A part two, you could say, of my previous question.
I was watching a Numberphile video about the favorite number of some mathematicians, and at one point, the creator of ViHart said the following –
Star. Which, it's not a number. It's like a number. The thing is, it's
smaller than all the positive numbers, and it's larger than all the
negative numbers. But it's not zero. It gets confused with zero.This is one of those weird things you get in combinatorial game
theory, and, I like it. I think the very non-numberness of it, I think
that's what appeals about a lot of these numbers.
The video itself can be found here.
I've searched the internet for anything about star, or $\ast$, as she wrote, but I can't seem to find anything that explains it well. All I understand is that it has something to do with combinatorics.
Would anyone know what it is, it's application, it's properties, or what it means, in terms that are possibly more clear than those found on Wikipedia, or any other sources that pop up from searching for star?
Best Answer
Don't think of $\ast$ as a number to begin with; that will only confuse you. $\ast$ is a mathematical object called a combinatorial game. An example of a combinatorial game you may have played before is Nim. In general, combinatorial games are characterized by having two players who take turns making moves, and also by the fact that both players have perfect information about the game (so a card game where you don't know your opponent's hand is not a combinatorial game).
$\ast$ itself is a pretty boring game: the player who moves first wins.
So:
Why does anyone care about $\ast$? In combinatorial game theory, combinatorial games admit a very elegant recursive definition: they are defined in terms of other games! Using the methods of combinatorial game theory, you can analyze complicated games in terms of simpler games, and sometimes when doing this $\ast$ appears as one of the simpler games you analyze.
Why did Vi Hart say that $\ast$ is "like a number"? It turns out that combinatorial games behave like numbers in many ways: you can add them as well as say when one game is greater than or equal to another game. There are also some combinatorial games that behave like the usual numbers you're used to, but others that behave very differently, like $\ast$.
To really understand this situation I recommend that you learn a little abstract algebra and number theory, which will teach you to think about various kinds of numbers. As far as combinatorial game theory itself, my understanding is that the classical texts here are Conway's On Numbers and Games and Berlekamp, Conway, and Guy's Winning Ways for your Mathematical Plays.
If you play chess or go, combinatorial game theory can be applied to understanding chess and go endgames (see Elkies for the former and these resources for the latter). In particular, you can use chess or go positions to write down some of the weirder combinatorial games that don't behave like numbers.