1
00:00:00,940 --> 00:00:02,620
PROFESSOR: The well
ordering principle
2
00:00:02,620 --> 00:00:05,500
is one of those
facts in mathematics
3
00:00:05,500 --> 00:00:08,119
that's so obvious that
you hardly notice it.
4
00:00:08,119 --> 00:00:11,210
And the objective of
this brief introduction
5
00:00:11,210 --> 00:00:13,790
is to call your attention to it.
6
00:00:13,790 --> 00:00:15,675
We've actually used it already.
7
00:00:15,675 --> 00:00:19,360
And in subsequent segments
of this presentation,
8
00:00:19,360 --> 00:00:23,340
I'll show lots of
applications of it.
9
00:00:23,340 --> 00:00:25,860
So here's a statement of
the well ordering principle.
10
00:00:25,860 --> 00:00:28,680
Every nonempty set of
nonnegative integers
11
00:00:28,680 --> 00:00:30,940
has a least element.
12
00:00:30,940 --> 00:00:33,694
Now this is probably familiar.
13
00:00:33,694 --> 00:00:35,360
Maybe you haven't
even thought about it.
14
00:00:35,360 --> 00:00:38,460
But now that I mentioned it,
I expect it's a familiar idea.
15
00:00:38,460 --> 00:00:40,620
And it's pretty
obvious too if you
16
00:00:40,620 --> 00:00:41,950
think about it for a minute.
17
00:00:41,950 --> 00:00:44,780
Here's a way to think about it.
18
00:00:44,780 --> 00:00:47,470
Given an nonempty set of
integers, you could ask,
19
00:00:47,470 --> 00:00:49,300
is 0 the least element in it?
20
00:00:49,300 --> 00:00:51,880
Well, if it is,
then you're done.
21
00:00:51,880 --> 00:00:55,015
Then you could say, is 1
the least element in it?
22
00:00:55,015 --> 00:00:56,310
And if it is, you're done.
23
00:00:56,310 --> 00:00:59,420
And if it isn't, you could
say 2, is 2 the least element?
24
00:00:59,420 --> 00:01:00,390
And so on.
25
00:01:00,390 --> 00:01:02,700
Given that it's not
empty, eventually you're
26
00:01:02,700 --> 00:01:04,510
to hit the least element.
27
00:01:04,510 --> 00:01:07,140
So, if it wasn't
obvious before, there
28
00:01:07,140 --> 00:01:09,690
is something of a
hand-waving proof of it.
29
00:01:09,690 --> 00:01:12,710
But I want to get you
to think about this well
30
00:01:12,710 --> 00:01:16,660
ordering principle a little bit
because it's not-- there are
31
00:01:16,660 --> 00:01:19,130
some technical parts
of it that matter.
32
00:01:19,130 --> 00:01:24,810
So for example, suppose I
replace nonnegative integers
33
00:01:24,810 --> 00:01:26,800
by nonnegative rationals.
34
00:01:26,800 --> 00:01:32,120
And I asked does every nonempty
set of nonnegative rationals
35
00:01:32,120 --> 00:01:33,700
have a least element?
36
00:01:33,700 --> 00:01:37,900
Well, there is a least
nonnegative rational, namely 0.
37
00:01:37,900 --> 00:01:42,480
But not every nonnegative set of
rationals has a least element.
38
00:01:42,480 --> 00:01:46,370
I'll let you think
of an example.
39
00:01:46,370 --> 00:01:49,580
Another variant is
when, instead of talking
40
00:01:49,580 --> 00:01:51,350
about the nonnegative
integers, I just
41
00:01:51,350 --> 00:01:52,910
talk about all the integers.
42
00:01:52,910 --> 00:01:54,520
Is there a least integer?
43
00:01:54,520 --> 00:01:57,790
Well, no, obviously because
minus 1 is not the least.
44
00:01:57,790 --> 00:01:59,140
And minus 2 is not the least.
45
00:01:59,140 --> 00:02:01,262
And there isn't
any least integer.
46
00:02:04,560 --> 00:02:08,120
We take for granted the
well ordering principle just
47
00:02:08,120 --> 00:02:09,340
all the time.
48
00:02:09,340 --> 00:02:13,024
If I ask you, what was the
youngest age of an MIT graduate
49
00:02:13,024 --> 00:02:15,190
well, you wouldn't for a
moment wonder whether there
50
00:02:15,190 --> 00:02:16,910
was a youngest age.
51
00:02:16,910 --> 00:02:19,440
And if I asked you for the
smallest number of neurons
52
00:02:19,440 --> 00:02:21,240
in any animal, you
wouldn't wonder
53
00:02:21,240 --> 00:02:24,750
whether there was or wasn't
a smallest number of neurons.
54
00:02:24,750 --> 00:02:25,940
We may not know what it is.
55
00:02:25,940 --> 00:02:28,500
But there's surely a
smallest number of neurons
56
00:02:28,500 --> 00:02:30,890
because neurons are
nonnegative integers.
57
00:02:30,890 --> 00:02:33,740
And finally, if I ask you what
was the smallest number of US
58
00:02:33,740 --> 00:02:37,030
coins that could
make $1.17, again, we
59
00:02:37,030 --> 00:02:38,600
don't have to worry
about existence
60
00:02:38,600 --> 00:02:41,340
because the well ordering
principle knocks that
61
00:02:41,340 --> 00:02:42,185
off immediately.
62
00:02:44,982 --> 00:02:46,440
Now for the remainder
of this talk,
63
00:02:46,440 --> 00:02:48,690
I'm going to be talking about
the nonnegative integers
64
00:02:48,690 --> 00:02:50,616
always, unless I
explicitly say otherwise.
65
00:02:50,616 --> 00:02:52,240
So I'm just going to
is the word number
66
00:02:52,240 --> 00:02:54,340
to mean nonnegative integer.
67
00:02:54,340 --> 00:02:56,260
There's a standard
mathematical symbol
68
00:02:56,260 --> 00:02:58,810
that we use to denote
the nonnegative integers.
69
00:02:58,810 --> 00:03:01,880
It's that letter N at
the top of the slide
70
00:03:01,880 --> 00:03:05,600
with a with a
diagonal double bar.
71
00:03:05,600 --> 00:03:07,640
These are sometimes called
the natural numbers.
72
00:03:07,640 --> 00:03:10,190
But I've never been able
to understand or figure out
73
00:03:10,190 --> 00:03:11,870
whether 0 is natural or not.
74
00:03:11,870 --> 00:03:13,950
So we don't use that phrase.
75
00:03:13,950 --> 00:03:17,620
Zero is included in N,
the nonnegative integers.
76
00:03:17,620 --> 00:03:21,680
And that's what we call
them in this class.
77
00:03:21,680 --> 00:03:23,590
Now, I want to point
out that we've actually
78
00:03:23,590 --> 00:03:25,920
used the well ordering
principle already
79
00:03:25,920 --> 00:03:28,060
without maybe not
noticing it, even
80
00:03:28,060 --> 00:03:32,120
in the proof that the square
root of 2 was not rational.
81
00:03:32,120 --> 00:03:35,490
That proof began by saying,
suppose the square root of 2
82
00:03:35,490 --> 00:03:40,530
was rational, that is, it was a
quotient of integers m over n.
83
00:03:40,530 --> 00:03:44,840
And the remark was that you
can always express a fraction
84
00:03:44,840 --> 00:03:46,550
like that in lowest terms.
85
00:03:46,550 --> 00:03:51,480
More precisely, you can always
find positive numbers m and n
86
00:03:51,480 --> 00:03:55,310
without common factors, such
that the square root of 2
87
00:03:55,310 --> 00:03:56,710
equals m over n.
88
00:03:56,710 --> 00:04:00,360
If there's any fraction equal
to the square root of 2,
89
00:04:00,360 --> 00:04:03,220
then there is a lowest
terms fraction m
90
00:04:03,220 --> 00:04:07,100
over n with no common factors.
91
00:04:07,100 --> 00:04:09,140
So now we can use well
ordering to come up
92
00:04:09,140 --> 00:04:14,560
with a simple, and hopefully
very clear and convincing,
93
00:04:14,560 --> 00:04:17,750
argument for why
every fraction can
94
00:04:17,750 --> 00:04:19,610
be expressed in lowest terms.
95
00:04:19,610 --> 00:04:22,890
In particular, let's
look at numbers m and n
96
00:04:22,890 --> 00:04:26,030
such that the square root
of 2 is equal to m over n--
97
00:04:26,030 --> 00:04:27,190
that fraction.
98
00:04:27,190 --> 00:04:30,780
And let's just choose the
smallest numerator that works.
99
00:04:30,780 --> 00:04:34,910
Find the smallest numerator
m, such that squared of 2
100
00:04:34,910 --> 00:04:36,700
is equal to m over n.
101
00:04:36,700 --> 00:04:40,440
Well, I claim that
that fraction, which
102
00:04:40,440 --> 00:04:43,640
uses the smallest
possible numerator,
103
00:04:43,640 --> 00:04:46,780
has got to be in lowest
terms because suppose
104
00:04:46,780 --> 00:04:50,470
that m and n had a common
factor c that was greater
105
00:04:50,470 --> 00:04:53,760
than 1-- a real common factor.
106
00:04:53,760 --> 00:04:57,060
Then you could replace
m over n by m over c,
107
00:04:57,060 --> 00:04:59,440
the numerator is a
smaller numerator that's
108
00:04:59,440 --> 00:05:00,960
still an integer, and n over c.
109
00:05:00,960 --> 00:05:03,360
The denominator is
still an integer.
110
00:05:03,360 --> 00:05:05,530
And we have a numerator
that's smaller than m
111
00:05:05,530 --> 00:05:10,197
contradicting the way that we
chose m in the first place.
112
00:05:12,940 --> 00:05:14,360
And this contradiction,
of course,
113
00:05:14,360 --> 00:05:17,410
implies that m and n
have no common factors.
114
00:05:17,410 --> 00:05:21,410
And therefore, as claimed,
m over n is in lowest terms.
115
00:05:21,410 --> 00:05:23,340
And of course, the
way I formulated
116
00:05:23,340 --> 00:05:26,720
this was for our application
of the fraction that was
117
00:05:26,720 --> 00:05:27,970
equal to the square root of 2.
118
00:05:27,970 --> 00:05:31,670
But this proof actually shows
that any rational number,
119
00:05:31,670 --> 00:05:35,560
any fraction, can be
expressed in lowest terms.