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I know that it isn't actually a number but I do think it's a concept in mathematics. So the question is, is there a symbol representing this concept? I thought maybe it was Phi but I couldn't find it for sure anywwhere.

Answer I was looking for (but inadvertantly phrased in such a way that caused much distress to this community):

"In floating-point computing that would be epsilon." – uncle brad

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Why do you think it is a concept in mathematics? –  Tobias Kildetoft May 2 '13 at 18:53
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In floating-point computing that would be epsilon. –  uncle brad May 2 '13 at 18:57
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Even if we introduce infinitesimals (which is probably at this time not good for your mathematical health), there is no smallest positive infinitesimal. –  André Nicolas May 2 '13 at 18:58
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@WilliamStagner Note that the hyperreals do not contain a smallest number greater than $0$. They contain elements that are greater than $0$ but smaller than all reals. –  Tobias Kildetoft May 2 '13 at 18:59
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thank you @uncle brad. Epsilon is what i was looking for. I guess i meant to limit the domain to floating point computation. –  Ramy May 2 '13 at 19:20
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4 Answers

up vote 1 down vote accepted

In floating-point computing that would be epsilon.

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Before you can give something a name you must make sure there is something to name. You claim (correctly) that it (whatever it is) is not a number. Then before running off to try and find a name for it, start by stipulating what it might be. Some of the comments to your question mention infinitesimals (in the form of the hyperreal numbers) and an algebraic version (in the form nilpotent elements in a ring (e.g., the ring of dual numbers)). Whatever it is you are looking for won't be found there, as (for pretty much the same reason that there isn't a smallest positive real number) there is no smallest positive element in those number systems.

The fact is that the ordering of the reals simply does not allow for such entities and no known useful extension of the real numbers does either. You are always free to invent and ideal new entity, declare it to be a smallest positive element and see what kind of system you get. It won't be pretty or particularly useful. But once you've done that, you can give that entity any name you like. In mathematics there is no concept of smallest positive number.

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In the theory of finite loopy stopper combinatorial game values, which, in some sense, includes a copy of the ring of dyadic rationals (fractions with denominators that are a power of 2), there is a "smallest positive game value". It is denoted +_on, although Conway also called it tiny.

Without developing all of the formal theory, I'll try to describe the idea: Two players, Left and Right are playing a bunch of games together (on each turn, they choose one of the games to make a move in), alternating turns. If it's their turn to move but they have no legal move, they lose. Now, if tiny is one of the games, and it's Left's turn, she can move to end the tiny game (so that component has no more legal moves for anyone), and Right doesn't have this luxury, so tiny confers some advantage to Left (measured by the game value). However, if it's Right's turn to move and tiny is available, he can make a move that Left essentially must respond to immediately (because if she doesn't, Right will gain the ability to "pass" and never run out of moves). This would be the biggest possible threat against Left, which makes tiny the smallest possible advantage for Left.

Now, it's really important to note that the extent to which these games act like numbers is extremely limited, although some games are just like numbers we know (1/2, 3, -4+1/8, etc.), and everyone else's comments to the effect of "this doesn't make sense when dealing with numbers" are absolutely valid. Even in the Surreal Numbers (which, in some sense, include everything that you'd want to call a "positive number", in any context), there is no such thing as a smallest positive number.

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A good account of this is in Siegel's "Combinatorial Game Theory", although it is also covered in the second edition of Winning Ways, and Siegel's "Coping with Cycles" (msri.org/people/staff/levy/files/Book56/12siegel.pdf) –  Mark S. Nov 13 '13 at 2:03
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In Volume 2 of the original Winning Ways, Berlekamp, Conway and Guy use the concept 1/ON in the evaluation of Fox and Geese. (ON is supposed to be the collection of Ordinal Numbers). p645 includes

Fox and Geese = 1 + $\frac 1 {ON}$ in which the left hand side isn't a genuine game, and the right-hand side isn't a genuine number

1/ON is a concept larger than zero, but smaller than any positive number. Sadly, the arguments which lead to this "value" for the game have been shown to be somewhat flawed, but the discussion in context does illuminate the kind of issue involved here, and why 1/ON isn't a number. And it is a fun way of looking at things ...

I did look to see if I could find out the latest on the value of the game. If anyone knows, please comment - else I'll post a question.

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Fox and Geese has value 2+ over and over is called 1/ on in Winning Ways. But while over is infinitesimal, it is not as small as the tiny I describe in my answer. –  Mark S. Nov 13 '13 at 2:01
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