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I was looking at this video and i wondered what she means by star. How is this number defined and where does it come up. Thanks in advance.

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up vote 1 down vote accepted

In the field of combinatorial game theory, certain two player games are assigned "values" that contain all of the information one might want to know about them. Some games' values are familiar numbers, which are similar to (but not quite the same as) points for one of the two players (traditionally named "Left" and "Right" but you can think of them as "white" and "black" in some games): 2 is similar to 2 points for Left, -3 is similar to 3 points for Right, etc. Technically, in the game represented by the value 2, Right can't move at all (so they would lose on their turn for having no legal moves), and Left has a move to the game represented by 1, where Left has a further move to "0", where neither player has a legal move.

"Star", often written "*", represents a particularly boring game where the player whose turn it is now (whether they're Left or Right) gets to move once, and then there are no legal moves anymore. Left isn't guaranteed a win, so it's not like a positive number, and Right isn't guaranteed a win, so it's not like a negative number (and more can be said along these lines), but unlike 0, the person to move now wins instead of loses, so it's not 0. It's a very weird value indeed.

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As an aside, it's worth noting that she is Vi Hart, the owner of a famous youtube channel with lots of explanatory recreational math videos: – Mark S. Nov 12 '12 at 21:27

Star signifies a symmetric first player win, while $0$ signifies a symmetric second player win.

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Star is not the only symmetric first player win. *2, *3, etc. are as well. – Mark S. Nov 12 '12 at 4:27

She's probably talking about the surreal numbers which is denoted $S*$.

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While the Surreal numbers are tied to combinatorial game theory, the * she had in mind is not a Surreal number. – Mark S. Nov 12 '12 at 4:27

She says that it is smaller than all positive numbers and larger than all negative numbers and not zero, so I would interpret it as an infinitesimal - something greater than zero that does not satisfy Archimede's axiom (for any two positive reals x and y, there is a positive integer n such that n times x is greater than y). In this case, for all positive integers n, n times "star" is less than one (or any other positive real).

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Unfortunately, as mentioned in the video, * is "confused with zero": neither greater than zero nor less than zero, but incomparable to it. – Mark S. Nov 12 '12 at 4:29

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