Surreal numbers actually have a close connection to games.
Chess is a game. But if we sit down to play Chess and I make a move, then the result is in some sense a new game. We could have set up the board that way originally (like a Chess problem in a newspaper) but chose not to in the interests of fairness.
This observation lets us develop a notation for games where we can write any game like this: [L,R]. Here the set L is the set of all states (that is to say other games) which can result from the player on the left making a move, whilst R is the corresponding set available to the player on the right. (In a real game of Chess after I had moved it would be your turn, but clearly it makes sense to imagine a game of Chess where White still plays first but the board is not in the traditional starting position.)
What would happen if we gave a player two games of Chess to play but told them they could only make a move in one (and the same for their opponent)? Each turn they would need to determine the value of moving on each board.
Once you start looking at a wide variety of different games, it turns out that this idea of assigning a value to a game has interesting properties. Some games have an obvious value. For example, we could play a game called "Take the point!" where whoever plays first gets a point. Pretty easy to assess! Some games have values which are much harder to find.
Where it all gets interesting is when you start doing sums with games. So in our "two games of Chess" example above, what we have really done is to add the games! And as you look deeper into the possible values of different types of games, something interesting turns up. There are some kinds of games, it turns out, which have values which can be compared to numbers, but don't actually correspond to anything we've seen before. So stuff like (informally) "a number that is smaller than all the numbers that are bigger than zero".
Very approximately, these weird possible game values that turn up are the Surreal Numbers! (More precisely, there are some other values which turn up that aren't numbers at all, Surreal or otherwise.)
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Chess is a game. But if we sit down to play Chess and I make a move, then the result is in some sense a new game. We could have set up the board that way originally (like a Chess problem in a newspaper) but chose not to in the interests of fairness.
This observation lets us develop a notation for games where we can write any game like this: [L,R]. Here the set L is the set of all states (that is to say other games) which can result from the player on the left making a move, whilst R is the corresponding set available to the player on the right. (In a real game of Chess after I had moved it would be your turn, but clearly it makes sense to imagine a game of Chess where White still plays first but the board is not in the traditional starting position.)
What would happen if we gave a player two games of Chess to play but told them they could only make a move in one (and the same for their opponent)? Each turn they would need to determine the value of moving on each board.
Once you start looking at a wide variety of different games, it turns out that this idea of assigning a value to a game has interesting properties. Some games have an obvious value. For example, we could play a game called "Take the point!" where whoever plays first gets a point. Pretty easy to assess! Some games have values which are much harder to find.
Where it all gets interesting is when you start doing sums with games. So in our "two games of Chess" example above, what we have really done is to add the games! And as you look deeper into the possible values of different types of games, something interesting turns up. There are some kinds of games, it turns out, which have values which can be compared to numbers, but don't actually correspond to anything we've seen before. So stuff like (informally) "a number that is smaller than all the numbers that are bigger than zero".
Very approximately, these weird possible game values that turn up are the Surreal Numbers! (More precisely, there are some other values which turn up that aren't numbers at all, Surreal or otherwise.)