1
00:00:00,012 --> 00:00:04,537
>> Games without chance.
Let's, let's shuffle the cards and see

2
00:00:04,537 --> 00:00:09,006
what I get this morning.
This is what I get this morning.

3
00:00:09,006 --> 00:00:14,552
That's a really mm not very good hand.
Unless you're playing High-Low.

4
00:00:14,552 --> 00:00:20,102
In which case, it's probably good.
But that's a chance move, and there's no

5
00:00:20,102 --> 00:00:29,003
chance moves in this course.
The Section we're doing now is entitled

6
00:00:29,003 --> 00:00:38,254
More Numbers.
So thus far, we have dyadic rationals.

7
00:00:38,254 --> 00:00:47,213
But it turns out there's many more
numbers.

8
00:00:47,213 --> 00:00:59,366
Now, one of the things we've been kind of.
Somewhat ambiguous about, in this whole

9
00:00:59,366 --> 00:01:06,103
business, is how do you feel about games
that have infinitely many options?

10
00:01:06,103 --> 00:01:08,121
Do you like them?
I do.

11
00:01:08,121 --> 00:01:14,037
Mathematicians tend to like games with
infinitely many things.

12
00:01:14,037 --> 00:01:20,633
Sometimes, computer scientists don't.
Because it takes too much space to store

13
00:01:20,633 --> 00:01:26,024
infinitely many things.
But, suppose you had square root of 2.

14
00:01:26,024 --> 00:01:31,913
How would you get that as a game?
Square root of 2 is not even a rational

15
00:01:31,913 --> 00:01:35,970
number.
So how, how can we express it as a dyadic

16
00:01:35,970 --> 00:01:38,597
rational.
Well, what you do.

17
00:01:38,598 --> 00:01:44,172
Is, is, is something, is a trick due to
[foreign], 19th century German

18
00:01:44,172 --> 00:01:49,752
mathematician, which is you put all the
numbers on the left, you put all the

19
00:01:49,752 --> 00:01:55,502
numbers less than square root of 2.
And on the right you put all the numbers

20
00:01:55,502 --> 00:02:01,251
bigger than square root of 2.
So you look at all the dyadic rationals.

21
00:02:01,252 --> 00:02:05,906
1 quarter, well let's see.
What is, what is square root of 2?

22
00:02:05,906 --> 00:02:11,987
Square root of 2 is 1.4 something, right?
So, so 1 and a quarter is less than square

23
00:02:11,987 --> 00:02:17,265
root of 2, so 1 and a quarter goes over
here, in addition to a lot of other

24
00:02:17,265 --> 00:02:21,971
numbers.
And over here 2's 1 and a half 1 and a

25
00:02:21,971 --> 00:02:26,551
half is 1.5 which is bigger than square
root of 2.

26
00:02:26,551 --> 00:02:32,881
So 1 and a half goes over here in addition
to a lot of other numbers.

27
00:02:32,881 --> 00:02:38,691
So, you look at all the dyadic rationals,
every time square it.

28
00:02:38,691 --> 00:02:42,985
If the dyadic ration-, if the square of
the dyadic rational is less than 2, you

29
00:02:42,985 --> 00:02:46,188
put if over here.
And if the square of the dyadic square is

30
00:02:46,188 --> 00:02:50,057
bigger than 2, you put it over here.
And it turns out this infinite gain

31
00:02:50,057 --> 00:02:53,091
because then it has infinitely many
options over here.

32
00:02:53,091 --> 00:02:56,783
And infinitely many options over here is
equal to square root of 2.

33
00:02:56,784 --> 00:03:00,221
And that way you can get all the
irrationals.

34
00:03:00,221 --> 00:03:05,987
By the way you need this actually to get
one third also because one third is not

35
00:03:05,987 --> 00:03:11,867
expressible in dyadic rationals.
Now if you're a mathematician you can have

36
00:03:11,867 --> 00:03:16,961
a lot of fun here because you can do this.
Take a look at this game.

37
00:03:16,961 --> 00:03:23,223
Take a look at the game whose left options
are zero, one, two, three, four, and keep

38
00:03:23,223 --> 00:03:27,076
on going forever, and has no right options
at all.

39
00:03:27,076 --> 00:03:30,591
Mathematicians actually have a name for
this.

40
00:03:30,591 --> 00:03:37,791
This is infinity, but in mathematics
there's millions of, there's infinitely

41
00:03:37,791 --> 00:03:43,185
many infinities.
And the infinity that this is, is called

42
00:03:43,185 --> 00:03:46,666
omega.
And you can even have more fun.

43
00:03:46,666 --> 00:03:54,797
Take a look at the game whose left options
are zero, and your right options are 1

44
00:03:54,797 --> 00:04:02,941
half, 1 4th, 1 8th, 1 16th, etc., etc.
Then this is a game that's in between here

45
00:04:02,941 --> 00:04:10,877
and here and so this game is positive,
left always wins but its less than any

46
00:04:10,877 --> 00:04:21,100
pris, any direct rational.
And so actually this game is called one

47
00:04:21,100 --> 00:04:30,945
over omega.
And so this is infintesinal where as this

48
00:04:30,945 --> 00:04:37,263
is infinite.
So with games like this, one gets whole

49
00:04:37,263 --> 00:04:42,311
bunches of numbers that aren't ordinary
real numbers.

50
00:04:42,311 --> 00:04:48,983
And, in fact, Donald Knuth gave a name to
all of these numbers that you get and

51
00:04:48,983 --> 00:04:58,601
they're called surreal numbers.
Because they contain the real numbers, but

52
00:04:58,601 --> 00:05:08,052
all kinds of other weird things.
Okay, that's all I have to say about, more

53
00:05:08,052 --> 00:05:09,311
numbers.