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First of all I want to say that when coming to math - I know absolutely nothing - so please forgive me if my question is not "scientifically" correct, if it is not "syntax-correct" - or even too trivial or "stupid".

Is there a scientific name for 0.infinity ? (ZERO point INFINITY)

And what would this value be ?

0.000000000000000....(infinity n) ..... 01

or

0.000000000000000....(infinity n) ..... 09

(I guess it is a kind of chicken-egg question)

or even 0.9999999999....(infinity n) ..... 99

I guess I can already know that the answer is that 0.infinity is none of the above and it is simply 0.infinity .

But then, how can I pronounce the smallest possible fracture ? (numeric fracture)

This doubt was established when I was trying to write some DESIGN related algorithm for my PHD and during that - I wanted to define to myself what would be the SMALLEST possible theoretical fracture of any given "whole"..

would 1-(0.infinity) produce the smallest fracture ?

I know that "infinity" is a bit more "philosophical" than "mathematical" - but then again I always saw math as a kind of philosophy (maybe wrongly so..) I also know that a lot of operations with infinity will result in "undetermined" (which is another big philosophical term) - but still I would like to define this to myself.

I am sorry if this question is a bit confused - but any help on any of the above doubts would be greatly appreciated.

P.s.

Please explain it like you would explain it to your 2-years-old-neighbor´s child - or in other words - in lame man´s words ..

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It makes no sense to talk about a number that has a 1 "after" infinitely many zeros. There is no "after" position for that 1 in the usual decimal notation of numbers. Something like "$0.\infty$" also is nonsensical as is in the usual setting of numbers, because the decimal notation requires each position to be occupied by a digit, and $\infty$ is not a digit. –  Arturo Magidin Mar 18 '12 at 5:38
    
@arturo - but 0 is a digit .. so an infinite series of 0 followed by something ? –  Obmerk Kronen Mar 18 '12 at 5:41
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But you can't have an infinite sequence of $0$s "followed" by something. There is no end to an infinite sequence of $0$s, so there is no place where you can put whatever it is you want to "follow it" with. Not in the usual decimal notation, where each digit must be in a location that is finitely many positions away from the decimal point. –  Arturo Magidin Mar 18 '12 at 5:42
    
All the numbers you can write down in a decimal expansions will have either finite digits (e.g. $9.78$) or infinite expansion (e.g. $1.333333\ldots$), but nothing transfinite; nothing beyond those magic three dots. –  user2468 Mar 18 '12 at 5:44
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@Obmerk: Just because you can think of something does not mean that it can be put into solid logical foundation; in fact, that is precisely the problem that Russell's paradox highlights (look it up if you are not familiar with it). In the usual setting of the real numbers, there is no such thing as "smallest possible part of the whole", regardless of the fact that you can think of. There is no "last integer", there is no "first positive real number." –  Arturo Magidin Mar 18 '12 at 20:01

4 Answers 4

up vote 5 down vote accepted

Too long for a comment.

The smallest possible number you are looking for depends on the context.

For example, in a computing context where the storage and representation bits are finite, you might want to consider the smallest floating-point number with no rounding errors. This is dependent on the finite capacity of your machine to represent really small numbers with finite amount of bits. Typically, that is called machine-epsilon.

In an absolute sense, then it's a little bit harder to define a smallest number. For example if Joe claims that particular real number, $\epsilon \in \mathbb{R},$ is the smallest positive number ever. Then I can refute Joe's claim simply because $\dfrac{\epsilon}{2}$ is still a positive real number, and it is smaller than what Joe proposed. And so on. We can always find smaller and smaller real numbers. This property is called density.

As a solution to your problem, you can define your smallest number $\epsilon$ to be an infinitesimally small (read: really small, indistinguishable from zero). Alternatively, define your $\epsilon$ to be arbitrary small number (read: smaller than all variables and quantities in your equations/algorithms). If you need to prove some results, then you might want to consider studying the behaviour of your equations/algorithms under $\lim_{\epsilon \to 0}.$

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Thanks . I guess I will just use words to describe what I need which is "a potion that is equal in size to the infinite division of a given defined space". –  Obmerk Kronen Mar 19 '12 at 7:14

Infinite digital expansions represent infinite sums (that's literally what the notation means to mathematicians) in standard real analysis. This means that there is some sequence of partial sums

$$a_0, \quad a_0+\frac{a_1}{10}, \quad a_0+\frac{a_1}{10}+\frac{a_2}{10^2}, \quad a_0+\frac{a_1}{10}+\frac{a_2}{10^2}+\frac{a_3}{10^3}, \quad \cdots$$

with each $a_i\in\{0,1,\dots,9\}$ at the heart of every number $a_0.a_1a_2a_3\cdots$. There is no room for e.g.

$$0.\underbrace{000\cdots0}_\infty1$$

because expansions can only encode what happens "before infinity." That isn't to say you can't construct such "numbers" and their ordering, they just aren't the real numbers then. (See here.)

The moral to take away here is that irrational real numbers are the result of a limiting process; you never get to define the "end" of the process per se, but only encode the process itself as a number, very loosely speaking. This is seen at a higher conceptual elevation with something called completions, which is formed here by equivalence classes of Cauchy sequences.

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First, the notion you have in your head is antithetical to the familiar notion of arithmetic. In particular, if there was a smallest positive number, then you get a contradiction after multiplying it by (1/2).

You can always invent all sorts of new structures, even including new arithmetic systems with smallest positive numbers. (The integers are such a system, by the way....)

In fact, it is rather common behavior in mathematics for one to write down a list of properties they want some mathematical structure, then try to reason with that structure, and to try and prove it consistent by constructing a model of that structure: e.g. how Cauchy and Dedekind constructed models of the real numbers using set theory and rational numbers)

Of course, unless you give them reason to, people probably won't be excited about the new system. One mistake you should try very, very hard to avoid: don't confuse the new structure you invent with any pre-existing structures. You shouldn't go around saying things like "there's a smallest positive fraction" unless you make it clear you're not talking about the real* numbers, but instead the system you invent.

*: "real" is a technical term for a particular number system. Don't confuse it with the English word "real".


Second, you have a misunderstanding about decimal notation. The places are indexed by integers: for every place, there are only finitely many places between it and the decimal point, and there is no "last" decimal place.

One can, of course, invent new notational systems as they please. But it won't be notation for the real numbers. Again, people are unlikely to be excited unless you can show your new notational system is interesting.

Some particular keywords that you may find interesting in this regard are "order type" and "ordinal number".

(and maybe "hypernatural number", but that is likely to lead you through a minefield of new ways to misunderstand things, since the relevant theory requires some amount of care and precision to use correctly)

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Thanks for the explanation. It is actually a very interesting field . It is still hard for me to accept that there are issues that can not be described by mathematical terms - but I guess i will just have to live with it . I have absolutely no intention nor the capability or the desire of creating any new notational systems . I would be content to first fully understand what a notational system is :-) –  Obmerk Kronen Mar 19 '12 at 7:18

The scientific name for $0\cdot\infty$ is indeterminate form in the context of the calculus. What this usually means is that you have one term tending to zero, another term tending to infinity, and you are interested in the behavior of their product. It is called an indeterminate form because apriori it cannot be determined what the behavior of the product will be without having additional information on the two terms. Namely, the product may tend to zero, stay bounded, tend to infinity, or exhibit other behaviors depending on the specific factors. The expression indeterminate form appears to have been introduced originally by Cauchy's student, Abbot Moigno.

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