>> 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.