>> Games without chance. Let's, let's shuffle the cards and see

what I get this morning. This is what I get this morning.

That's a really mm not very good hand. Unless you're playing High-Low.

In which case, it's probably good. But that's a chance move, and there's no

chance moves in this course. The Section we're doing now is entitled

More Numbers. So thus far, we have dyadic rationals.

But it turns out there's many more numbers.

Now, one of the things we've been kind of. Somewhat ambiguous about, in this whole

business, is how do you feel about games that have infinitely many options?

Do you like them? I do.

Mathematicians tend to like games with infinitely many things.

Sometimes, computer scientists don't. Because it takes too much space to store

infinitely many things. But, suppose you had square root of 2.

How would you get that as a game? Square root of 2 is not even a rational

number. So how, how can we express it as a dyadic

rational. Well, what you do.

Is, is, is something, is a trick due to [foreign], 19th century German

mathematician, which is you put all the numbers on the left, you put all the

numbers less than square root of 2. And on the right you put all the numbers

bigger than square root of 2. So you look at all the dyadic rationals.

1 quarter, well let's see. What is, what is square root of 2?

Square root of 2 is 1.4 something, right? So, so 1 and a quarter is less than square

root of 2, so 1 and a quarter goes over here, in addition to a lot of other

numbers. And over here 2's 1 and a half 1 and a

half is 1.5 which is bigger than square root of 2.

So 1 and a half goes over here in addition to a lot of other numbers.

So, you look at all the dyadic rationals, every time square it.

If the dyadic ration-, if the square of the dyadic rational is less than 2, you

put if over here. And if the square of the dyadic square is

bigger than 2, you put it over here. And it turns out this infinite gain

because then it has infinitely many options over here.

And infinitely many options over here is equal to square root of 2.

And that way you can get all the irrationals.

By the way you need this actually to get one third also because one third is not

expressible in dyadic rationals. Now if you're a mathematician you can have

a lot of fun here because you can do this. Take a look at this game.

Take a look at the game whose left options are zero, one, two, three, four, and keep

on going forever, and has no right options at all.

Mathematicians actually have a name for this.

This is infinity, but in mathematics there's millions of, there's infinitely

many infinities. And the infinity that this is, is called

omega. And you can even have more fun.

Take a look at the game whose left options are zero, and your right options are 1

half, 1 4th, 1 8th, 1 16th, etc., etc. Then this is a game that's in between here

and here and so this game is positive, left always wins but its less than any

pris, any direct rational. And so actually this game is called one

over omega. And so this is infintesinal where as this

is infinite. So with games like this, one gets whole

bunches of numbers that aren't ordinary real numbers.

And, in fact, Donald Knuth gave a name to all of these numbers that you get and

they're called surreal numbers. Because they contain the real numbers, but

all kinds of other weird things. Okay, that's all I have to say about, more

numbers.