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Short question:

What is the name of the number closest but not equal to zero?

Long question:

Some programmers were discussing about the smallest number close to zero, which is Math.Epsilon or Math.MinValue depending on the programming language. It's value is limited by the definition of the IEEE float or double number format.

The discussion arose whether there is a similar number in Mathematics. We came up with the following idea of its value:

$$0.\overline{0}1$$

Is there a name of that number, the closest number to zero, not equal to zero?

Asaf Karagila
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Thomas Weller
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    Seem to remember this was already discussed to death on MSE. – Did Feb 02 '14 at 10:37
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    As for your decimal representation, that's, to put it in programming terms, a syntax error. It's meaningless to put the "and so on and so on" bar in the middle of the expression. It's supposed to be a shorthand for an infinite sequence of digits, but what sequence of digits would that map onto? A sequence with all zeros... and a one? That's contradictory. I suppose you could index the sequence with some strange ordered set other than $\mathbb N$ if you wanted to get really deviant. – Jack M Feb 02 '14 at 11:11
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    Either vote to close or post as duplicate, or comment ; but do not pointlessly downvote and -not- comment. That is just harassment. – Patrick Da Silva Feb 02 '14 at 23:38
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    This number is also known as $0$. – JLA Feb 02 '14 at 23:55
  • @JLA: Maybe $0.\overline{0}1$ is not the correct value of the number I'm looking for. If you know the value, but not the name, feel free to edit the question and specify the value precisely. – Thomas Weller Feb 03 '14 at 08:19
  • This seems related: http://math.stackexchange.com/questions/379544/is-there-a-symbol-for-the-idea-of-the-smallest-value-greater-than-zero – Martin Sleziak Feb 03 '14 at 14:52
  • @ThomasW. According to the definition of the real numbers, the number you wrote cannot be anything but $0$. There is no real number that is closest to $0$ but not equal to it. – JLA Feb 03 '14 at 18:10

5 Answers5

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There is no such real number. This is simply because if you would have such a number, you could divide it by two and again get another number closer to your previous one, and still not zero.

Patrick Da Silva
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    Got me by 10 seconds ! – Traklon Feb 02 '14 at 10:37
  • Not to mention the fact that you could also have something like $0.\overline{00}1$: even closer to zero. :-D – Agustí Roig Feb 02 '14 at 10:47
  • Well, it may be similar to $\infty$, where you can also add 1 and have a number larger than $\infty$. Still, $\infty$ it has its own name (or symbol). – Thomas Weller Feb 02 '14 at 12:57
  • Yep, but $\infty$ is NOT a real number. And nope, $\infty +1$ is not larger than $\infty$. Mathematicians are strange guys, aren't they? – Agustí Roig Feb 02 '14 at 16:06
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    @Thomas W. : If you want to go into extensions of the fields of the real numbers, this is an entire different story, and the notion of "number" is not what we usually refer to there. I'd rather be explicit when I refer to such extensions and not call the newly obtained so-called "elements", "numbers". This would lead to confusion. For instance, one could consider the set of all formal power series in $\varepsilon$ : $$ \mathbb R[\varepsilon] \overset{def}= \left{ \left. \sum_{n \ge 0} a_i \varepsilon^i , \right| , a_i \in \mathbb R \right} $$ – Patrick Da Silva Feb 02 '14 at 23:32
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    Where $\varepsilon$ has the property that $0 < \varepsilon < x$ for all $x > 0$. However, turning this into an ordered field requires a bit of work. – Patrick Da Silva Feb 02 '14 at 23:33
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This number is known as an infinitesimal. In modern analysis it doesn't exist as part of the real numbers, but in the days of Newton and Leibniz it was used quite often. Note, however, that the concept of "the closest number to zero" is internally inconsistent, because then $\frac12x=x$, so $2x=x$, and that violates the rule that if $0<a$ then $a<a+a$.

In the mid-19th century it was shown that calculus can be based on finitary means which does not require infinitesimals, but in the mid-20th century it was shown that calculus can be constructed in such way where infinitesimals exist. However this will not add a lot of new theorems to calculus, because what is provable with infinitesimals will be provable without it; and it will cause the field of numbers that we get with infinitesimals not to be Dedekind-complete, which is a very nice property that we use often.

The field where infinitesimals are used is called non-standard analysis, and it uses fields of numbers called hyperreal numbers (but also others).

Asaf Karagila
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  • Indeed, the number would not be divisible any more. Both, Newton and Leibnitz, did not give the number a name? – Thomas Weller Feb 03 '14 at 08:23
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    From Wikipedia I gather that it was something like "evanescent quantity". Quite poetic. If you want to be sure for yourself, find a copy of Newton's Principia. – Asaf Karagila Feb 03 '14 at 08:27
  • @Asaf, thanks for your answer. Note that in many contexts such as integration the relevant piece of information is a hyperfinite grid with infinitesimal increment which is a fixed infinitesimal $\epsilon>0$. As fas as a name goes, one can certainly work with $dx$ which is what Leibniz did. – Mikhail Katz Feb 03 '14 at 13:26
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    Although not the highest ranked answer, it's my preferred one. It's good to understand and includes the fact that some genius has already tried to find the number and maybe didn't give it a special name. – Thomas Weller Feb 03 '14 at 15:24
  • @user72694: I'm not saying that non-standard analysis is not useful. Its techniques can be illuminating in understanding how integration works. But as far as proving new theorems... – Asaf Karagila Feb 03 '14 at 17:10
  • @Asaf, we have a new article in the Notices AMS where we describe a new aspect of decreasing rearrangements of real functions using the hyperreals. Such a description is not available in the A-track. – Mikhail Katz Feb 03 '14 at 17:19
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    @Thomas W. I agree with you that Asaf Karagila's answer is a great one and I'm sorry if it seemed I was laughing at you at certain points in mine. Nevertheless, I think you should bear in mind that this infinitesimals that Karagila's is talking about are NOT numbers in the standard meaning of the word "number". As other answers have pointed you out, there is an inherent contradiction trying to work with infinitesimals as everyone does with "usual" numbers. – Agustí Roig Feb 03 '14 at 18:33
  • Aside: the hyperreals are internally Dedekind complete, which is often enough. –  Apr 06 '14 at 20:33
  • @Hurkyl: A lot of things are "often enough", but we're usually interested in "comfortable" rather than "sufficient". – Asaf Karagila Apr 06 '14 at 20:34
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Besides what Patrick Da Silva already told you, you should understand that there is a difference between the limits of a programming language -the smallest (positive) number that it can handle-, due to physical limits of our computers, or of our software, and mathematical objects, such numbers, that don't have such limits.

For instance, according to Mathworks, the smallest positive number that Matlab can work with is

$$ 2.2251e-308 = 2.2251 \times 10^{-308} \ , $$

which is pretty small, isn't it?

But your mind is more powerful than Matlab and has no problem to divide it by $2$, getting

$$ 1.11255 \times 10^{-308} \ . $$

So, yeah, you've approached zero better than Matlab can: isn't that amazing?

Ok, on a more serious tone, you could talk about something similar to the "smallest positive number" with non-standard analysis and its notion of infinitesimal number, but I'm not sure you would like it.

Agustí Roig
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There is the concept of "machine zero" which is useful. In many programming languages, if you start with a nonzero floating point value $x=1.0$ and repeatedly divide by (say) $2.0$, as in the assignment $x\leftarrow x/2.0$, you will eventually reach a point where the machine rounds the answer to zero. The value immediately before the division was rounded to $0$ can be regarded as the smallest positive number you can work with. This sort of thing is used to avoid accidentally using an "increment" or "step" value that doesn't really change a number when you add it to the number, so you don't fall into an endless loop when trying to step across a finite interval.

MPW
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  • I could hardly believe that this works for starting points different to $2^{x}$. But I tried in C# for 1...50 as well as $\frac{1}{2}$ ... $\frac{1}{50}$ and it works. Thanks. – Thomas Weller Feb 03 '14 at 20:26
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In terms of the extended decimal expansion of a hyperreal introduced by Lightstone, the number you wrote down makes sense as the infinitesimal with the first nonzero digit "1" at a suitable infinite hypernatural rank. You might want to think what $1-0.\overline{0}1$ would look like but this tends to provoke passions at SE :-)

As far as names are concerned, Leibniz referred to such numbers as being inassignable as opposed to usual numbers we could measure (the assignable ones).

Mikhail Katz
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