:17,180
for 'n' equals 1.
263
00:13:17,180 --> 00:13:19,730
And, for good measure, we also
showed that it was true for
264
00:13:19,730 --> 00:13:20,710
'n' equals 2.
265
00:13:20,710 --> 00:13:21,940
So, we can check this off.
266
00:13:21,940 --> 00:13:23,010
We've done that.
267
00:13:23,010 --> 00:13:24,010
Now, what do we do?
268
00:13:24,010 --> 00:13:27,670
We assume the conjecture is
true for 'n' equals 'k'.
269
00:13:27,670 --> 00:13:28,920
That means, what?
270
00:13:31,200 --> 00:13:32,670
Well, just replace 'n' by 'k'.
271
00:13:38,190 --> 00:13:40,700
We're assuming that
this is true.
272
00:13:40,700 --> 00:13:44,750
From the truth of this,
what must we do next?
273
00:13:44,750 --> 00:13:48,710
Well, what we must do next is
investigate what happens if
274
00:13:48,710 --> 00:13:50,030
you add, what?
275
00:13:50,030 --> 00:13:52,660
Not 'k' numbers, but 'k + 1'.
276
00:13:52,660 --> 00:13:54,960
So, in other words, what happens
when you replace 'n'
277
00:13:54,960 --> 00:13:56,470
by 'k + 1'?
278
00:13:56,470 --> 00:13:58,120
Now, watch how we do this.
279
00:13:58,120 --> 00:14:01,070
The same thing that we did in
theory before, we say, look
280
00:14:01,070 --> 00:14:04,620
at, we already know how
to handle this amount.
281
00:14:04,620 --> 00:14:07,742
We're told that that's going to
be ''k' times 'k + 1' over
282
00:14:07,742 --> 00:14:12,200
2', so let's rewrite
this in this way.
283
00:14:12,200 --> 00:14:16,200
Now, we can replace the
bracketed expression by ''k'
284
00:14:16,200 --> 00:14:18,620
times 'k + 1' over 2'.
285
00:14:18,620 --> 00:14:21,490
We add on, of course, 'k + 1'
because that's the last term
286
00:14:21,490 --> 00:14:22,880
that's over here.
287
00:14:22,880 --> 00:14:29,050
Now, we factor out 'k + 1'
from this factor here.
288
00:14:29,050 --> 00:14:30,950
That leaves us with, what?
289
00:14:30,950 --> 00:14:34,720
'k/2 + 1'.
290
00:14:34,720 --> 00:14:37,330
And this, in turn, says, what?
291
00:14:37,330 --> 00:14:43,560
That the sum of the first 'k +
1' numbers is ''k + 1' times
292
00:14:43,560 --> 00:14:44,810
'k + 2' over 2'.
293
00:14:48,120 --> 00:14:51,350
And notice that that's exactly
what the conjecture should say
294
00:14:51,350 --> 00:14:54,790
when 'n' equals 'k +
1', namely, what?
295
00:14:54,790 --> 00:14:57,230
The sum of the first 'n'
numbers, no matter how many
296
00:14:57,230 --> 00:14:58,260
you have, is, what?
297
00:14:58,260 --> 00:15:01,390
The last number times one
more than the last
298
00:15:01,390 --> 00:15:03,480
number divided by 2.
299
00:15:03,480 --> 00:15:07,350
The sum of the first 'k
+ 1' numbers is, what?
300
00:15:07,350 --> 00:15:12,110
It's 'k + 1', the last one,
times 'k + 2', which is one
301
00:15:12,110 --> 00:15:15,240
more than the last one,
divided by 2.
302
00:15:15,240 --> 00:15:18,940
And now, what we've done is we
have verified the second part
303
00:15:18,940 --> 00:15:23,520
of our mathematical induction
setup, namely, if we go back
304
00:15:23,520 --> 00:15:26,490
to our basic definition over
here, we have to show, what?
305
00:15:26,490 --> 00:15:30,170
Prove that the truth for 'n'
equals 'k' implies the truth
306
00:15:30,170 --> 00:15:34,670
for 'n' equals 'k + 1', which
is exactly what we did.
307
00:15:34,670 --> 00:15:37,890
And, while you're thinking about
that, let's take a break
308
00:15:37,890 --> 00:15:41,170
for a few more asides which, I
think, may cement down this
309
00:15:41,170 --> 00:15:43,480
idea a little bit
more strongly.
310
00:15:43,480 --> 00:15:46,430
I mentioned before that
induction is something that
311
00:15:46,430 --> 00:15:50,230
one uses when one already has
a suspicion as to what the
312
00:15:50,230 --> 00:15:51,540
right answer is.
313
00:15:51,540 --> 00:15:54,090
I don't know how this grabs
you, buy my own particular
314
00:15:54,090 --> 00:15:57,050
feeling is that you do not look
at the sum of the first
315
00:15:57,050 --> 00:16:01,250
'n' numbers and say aha, it's
the last one times one more
316
00:16:01,250 --> 00:16:03,800
than the last one
divided by 2.
317
00:16:03,800 --> 00:16:06,050
You see, that's the nice thing
about textbook problems.
318
00:16:06,050 --> 00:16:09,270
When they give you a problem
on induction and they say
319
00:16:09,270 --> 00:16:13,060
prove this conjecture, notice
that they've already given you
320
00:16:13,060 --> 00:16:15,920
a tremendous hint, namely,
they've told you what the
321
00:16:15,920 --> 00:16:17,670
conjecture is.
322
00:16:17,670 --> 00:16:20,330
You see, in the textbook of real
life, one usually has to
323
00:16:20,330 --> 00:16:23,350
find out what the conjecture
is for oneself.
324
00:16:23,350 --> 00:16:26,150
In fact, in the form of a rather
interesting aside,
325
00:16:26,150 --> 00:16:29,050
there is a very interesting
mathematical anecdote
326
00:16:29,050 --> 00:16:31,460
connected with this particular
problem.
327
00:16:31,460 --> 00:16:35,370
It's an anecdote attributed to
the mathematician, Gauss, who,
328
00:16:35,370 --> 00:16:37,460
when he was a young chap, was a
329
00:16:37,460 --> 00:16:39,290
discipline problem in school.
330
00:16:39,290 --> 00:16:45,870
And the story is that his
teacher, as a punishment,
331
00:16:45,870 --> 00:16:48,610
asked him to add the
first 100 numbers.
332
00:16:48,610 --> 00:16:52,140
And Gauss wrote down the answer
very, very rapidly.
333
00:16:52,140 --> 00:16:54,040
And what he did was, he didn't
add these all up.
334
00:16:54,040 --> 00:16:56,070
What he observed was, what?
335
00:16:56,070 --> 00:17:00,010
The first one plus the last
one added up to 101.
336
00:17:00,010 --> 00:17:02,110
The second plus the next
to the last added
337
00:17:02,110 --> 00:17:04,970
up to 101, you see?
338
00:17:04,970 --> 00:17:08,790
And each pair going in this
way added up to 101.
339
00:17:08,790 --> 00:17:11,430
And how many pairs were
there, all together?
340
00:17:11,430 --> 00:17:14,400
Well, there were 100 numbers,
so there were 50 pairs.
341
00:17:14,400 --> 00:17:15,970
In other words, there
were, what?
342
00:17:15,970 --> 00:17:21,560
100 divided by 2 pairs, each
pair adding up to 101.
343
00:17:21,560 --> 00:17:25,660
And now, notice the recipe: 100,
namely the last number,
344
00:17:25,660 --> 00:17:30,540
times one more than the last
number, 101, divided by 2.
345
00:17:30,540 --> 00:17:36,600
By the way, in the exercises
on this assignment we have
346
00:17:36,600 --> 00:17:40,010
another problem, and that is,
if you think this one was
347
00:17:40,010 --> 00:17:43,370
already cumbersome, try guessing
what the recipe for
348
00:17:43,370 --> 00:17:47,000
this one is: what is the sum
of the first n squares?
349
00:17:47,000 --> 00:17:50,740
In other words, not 1, plus 2,
plus 3, et cetera, but 1
350
00:17:50,740 --> 00:17:54,710
squared, plus 2 squared, plus
3 squared, et cetera.
351
00:17:54,710 --> 00:17:58,280
I give you the answer, but try
to think for a while as to how
352
00:17:58,280 --> 00:18:00,920
likely it is that you would have
conjectured this in the
353
00:18:00,920 --> 00:18:01,950
first place.
354
00:18:01,950 --> 00:18:05,700
It turns out to be ''n'
times 'n + 1' times
355
00:18:05,700 --> 00:18:08,760
'2 n + 1' over 6'.
356
00:18:08,760 --> 00:18:11,560
You see, the reason I bring this
out is that I call this a
357
00:18:11,560 --> 00:18:13,090
contrived example.
358
00:18:13,090 --> 00:18:16,170
It is not the case where the
mathematician would most
359
00:18:16,170 --> 00:18:19,320
likely have invented
mathematical induction.
360
00:18:19,320 --> 00:18:21,440
The case where he would have
invented mathematical
361
00:18:21,440 --> 00:18:24,450
induction is the case that we
did earlier, for example,
362
00:18:24,450 --> 00:18:26,780
where we talk about the
limit of a sum being
363
00:18:26,780 --> 00:18:27,510
the sum of the limits.
364
00:18:27,510 --> 00:18:30,690
You can actually see what's
happening, how the truth that
365
00:18:30,690 --> 00:18:33,630
'k' implies the truth
for 'k + 1'.
366
00:18:33,630 --> 00:18:36,340
And by the way, let me make
one more aside here that I
367
00:18:36,340 --> 00:18:37,950
forgot to mention earlier.
368
00:18:37,950 --> 00:18:41,150
In our definition of
mathematical induction, we
369
00:18:41,150 --> 00:18:44,090
said show the conjecture is
true for 'n' equals 1.
370
00:18:44,090 --> 00:18:47,270
This was quite hypocritical
because the very first example
371
00:18:47,270 --> 00:18:50,160
that I picked didn't even
start until 'n' was 2.
372
00:18:50,160 --> 00:18:52,790
We talked about the limit of
a sum being the sum of the
373
00:18:52,790 --> 00:18:55,250
limits, and the smallest
sum we talked about was
374
00:18:55,250 --> 00:18:57,150
the sum of two terms.
375
00:18:57,150 --> 00:18:59,630
The point that I wanted to
mention is, for example,
376
00:18:59,630 --> 00:19:03,230
suppose that the first number
you can prove the conjecture
377
00:19:03,230 --> 00:19:05,820
for is, for example,
'n' equals 7.
378
00:19:05,820 --> 00:19:07,010
I don't know why I picked 7.
379
00:19:07,010 --> 00:19:07,720
I had to pick something.
380
00:19:07,720 --> 00:19:09,180
Let's just call it
'n' equals 7.
381
00:19:09,180 --> 00:19:12,640
Suppose I can also show that, if
the conjecture is true for
382
00:19:12,640 --> 00:19:16,200
'k', it's true for 'k + 1'.
383
00:19:16,200 --> 00:19:20,060
Then, you see, what I can
conclude is I can conclude
384
00:19:20,060 --> 00:19:23,390
that it's true for 'n' greater
than or equal to 7.
385
00:19:23,390 --> 00:19:26,440
Namely, if it's true for 7, this
says it will be true for
386
00:19:26,440 --> 00:19:28,310
one more than 7, which is 8.
387
00:19:28,310 --> 00:19:31,070
If it's true for 8, this says
it will be true for 9.
388
00:19:31,070 --> 00:19:32,860
If it's true for 9, this
says it will be
389
00:19:32,860 --> 00:19:34,730
true for 10, et cetera.
390
00:19:34,730 --> 00:19:36,270
And we go on this way.
391
00:19:36,270 --> 00:19:39,250
Now, again, this may be a very
naive way of looking at it,
392
00:19:39,250 --> 00:19:42,630
but I always look at
mathematical induction as a
393
00:19:42,630 --> 00:19:47,640
bunch of toy soldiers stacked
up in a line in such a way
394
00:19:47,640 --> 00:19:50,910
that, if any one of the toy
soldiers falls down, he knocks
395
00:19:50,910 --> 00:19:54,340
down the one that's immediately
behind him, OK?
396
00:19:54,340 --> 00:19:56,250
You see what I'm driving
at here?
397
00:19:56,250 --> 00:19:59,120
If the first falls-- and, by
the way, that's a big if--
398
00:19:59,120 --> 00:20:02,130
if the first falls, he knocks
down the second.
399
00:20:02,130 --> 00:20:04,270
The second falling knocks
down the third.
400
00:20:04,270 --> 00:20:06,430
The third falling knocks
down the fourth.
401
00:20:06,430 --> 00:20:09,720
The fourth falling down knocks
down the fifth, et cetera.
402
00:20:09,720 --> 00:20:11,920
Notice that, if the first
one doesn't fall,
403
00:20:11,920 --> 00:20:12,920
none of them fall.
404
00:20:12,920 --> 00:20:16,350
Or, for that matter, going back
to my previous analogy,
405
00:20:16,350 --> 00:20:19,590
if the seventh one is the first
one that falls, all the
406
00:20:19,590 --> 00:20:22,330
ones behind him fall down.
407
00:20:22,330 --> 00:20:23,560
Now, be very careful.
408
00:20:23,560 --> 00:20:27,470
Mathematical induction doesn't
say the first 50 fall down, or
409
00:20:27,470 --> 00:20:30,230
the first 100 fall down,
it says they all
410
00:20:30,230 --> 00:20:31,490
have to fall down.
411
00:20:31,490 --> 00:20:35,120
For example, here's a case where
several fall down, but,
412
00:20:35,120 --> 00:20:38,303
all of a sudden, one isn't
knocked down by the one in
413
00:20:38,303 --> 00:20:38,890
front of him.
414
00:20:38,890 --> 00:20:41,220
In other words, what
mathematical induction really
415
00:20:41,220 --> 00:20:46,460
involves is the idea not just
that something is true, but
416
00:20:46,460 --> 00:20:50,100
that it's true because the
previous one was true.
417
00:20:50,100 --> 00:20:54,570
You see, for example, is it
possible that all of the
418
00:20:54,570 --> 00:20:58,700
soldiers fall down even though
the one in front didn't knock
419
00:20:58,700 --> 00:21:00,040
the other guy down?
420
00:21:00,040 --> 00:21:02,510
I mean, it's possible all of
these go down, but for
421
00:21:02,510 --> 00:21:03,600
different reasons.
422
00:21:03,600 --> 00:21:06,180
Mathematical induction says
much more than that.
423
00:21:06,180 --> 00:21:09,910
Mathematical induction says
yes, they all go down, but
424
00:21:09,910 --> 00:21:12,980
each goes down because
of the one before.
425
00:21:12,980 --> 00:21:16,410
And, by the way, one
more little aside.
426
00:21:16,410 --> 00:21:19,690
Notice that, in our analog,
we assumed that
427
00:21:19,690 --> 00:21:21,930
there was a next one.
428
00:21:21,930 --> 00:21:22,970
You say, well, what do
you mean you assume
429
00:21:22,970 --> 00:21:23,900
there was a next one?
430
00:21:23,900 --> 00:21:25,820
Obviously, there has
to be a next one.
431
00:21:25,820 --> 00:21:29,610
But, the concept of next depends
on whole numbers.
432
00:21:29,610 --> 00:21:33,510
For example, when you
deal with fractions,
433
00:21:33,510 --> 00:21:34,840
there is no next one.
434
00:21:34,840 --> 00:21:36,260
Let me show you what
I mean by that.
435
00:21:36,260 --> 00:21:38,190
Let's talk about the real
numbers in general.
436
00:21:38,190 --> 00:21:39,480
On the number line, here's 0.
437
00:21:39,480 --> 00:21:42,750
What is the first fraction,
the first real number?
438
00:21:42,750 --> 00:21:43,680
I don't care what you call it.
439
00:21:43,680 --> 00:21:47,190
What is the first number which
is greater than 0?
440
00:21:47,190 --> 00:21:49,950
What is the first number which
is greater than 0?
441
00:21:49,950 --> 00:21:52,680
And the answer is, there
is none because
442
00:21:52,680 --> 00:21:54,540
whichever one you pick--
443
00:21:54,540 --> 00:21:56,060
call it 'r'.
444
00:21:56,060 --> 00:21:59,130
Let 'r' stand for the first
number which you think is
445
00:21:59,130 --> 00:22:01,430
bigger than 0, OK?
446
00:22:01,430 --> 00:22:04,855
How about 'r/2'?
447
00:22:04,855 --> 00:22:10,340
'r/2' is still bigger than 0,
but 'r/2' is less than 'r'.
448
00:22:10,340 --> 00:22:14,340
In other words, given any number
you pick that's bigger
449
00:22:14,340 --> 00:22:17,110
than 0, you can fit
in another one.
450
00:22:17,110 --> 00:22:20,550
So, there is no number which
is immediately next to 0.
451
00:22:20,550 --> 00:22:23,540
In other words, as a caution,
notice that mathematical
452
00:22:23,540 --> 00:22:26,410
induction is used when
we're dealing with a
453
00:22:26,410 --> 00:22:29,440
whole number of objects.
454
00:22:29,440 --> 00:22:34,280
Now, let's emphasize some of
these little asides from a
455
00:22:34,280 --> 00:22:37,060
more specific point of view.
456
00:22:37,060 --> 00:22:40,150
Let me, first of all, give you
an example in which something
457
00:22:40,150 --> 00:22:43,370
is true for a whole bunch of
numbers, but, all of a sudden,
458
00:22:43,370 --> 00:22:45,100
isn't true in general.
459
00:22:45,100 --> 00:22:47,470
Now, how anybody ever stumbled
across the example I'm going
460
00:22:47,470 --> 00:22:50,670
to give you next, I have no
idea, but I find it's a very
461
00:22:50,670 --> 00:22:52,380
interesting concept.
462
00:22:52,380 --> 00:22:55,340
Let's look at this.
463
00:22:55,340 --> 00:22:59,120
Let's write down the following
function: 'p' of 'n', where
464
00:22:59,120 --> 00:23:02,180
'n' is any positive whole
number, will be defined to be
465
00:23:02,180 --> 00:23:05,530
'n' squared, minus
'n', plus 41.
466
00:23:05,530 --> 00:23:10,610
For example, 'p' of 1 will be
1 squared, minus 1, plus 41,
467
00:23:10,610 --> 00:23:12,510
which happens to be 41.
468
00:23:12,510 --> 00:23:15,950
41 happens to be a prime number:
a number which has no
469
00:23:15,950 --> 00:23:18,780
factors other than
itself and 1.
470
00:23:18,780 --> 00:23:20,540
Let's try 2 in here.
471
00:23:20,540 --> 00:23:26,570
2 squared, minus 2, plus 41,
is 43: also a prime number.
472
00:23:26,570 --> 00:23:29,130
Let's try 3 in here.
473
00:23:29,130 --> 00:23:34,570
3 squared is 9, minus 3
is 6, plus 41 is 47:
474
00:23:34,570 --> 00:23:36,290
also a prime number.
475
00:23:36,290 --> 00:23:40,290
The amazing thing is that, as
you go all the way through to
476
00:23:40,290 --> 00:23:44,110
40, you get nothing
but prime numbers.
477
00:23:44,110 --> 00:23:47,530
And you say, ah, it's right
so far, it must be
478
00:23:47,530 --> 00:23:48,790
right all the time.
479
00:23:48,790 --> 00:23:50,290
And this is wishful thinking.
480
00:23:50,290 --> 00:23:53,150
This is like the man who wants
to count the deck of cards to
481
00:23:53,150 --> 00:23:55,890
see if the cards are all there,
and he says one, two,
482
00:23:55,890 --> 00:23:57,090
three, four, five.
483
00:23:57,090 --> 00:23:58,620
Well, so far, so good.
484
00:23:58,620 --> 00:24:00,450
They must all be here.
485
00:24:00,450 --> 00:24:02,430
It's like falling off the Empire
State building, and
486
00:24:02,430 --> 00:24:04,900
halfway down, somebody says,
"How you doing?" You say, "So
487
00:24:04,900 --> 00:24:07,490
far, so good." No, we're in a
little bit of trouble here
488
00:24:07,490 --> 00:24:11,510
because, as soon as we pick
'n' to be 41, watch what
489
00:24:11,510 --> 00:24:12,520
happens here.
490
00:24:12,520 --> 00:24:18,890
This becomes 41 squared,
minus 41, plus 41.
491
00:24:18,890 --> 00:24:22,750
And this is 41 squared, which,
obviously, is not a prime.
492
00:24:22,750 --> 00:24:25,000
It's 41 times 41.
493
00:24:25,000 --> 00:24:27,700
And here is an interesting
example where a certain
494
00:24:27,700 --> 00:24:31,750
formula generates nothing but
primes for the first 40
495
00:24:31,750 --> 00:24:35,190
integers, but fails on
the 41st integer.
496
00:24:35,190 --> 00:24:37,020
Why did this happen?
497
00:24:37,020 --> 00:24:41,080
Because, evidently, the fact
that this was a prime in no
498
00:24:41,080 --> 00:24:43,600
way depended structurally
on the fact that
499
00:24:43,600 --> 00:24:44,930
this one was a prime.
500
00:24:44,930 --> 00:24:47,990
There is our mathematical
induction again, that not only
501
00:24:47,990 --> 00:24:51,730
must the conjecture be true, but
it must follow inescapably
502
00:24:51,730 --> 00:24:53,510
from the case before.
503
00:24:53,510 --> 00:24:56,010
If you'd like a more
fascinating, realistic
504
00:24:56,010 --> 00:24:59,000
example, it's something that
we call the unique
505
00:24:59,000 --> 00:25:02,890
factorization theorem of
elementary number theory.
506
00:25:02,890 --> 00:25:08,470
This says that every positive
whole number greater than one
507
00:25:08,470 --> 00:25:12,480
can be factored uniquely into a
product of primes, unique up
508
00:25:12,480 --> 00:25:14,640
to the order in which
you write them.
509
00:25:14,640 --> 00:25:17,260
For example, 2 is already
a prime, it's 2.
510
00:25:17,260 --> 00:25:19,600
3 is already a prime: 3.
511
00:25:19,600 --> 00:25:22,810
4 can be factored
as 2 times 2.
512
00:25:22,810 --> 00:25:24,900
5 can be factored as--
513
00:25:24,900 --> 00:25:27,220
well, it's already
a prime, it's 5.
514
00:25:27,220 --> 00:25:28,780
6 can be written as, what?
515
00:25:28,780 --> 00:25:30,440
2 times 3.
516
00:25:30,440 --> 00:25:32,300
7 can be written,
of course, as 7.
517
00:25:32,300 --> 00:25:33,670
It's already a prime.
518
00:25:33,670 --> 00:25:37,740
8 is 2 times 2 times 2.
519
00:25:37,740 --> 00:25:40,090
9 is 3 times 3.
520
00:25:40,090 --> 00:25:44,860
10 is 2 times 5, and 11
is already a prime.
521
00:25:44,860 --> 00:25:46,280
Well, this doesn't
prove anything.
522
00:25:46,280 --> 00:25:48,820
I'm just trying to demonstrate
what the theorem says.
523
00:25:48,820 --> 00:25:50,460
The interesting thing is do you
524
00:25:50,460 --> 00:25:53,510
notice how 'n + 1' factors--
525
00:25:53,510 --> 00:25:56,470
I don't know what phrase to use
here, but let's call it
526
00:25:56,470 --> 00:25:58,860
considerably different
than 'n'.
527
00:25:58,860 --> 00:26:02,060
In other words, look at what
happened to 10 when I
528
00:26:02,060 --> 00:26:03,190
added 1 onto it.
529
00:26:03,190 --> 00:26:05,170
It factored into 2 times 5.
530
00:26:05,170 --> 00:26:08,710
I add 1 onto it, and, all of
a sudden, the factorization
531
00:26:08,710 --> 00:26:10,420
properties change.
532
00:26:10,420 --> 00:26:11,870
Let me give you an example.
533
00:26:11,870 --> 00:26:13,770
Look at 59.
534
00:26:13,770 --> 00:26:16,240
59 happens to be a prime.
535
00:26:16,240 --> 00:26:18,390
Look at 61.
536
00:26:18,390 --> 00:26:21,140
61 also happens to be a prime.
537
00:26:21,140 --> 00:26:24,350
By the way, for those of you are
number theory buffs, these
538
00:26:24,350 --> 00:26:26,410
are called twin primes.
539
00:26:26,410 --> 00:26:29,770
Consecutive odd numbers, which
are both prime, are called
540
00:26:29,770 --> 00:26:31,580
twin primes.
541
00:26:31,580 --> 00:26:34,670
By the way, 2 is the only even
prime, of course, because, if
542
00:26:34,670 --> 00:26:38,550
a number is greater than
2, and it's even, it's
543
00:26:38,550 --> 00:26:39,540
divisible by 2.
544
00:26:39,540 --> 00:26:41,525
So, 2 is the only even prime.
545
00:26:41,525 --> 00:26:42,850
Twin primes are, what?
546
00:26:42,850 --> 00:26:45,520
Consecutive odd numbers both
of which are primes.
547
00:26:45,520 --> 00:26:49,230
So, 59 and 61 are a pair
of twin primes.
548
00:26:49,230 --> 00:26:52,620
One might intuitively suspect,
therefore, that the number
549
00:26:52,620 --> 00:26:56,070
between them must be sort
of a prime too, or
550
00:26:56,070 --> 00:26:57,200
whatever that means.
551
00:26:57,200 --> 00:26:59,830
Obviously, it can't be a prime
because in between them comes
552
00:26:59,830 --> 00:27:02,600
60, which is, in particular,
an even number.
553
00:27:02,600 --> 00:27:06,200
But look at all the nice
factors that sixty has.
554
00:27:06,200 --> 00:27:07,050
60 is, what?
555
00:27:07,050 --> 00:27:11,070
It's one more than 59, one less
than 61, but look at how
556
00:27:11,070 --> 00:27:12,320
different it factors.
557
00:27:12,320 --> 00:27:14,990
In fact, a pseudo induction-type
thing is look
558
00:27:14,990 --> 00:27:25,150
at the factors of 60:
1, 2, 3, 4, 5, 6.
559
00:27:25,150 --> 00:27:25,600
We say, what?
560
00:27:25,600 --> 00:27:27,700
7, experimental error?
561
00:27:27,700 --> 00:27:28,120
No, no.
562
00:27:28,120 --> 00:27:30,850
But, this is not induction,
by the way.
563
00:27:30,850 --> 00:27:35,730
The fact that 1, 2, 3, 4, 5,
and 6 are all factors of 60
564
00:27:35,730 --> 00:27:38,420
does not mean that 7 is going
to be a factor of 60.
565
00:27:38,420 --> 00:27:41,370
We do not say just because it
works so far, it's going to
566
00:27:41,370 --> 00:27:42,230
keep working.
567
00:27:42,230 --> 00:27:44,290
And I'm not going to belabor
this point anymore.
568
00:27:44,290 --> 00:27:48,610
Suffice it to say that look at
how differently 60 factors
569
00:27:48,610 --> 00:27:51,230
compare to the number the came
just before it and the number
570
00:27:51,230 --> 00:27:52,660
that came just after it.
571
00:27:52,660 --> 00:27:55,610
In other words, assuming that
the unique factorization
572
00:27:55,610 --> 00:28:00,350
theorem is true, the truth for
'n + 1', somehow or other, has
573
00:28:00,350 --> 00:28:02,900
nothing to do with the
truth for 'n'.
574
00:28:02,900 --> 00:28:05,760
And you see this is, again,
another weakness of, what's
575
00:28:05,760 --> 00:28:07,180
called, induction.
576
00:28:07,180 --> 00:28:10,650
Well, so, that's a weakness
of induction.
577
00:28:10,650 --> 00:28:13,710
The point that I'm making is why
shouldn't induction have
578
00:28:13,710 --> 00:28:14,770
some weaknesses?
579
00:28:14,770 --> 00:28:17,900
After all, if it could solve
every problem, what we would
580
00:28:17,900 --> 00:28:21,040
do is have a calculus book that
was three pages long.
581
00:28:21,040 --> 00:28:22,280
It would be called,
The Principle of
582
00:28:22,280 --> 00:28:23,840
Mathematical Induction.
583
00:28:23,840 --> 00:28:27,740
When we solved that problem by
induction, everything else
584
00:28:27,740 --> 00:28:28,940
would be done.
585
00:28:28,940 --> 00:28:30,450
No, there are problems
that do not lend
586
00:28:30,450 --> 00:28:31,820
themselves to induction.
587
00:28:31,820 --> 00:28:35,130
In summary, induction is a
particularly effective
588
00:28:35,130 --> 00:28:39,090
technique which one uses to
prove that something is true
589
00:28:39,090 --> 00:28:43,870
for all whole numbers provided
that one has a suspicion that
590
00:28:43,870 --> 00:28:45,600
this thing is true in
the first place.
591
00:28:45,600 --> 00:28:48,990
And secondly that, even if the
suspicion is true, the truth
592
00:28:48,990 --> 00:28:52,890
for the 'n' plus first case
follows inescapably from the
593
00:28:52,890 --> 00:28:55,350
truth for the n-th case.
594
00:28:55,350 --> 00:28:58,940
At any rate, this completes
our lecture for today.
595
00:28:58,940 --> 00:29:02,965
And, until next time,
good bye.
596
00:29:02,965 --> 00:29:05,940
MALE VOICE: Funding for the
publication of this video was
597
00:29:05,940 --> 00:29:10,660
provided by the Gabriella and
Paul Rosenbaum Foundation.
598
00:29:10,660 --> 00:29:14,830
Help OCW continue to provide
free and open access to MIT
599
00:29:14,830 --> 00:29:19,030
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at ocw.mit.edu/donate.