-1

For example the probability closest to 100 % or the number closest to infinity, does that have a representation that I can write?

I suppose it would be ∞-1/∞ ?

For example, it happens that somebody says that the probablity is 99.9999999999....%

Similarly, we have the concept of infinity in calculus, a infinitely large quantatiy or number, isn't there an expression for the smallest quantity or the smallest number so that I could more formally express the idea?

Niklas Rosencrantz
  • 2,265
  • Isn't $x$ closest to $x$? – Sil Jan 07 '17 at 16:40
  • 1
    Do you mean $\approx$ – Jason Born Jan 07 '17 at 16:43
  • The probability closest to 100% is 100%, and as many questions on this side tell you $99,\bar 9% = 100%$. And there is no number $x\in \mathbf R$ closest to infinity, as $x+1\in \mathbf R$ is still larger than $x$. – martini Jan 07 '17 at 16:43
  • But I mean including subtracting the infinitismal, similar to x-1/∞. – Niklas Rosencrantz Jan 07 '17 at 16:45
  • First worry about what you are trying to say, then about notation. The "number closest to infinity" is the greatest number (of a collection). There is no number closest to infinity unless you specify a collection to draw from. Similarly there is no number closest to $1$ unless it is $1$ itself. – Ross Millikan Jan 07 '17 at 16:45
  • Without context of the question in title, it is almost impossible to answer it. –  Jan 07 '17 at 16:45
  • The smallest increment before x? I.e. x-1/∞? – Niklas Rosencrantz Jan 07 '17 at 16:46
  • Also in "For example the probability closest to 100 % or the number closest to infinity", what how do you define "closest"? –  Jan 07 '17 at 16:47
  • @Jack An infinitesimal increment before. The name for that point. – Niklas Rosencrantz Jan 07 '17 at 16:49
  • In modern mathematics, one must normally have precise definitions of the things one wants to study before doing anything interesting with them. In particular it makes no sense to ask for such notation for a woolly concept like “the closest thing to $x$ [but not equal]” when such a thing cannot be precisely defined. Similarly, writing $x-\frac1\infty$ makes no sense. Without a precise definition of $\infty$ or division and addition with such “numbers,” it makes no sense to define something by that sort of sequence of symbols. – Dan Robertson Jan 07 '17 at 16:55
  • There exist some precise definitions/axiomatisations of extended real number systems which include infinites and infinitesimals. For example the surreal numbers or the hyper real numbers. These tend to require some advanced mathematics and plenty of mathematical maturity to work with but do lead to some nice elegant results. For example, they can make sense of the $\frac{\mathrm dy}{\mathrm d x}$ notation more than formally. As far as I am aware, in none of them could you have a concept like “the maximum number less than x.” Perhaps you should familiarise yourself with the idea of a supremum. – Dan Robertson Jan 07 '17 at 16:59
  • 1
    There is no number "closest" to any other number, since there is always the number halfway between them. This is regardless of whether you're using the usual real number line or any of a number of infinitesimal systems, like the hyperreals and surreals. – Arthur Jan 07 '17 at 17:05
  • I could accept the answer that the limit doesn't exist because it can always be split into two limits(?) – Niklas Rosencrantz Jan 07 '17 at 17:09
  • 1
    @DanRobertson Even in the Surreals (or anything else like an ordered field) there won't be a smallest positive number: you can always divide a positive number by 2. – Mark S. Jan 07 '17 at 17:12
  • @MarkS. I don't know a great deal about such systems and I was not banking on the ordered-field assumption. I thought perhaps a system could exist with some extra conditions on the types of thing that +,-,etc applied to. – Dan Robertson Jan 07 '17 at 17:56
  • IIUC, there can't be any "infinitesimal incremenent" because you can always divide it by 2(?) – Niklas Rosencrantz Jan 07 '17 at 18:01
  • @Dan Robertson, depending on what you want, there is something like this, as I mention in my answer. – Mark S. Jan 07 '17 at 18:15

1 Answers1

1

It sounds like you're looking for a "smallest positive number (in a context with subtraction)", and there are three interpretations that come to mind:

1. Is there a smallest positive real number?

No. For the real numbers, and indeed any other context (even weird ones with "infinitesimals") where you can divide and talk about "positive" numbers, you can always divide a positive number by $1+1$ to get a smaller positive number.

2. Is there a smallest positive number in any context with subtraction?

Yes. As Ross Millikan pointed out in a comment, $1$ is the smallest positive integer, so you can look at things like "$5-1$ is the integer right before $5$". Since you mentioned 100%, I doubt this case will be satisfying to you.

3. Is there a smallest positive number in any more interesting context where there are fractions like 1/2?

Yes. But it's really esoteric. In combinatorial game theory, a certain position where one player can make a threat to win everything but the other can shut that down right now has reason to be considered the smallest positive thing. In the relevant context, fractions like $\frac12$ make sense (and are other game positions). For a bit more detail, see my answer here.

Community
  • 1
Mark S.
  • 23,925
  • Doesn't this imply that there is no "infinitesimal" since you can always divide it by 2 to get a smaller? – Niklas Rosencrantz Jan 07 '17 at 18:32
  • @Dac Saunders, 1. "Positive Infinitesimal" usually means something like "less than 1/n for any positive integer n", not "a smallest positive thing". 2. In any standard context where division and declaring things positive/negative/zero is always possible, yes, you can just divide by 2 to make it smaller. This is true even of infinitesimals (you just get smaller infinitesimals). – Mark S. Jan 07 '17 at 18:54