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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

Ramy
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    Why do you think it is a concept in mathematics? – Tobias Kildetoft May 02 '13 at 18:53
  • Try reading this article: http://en.wikipedia.org/wiki/Hyperreal_number. There are mathematically rigorous treatments of infinitesimal quantities, like you are describing. – William Stagner May 02 '13 at 18:55
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    In floating-point computing that would be epsilon. – uncle brad May 02 '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 02 '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 02 '13 at 18:59
  • We talk about arbitrarily small numbers close to a given value...but "the smallest number greater than zero" does not exist, nor is it meaningful to try and conceptualize the existence of a *smallest* number $x$ greater than $0$. For any $x > 0$, there exists an $\epsilon$ such that $0 \lt \epsilon < x$ – amWhy May 02 '13 at 19:00
  • Taking a leaf out of Miles Reid's "Undergraduate Commutative Algebra," we can also talk about "really small numbers" algebraically, in terms of nilpotent elements of a ring. A nilpotent element $\epsilon$ in a ring $R$ is an element such that $\epsilon^n = 0$. Geometrically, one can think of a (nonzero) nilpotent element $\epsilon$ as "so small that $\epsilon^n = 0$, but $\epsilon\neq 0$." For example, if we're working in $\Bbb Z/4\Bbb Z$, $2\neq 0$, but $2^2 = 0$. – Stahl May 02 '13 at 19:01
  • Would you expect astronomers to have a name for the green cheese moon? – Thomas Andrews May 02 '13 at 19:07
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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 02 '13 at 19:20
  • "In mathematics (particularly calculus), an arbitrarily small positive quantity is commonly denoted ε; see (ε, δ)-definition of limit." -http://en.wikipedia.org/wiki/Epsilon – Ramy May 02 '13 at 19:25
  • What about a symbol for the concept of a largest natural number? – Adam Nov 13 '13 at 23:36
  • @unclebrad It sounds like if you post your comment as an answer, it will be accepted. Otherwise, Ramy should probably post that comment as an answer and accept it. – Mark S. Nov 14 '13 at 04:30
  • @MarkS. - Thanks, will do. – uncle brad Nov 14 '13 at 14:47
  • See also: http://math.stackexchange.com/questions/660559/what-is-the-name-of-0-overline01 – Martin Sleziak Feb 03 '14 at 19:13
  • a downvote for accepting a wrong answer. eps is not the smallest value greater than zero – miracle173 Feb 12 '21 at 07:45
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    @miracle173: if the answer is wrong, you should downvote the answer. Downvoting the question should usually be done if the question itself is somehow deficient, not because in your view the wrong answer was accepted. That'd be like me going to downvote your answers simply because I disagreed with your downvote here :-) I'm not going to do that, of course, just trying to point out the possible illogicality of your action. Having said that, I don't want to start an argument, I'm hoping that (assuming I can't convince you) we can just agree to disagree and move on. – paxdiablo Feb 19 '21 at 23:47

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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.

Mark S.
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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" (http://www.msri.org/people/staff/levy/files/Book56/12siegel.pdf) – Mark S. Nov 13 '13 at 02:03
  • If the downvoter has some criticism they would like to share either publicly or privately (contact info in profile), I would appreciate it so that I can improve this or future posts. – Mark S. Feb 16 '21 at 17:16
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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.

Ittay Weiss
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  • Isn't infinity also "not a number"? That hasn't stopped it being a rather useful concept. In fact, is not calculus built on the concept of limits, which relies on approaching zero but never quite reaching it? Not saying your answer is wrong, just questioning some of the presumptions behind it. – paxdiablo Feb 19 '21 at 20:52
  • I don't quite understand what you are trying to say. Firstly, which presumptions are you questioning? I haven't made any. Yes, infinity is not a number. Yes, it is a useful concept. Many concepts are useful and are not numbers. In fact, most concepts are not numbers. Yes, calculus is built on the limit notion. No, the limit notion does not rely on approaching zero but never getting there. The limit notion does not formalise any such vague idea. Rather it formalises the concept of $0$-order approximation. – Ittay Weiss Feb 19 '21 at 21:22
  • It was the comment "The fact is that the ordering of the reals simply does not allow for such entities". I may have been mistaken but that seemed to me to be saying that, unless a "number" is concrete (in that it can be ordered), it cannot exist (to which I proposed infinity as a counterpoint). If I've misunderstood what you were trying to say, I apologise, but it may be worthwhile clarifying that bit. Though, of course, that's entirely up to you and I don't feel strongly enough to downvote because of it :-) – paxdiablo Feb 19 '21 at 23:33
  • I am still perplexed. It is a theorem, not an opinion or presumption, that the real numbers system does not contain any entity with the property of being the smallest positive number nor an entity with the property of being infinity. – Ittay Weiss Feb 20 '21 at 07:43
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In floating-point computing that would be epsilon.

uncle brad
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  • Actually epsilon is the smallest value such that 1+epsilon is greater than 1. – miracle173 Jul 30 '20 at 06:20
  • @miracle173: if you subtract one from both sides of that equation, epsilon becomes the smallest value greater than zero :-) – paxdiablo Feb 09 '21 at 07:13
  • @paxdiablo: No, that is wrong. epsilon depends on the length of the mantissa (the size of the precision), The smallest number is related to the size of the exponent. float.h defines macro for this values. On a typical intel machine epsilon is about $10^{-7}$ and the smallest normal representable positive number is about $10^{-38}.$ "What Every Computer Scientist Should Know About Floating-Point Arithmetic" by DAVID GOLDBERG is a well known article about floating point arithmetic that describes these constants. – miracle173 Feb 12 '21 at 08:12
  • @miracle173: this is a *math* site, so you should probably be using the mathematical description of epsilon (from calculus). The vagaries of IEEE754 floating point representation is irrelevant in that case. – paxdiablo Feb 12 '21 at 12:00
  • @paxdiablo "In floating-point computing that would be epsilon" means that he talks about the epsilon defined in floatin-point computing. You ignore this so I think you are a troll. – miracle173 Feb 18 '21 at 08:16
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    @miracle173, not a troll although a troll would say that as well so you never can tell :-) If it helps, you can check my SO activity, far more active than here and indicative of non-trolliness. I was just stating that, given this is a math site, we would be better off using the math definitions. The "in floating point" comment was added to the question after a comment by uncle brad but that's just an indication of a related concept. It seems to me the question itself is not asking about computer-based implementations of math, rather it asks about a math concept providing the same thing. – paxdiablo Feb 19 '21 at 23:39
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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.

Mark Bennet
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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 02:01