Talk:0.999.../Arguments

Main Talk Archives
May 6, 2005 – November 16, 2005 November 16, 2005 – December 7, 2005 December 7, 2005 – December 9, 2005 December 9, 2005 – December 20, 2005 December 18, 2005 – February 11, 2006 February 11, 2006 – February 23, 2006 February 23, 2006 – June 1, 2006

Arguments Archives
February 24, 2006 – May 19, 2006

This page is for mathematical arguments concerning 0.999.... Previous discussions have been archived from the main talk page, which is now reserved for editorial discussions.

The number that "approaches 1 but isn't" does not exist

Copy Summarized from normal talk By E-Bod

Also for those of use that use "0.999 repeating" as a physical representation of an abstract number "approaching but not equal to" one we need a new variable. Has a new variable been given (No pun in tended).

For instant although .999¯ equals one ... when we graph the number line (-∞,1),(1,∞) or (x≠1) How do we illustrate the "conceptual number" before and after 1 or any other whole number. Before we proved this we just used .999¯ and .000¯1 to say this concept. Now that we Proved this number actually does not exist we need a new way of saying what we have to say. This Proof is hard to Grasp because we don't have an alternative to illustrate the idea.

Remember that "1" is a symbol for a number. And if ".999¯" equals one then it is just a #REDIRECT [[1]]. We now need a new Title for XYZ.--E-Bod 22:38, 11 June 2006 (UTC)

Somebody still needs to address what to call the number that approaches 1 but isn't. Just like X^0 is one not zero if you make a table and look at it. E-Bod

That's the point of the proof: The number that "approaches 1 but isn't" does not exist (at least, not in the standard reals we usually employ). As to the significance, I can't think of any "real world applications" right now. This proof should just show the non-existence of such an "approaching number" and thus clarify our mental image of the reals. . Yours, Huon 09:04, 12 June 2006 (UTC)

Ok I accept that The number that "approaches 1 but isn't" does not exist. In that Definition of exist it seems really Obvious. That's not a Tangible Number. However If Approaching but not Equal to Does not Exist then ∞ Does not Exist Eather. Lets say you make a tan graph Image:Tangent.svg There is a the Concept of the Last point you gen go to Approaching π/2 And still be in The Domain of the Function. We have represented this as π/2)(π/2 or ≠ π/2. As a Teaching tool we could say for a similar graph That went to infinity at alone as we can't have one but we can have .9 .99 .999 .9999 etc or 1.1 1.01 1.001 1.0001 1.00001. Just like the Final number does not Exist in the Real world we still have imaginary numbers with real life applications (I think rocket science or something). I know that in some Online games it is possible to get 2 points connect with a 0 distance rope. Ideal gasses (Which don't exist) take up 0 Volume at 0K (Kelvin). Even if the number does not exist that does not mean we can't still represent the non existing number. If somebody Does Find the replacement other than .9999Bar then we are laking something in this article. Or in Math Knowledge in General. Just remember No Original Research--E-Bod 00:07, 13 June 2006 (UTC)

First, I don't think you have given a concrete example where usage of .999¯, 0.000¯1 or the like really helps in any way; 0 did just fine for the ideal gas example (though if one also dicusses negative temperatures, the symbol +0 can be used to emphasize that this is the positive zero). Even if such an example is provided, I'm not sure it is within the scope of the article, as it is about a mathematical concept, where such vague explanations have no place. This doesn't mean that there isn't a mathematical way to address the ideas you describe - Just that they aren't really related to what the article is about. -- Meni Rosenfeld (talk) 15:45, 13 June 2006 (UTC)

The number you ask for exists, but it isn't in the reals. See infinitessimals. What you want is 1 - dx. Now, as stated before, could you name a single use for it? Or is this just to give you mental closure? -- He Who Is^{[ Talk ]} 19:21, 19 June 2006 (UTC)

From My Limited Point of View

I'm not a mathematician, so I can't really discuss this in terms of higher mathematics. I still, however, have a view. The way I see it, nothing can be stated as fact about a hypothetical number like infinity. The way I see it, we can only speculate about infinity from the view point of finite numbers.

The way I see it, if you subtracted 0.9 recurring from 1, you'd end up with 0.(infinity - 1)0s 1, because, if you subtract 0.9 from 1 you get 0.1 if you subtract 0.99999999 from 1, you get 0.00000001, so it is only logical that the result of 1 - 0.9 recurring should have the same decimal places as 0.9 recurring (infinite) and follow this pattern.

The fact that a third of one has to be defined with infinite nines after the decimal point, merely proves to me that the decimal system is inadequate for calculating a third of one.

To me, the answer will vary depending on one's view of infinity, which, from a philosophical standpoint, is open to interpretation. I'm only willing to accept 1 as equal to 0.9 recurring as one interpretation; though I tend to think sequentially.--Jcvamp 21:37, 23 June 2006 (UTC)

That's why we have the distinction between elementary and advanced proofs. The advanced proofs are probably more difficult to understand, but the important point is: They do not use infinity in any way. For example, the order proof characterizes 0.9999... by two properties:

• It is not greater than 1.
• For any finite 'n', it is greater or equal than 0.999...9 with 'n' nines.

That alone, without any statements about infinity, is enough to show 0.9999...=1. The elementary proofs, on the other hand, indeed are a little fast, and you did find the reason why the fraction proof is not as rigorous as the advanced proofs. Yours, Huon 07:26, 24 June 2006 (UTC)

'For any finite 'n', it is greater or equal than 0.999...9 with 'n' nines.' That makes no sense to me, firstly because 'equal than' is grammatically incorrect, and secondly because it seems as though you're saying that 0.9 recurring is greater or equal to itself by a number of nines. If that's not what you're saying, to what does 'it' actually refer?--Jcvamp 10:30, 24 June 2006 (UTC)

Well, I will first have to object to your statement "nothing can be stated as fact about a hypothetical number like infinity". Mathematics is about starting with definitions and axioms, and seeing what results from them. If we choose to define "infinity" in one way or another (and different definitions can be found in different contexts), we can state about it whatever follows logically from the definition. The philosophical arguments, or points of view, only arise when we try to decide which definition of infinity will best capture our intuitive feeling of what "infinity" should mean, and perhaps, the interrelations between the concept and phenomena in nature. But this is beside the point, partly because, as Huon mentions, "infinity" is not explicitly mentioned in the "real" proofs.

The article is about real numbers - A concept that can rigorously be either defined or characterized with axioms. Every approach leads to the same results, so it doesn't matter which one you choose. One of these results is that every non-negative real number can be represented as one, sometimes two, infinite sequences of digits - And conversely, every infinite sequence of digits represents a real number. The article is about the fact that the real number represented by the sequence 0.999... is the number 1 (the other sequence that represents it is, of course, 1.000...). That much isn't open to debate.

What is open to debate is whether we wish to restrict oursleves to real numbers. Usually we do, since real numbers have a lot of nice properties and they do a good job explaining nature. But if we are willing to accept other numbers, as is done in non-standard analysis, then it is possible to think of 0.999... as representing a number other than 1, such as 1 - 10^-ω. But this is beyond the scope of the article - which is, again, about real numbers.

About Huon's comment, the phrase "a is greater or equal than b" is, while grammatically incorrect, a common shorthand for "a is greater than b or equal to it", which is true whenever b is not greater than a. What Huon said was that the number 0.999... Where the 9's continue endlessly, is at least as great as any number 0.999...999 where the 9's end eventually - there are n 9's and then an infinite sequence of zeros. -- Meni Rosenfeld (talk) 12:56, 24 June 2006 (UTC)

Frank: I'm limited in knowledge but before I read this article I may of believed that 0.999... = 1 however as far as I can tell every single proof on this page is seriously flawed.

Fraction Proof: How can you claim 1/3 exactly equals 0.333...? I mean it is the same as claiming that 0.999... = 1

Algebra Proof: 10 × 0.9999... = 9.9999... I think it equals 9.999... (Notice the different in 9's) You have literaly added 1/Inf to it so the proof becomes 0.999... + 1/Inf = 1

Advanced proof: Some proof that 0.999... is a rational number is in order here.

Order proof: Errr playing with decaimals and 1/Inf is probaility not going to work :p Best proof so far.

Limit proof: You do understand why we say this right? "The limit of ...."

Geometric: As above

As far as I can tell if you claim 0.999... = 1 then you are claiming 1 / Inf = 0

For all values of X: X / X = 1

As X tends to infinity:

Inf / Inf = 1

X / X = 1 is the same as (1 / X) * X = 1

As X tends to infinity:

0 * Inf = 1

There you go a nice proof by condiction?

If you can explain the flaws above I'm willing to listen if you can't you should remove them from the page.

Infinity divided by infinty does not equal 1, and 0 times infinity does not equal 0. The limit of x to infinity of x/x = 1, and the limit of x to infinty of 0*x=0, but using infinity directly, those statements are undefined both over the projected reals and on the extended reals. -- He Who Is^{[ Talk ]} 03:20, 25 June 2006 (UTC)

The fraction proof and the algebra proofs are not really proofs; You shouldn't pay to much attention to them. "Advanced proofs" is just a title for the following 2 proofs. In any case, it is proven that 0.999... = 1 and 1 is of course rational. Now, the order and limit proofs rely on the construction of real numbers as dedekind cuts and cauchy sequences, respectively, of rational numbers. You must understand these constructions in order to understand these proofs. See also construction of real numbers. Now, can you show me a single place where, in either of these proofs, the word "infinity" is mentioned (and no, "infinitisemals" in the order proof doesn't count, it's just mentioned as a pleasent way of describing the Archimedean property)? You can't, because it isn't. There are all sorts of things you can do with the concept of infinity, but they aren't relevant as the proofs do not use this concept. They only use properties of natural numbers and rational numbers. So, in some contexts we would say that 1 / ∞ = 0; Here we don't. We only say that for every ε > 0 there exists a real (or rational, or natural) number M > 0 such that if x > M Then |1/x| < ε. (See Limit (mathematics)) -- Meni Rosenfeld (talk) 08:06, 25 June 2006 (UTC)

As far as I'm concerned, if someone says anything claiming it's fact, I won't necessarily believe it is true. I will ask for evidence and make a judgement based on the evidence. To me, that's simply me not being gullible, and trying to use common sense.

In regards to this, none of the arguments supporting the idea that 1=0.9 recurring, make any sense. Perhaps it's my limited understanding of mathematics. Until someone can explain this to me in a way that makes sense, I can't view this as a fact. This isn't me being arrogant, this is me not believing things unless they make sense to me. I'm still open to the possibiliy that 1 could equal 0.9 recurring, but so far the idea doesn't sound rational.

Also, people keep on saying that the concept of infinity isn't used in the advanced proofs. How can you exclude the concept of infinity from a number that has recurring (aka infinite) digits? Again, this might just be me being too stupid to comprehend what you're saying, but I would like to understand the reasoning behind the claim.--Jcvamp 13:39, 25 June 2006 (UTC)

One other thing. Why does infinity follow different rules from any other number? I don't understand how, when dealing with infinity, which is immeasurable, it can be used in calculation without viewing it in the same way we'd view a finite number. If we can assign a value to, for example, X in algebra, in order to test a formula, why can't we do the same thing with infinity?

Now I've just started thinking. If infinity is a limit, and nothing is higher than infinity, then infinity multiplied by 10 would be infinity... Can you see why I called this entry 'From My Limited Point of View'? I still think, for this reason, that the decimal system is simply inadequate to calculate a third of 1, and that it can only be correctly represented as a fraction.--Jcvamp 13:49, 25 June 2006 (UTC)

Infinity isn't a number - you can't do sums with it like "10 times infinity". Where infinity comes in is when dealing with a series - an infinite sum is defined as the limit of the sequence of partial sums. 0.999... is not defined as "having an infinite number of digits", rather it is, by definition, an infinite sum, namely, the infinite sum 0.9 + 0.09 + 0.009 + ...

So the problem boils down to showing that the series: 0.9, 0.99, 0.999, ... has a limit of 1. Note that nowhere in this do we treat infinity as a number - saying things like "infinite sum" is just a shorthand way of describing these methods.

I'd suggest reading up on:

• Recurring decimal
• Series (mathematics)
• Limit of a sequence

Mdwh 14:58, 25 June 2006 (UTC)

I suppose the most useful response would be addressing your questions one at a time. So here goes:

True, believing blindly everything you're told is probably not a good idea. But statements regarding a given subject should be given more weight when said by experts in that subject. You would believe your physician when he's giving you medical advice, right? Likewise, if mathematicians tell you something about a mathematical issue, you should at least be inclined to believe them. You have the right to challenge their claim - but only if you accept the obligation to learn enough about the topic so that you may have an informed opinion.

So, there is a problem in stating that you don't believe the proofs in the article because you lack the mathematical knowledge to understand them. Unless you specify the exact points in the proofs which make you uneasy (which I will be more than happy to hear), it is your responsibility to study the necessary background information required to understand them. Not everything can be satisfactorily explained in simple terms anyone could understand, definitely not in mathematics. Regarding the specific argument in the article, before we can sensibly discuss whether 0.999... is equal to 1 or not, we must first agree on what 0.999... is (arguing about the value of something is pointless if each of us thinks of a different thing). I would probably say that 0.999... is:

$\{A|A\subseteq \mathbb {N} \times \mathbb {Q} \;\land \;\forall n(n\in \mathbb {N} \rightarrow \exists q((n,q)\in A))\;\land \;\forall n\forall q_{1}\forall q_{2}((n,q_{1})\in A\land (n,q_{2})\in A\rightarrow q_{1}=q_{2})\;\land \;\forall \epsilon ((\epsilon \in \mathbb {Q} \land \epsilon >0)\rightarrow (\exists N\forall n\forall q(n>N\land (n,q)\in A\rightarrow -\epsilon <q-\sum _{i=1}^{n}(9\cdot 10^{-n})<\epsilon ))\}$

In other words, it is the equivalence class, in terms of Cauchy sequences of rational numbers, of the sequence $a_{n}=\sum _{i=1}^{n}(9\cdot 10^{-n})$ (did you see how the word "infinity" wasn't mentioned?). You would probably not understand what I meant. If we work on it, we can perhaps reach a definition we can both understand and agree on. But this is a thing we must do before we hope to discuss whether it is, or is not, equal to 1.

Why does infinity (when we choose to see it as a number) follow different rules from any other number? I'll ask a different question. Whenever you calculate a + b, where b is not 0, you get a result which is different from a : 5 + 3 ≠ 5, 8 + 9 ≠ 8, etc. But when you calculate a + 0, you get a: 4 + 0 = 4, 6 + 0 = 6. Why does 0 follow different rules from any other number? Because it is not any other number. 0 is 0, an other number is an other number. 0 follows its own rules, one of them being that a + 0 is always a. So does infinity has its own rules, one of them being that ∞ / ∞ is undefined.

10 times ∞, in some contexts, is indeed ∞. I can't see the relevance of this, though. It has been proven that the decimal system is adequate to represent any real number, including 1/3, or any other rational number. But again, in order to discuss this you must first understand what a real number is - the article real number might be a good start. -- Meni Rosenfeld (talk) 16:24, 25 June 2006 (UTC)

Another proof was given in the archives. This one is just as rigogous, but more detailed than the advanced proofs given in the article; thus, it might be easier to understand. Its main disadvantage is that it does not employ a definition of 0.9999..., but two properties 0.9999... should satisfy:

• For every natural number n, 0.9999... (with infinitely many nines) is greater than 0.999...9 (with n nines).
• 0.9999... is not greater than 1.

Furthermore, one property of the real numbers is needed: For any positive real number x, there exists a natural number n(x) such that n(x)*x > 1 (the Archimedean property, used in several equivalent variations). If you are prepared to accept those properties as preconditions, reading up the proof may be worthwile. On the other hand, showing these properties is more difficult, and would probably once more require university-level maths. I hope that helps. Huon 10:01, 26 June 2006 (UTC)

'So, there is a problem in stating that you don't believe the proofs in the article because you lack the mathematical knowledge to understand them.'

I'm not going to take anything as fact until I KNOW it's true. In this case, I lack the mathematical knowledge to say 'I know that is true', so I can't say I believe it as fact. I am, however, willing to admit that I don't understand it, and ask questions to help me learn about it.

'But statements regarding a given subject should be given more weight when said by experts in that subject. You would believe your physician when he's giving you medical advice, right? Likewise, if mathematicians tell you something about a mathematical issue, you should at least be inclined to believe them.'

Firstly, I'm not saying I don't give more weight to statements made by experts. Secondly, there are other people arguing against this issue who seem to know as much about mathematics as the people arguing for it. Thirdly, I have no ways of knowing who is an expert on here and who isn't. That's why, I prefer to look at things objectively.

'it is your responsibility to study the necessary background information required to understand them.'

Exactly. I'm not asking anyone to tutor me in mathematics. I thought I'd let you know my position on this so you'll understand why I'm questioning the idea of 0.9 recurring equalling 1. As it stands, I'm more open to the possibility of it being true, and I am willing read up on it on my own.

The main reason I decided to ask questions on this page is because a friend of mine said to me, the other day 'Isn't 0.9 recurring equal to 1?', and I said I didn't think it was. She said it was a mathematical fact that it was, and showed me the process of dividing 1 by 3, then multiplying it by 3 and getting 0.9 recurring.

I couldn't, based on that explanation, accept what she said was a 'mathematical fact'. She went on to tell me my view was wrong, and asked me how I'd feel if someone said 'The world is flat'. So, anyway, that's why I decided to ask you the same questions I asked her, and though I don't readily understand all of the answers, I can see more mathematical basis to them, and I now know what to study.

Thanks for your help.--Jcvamp 21:07, 26 June 2006 (UTC)

Very well. I agree with all of the above. I certainly do agree that saying "0.999... = 3 * 0.333... = 3 * (1/3) = 1" isn't very convincing, or mathematically accurate (unless, of course, one proves that the long division algorithm holds for infinite representations - which I believe is harder than our original question). I hope we've helped you get a feel of how this statement can be really proven. I also hope you've read the proof Huon referred to. It doesn't begin with the fundumentals, but rather with some statements that are very reasonable, and not too hard to prove based on the fundumentals - but from that starting point, it proceeds in a sufficiently rigorous and detailed way. The only initial assumption which may be hard to accept is the Archimedean property - but it may sound more plausible if you rephrase it as, "for every real number x, there is a whole number n such that n > x". Of course, it can be proven rigorously based on a precise definition of a real number, and it is in fact the main idea that separates the good old real numbers from other number systems, such as the hyperreal numbers and the surreal numbers. -- Meni Rosenfeld (talk) 09:45, 27 June 2006 (UTC)

Incorrect use of Algebra?

Aren't the algebraic proofs easily refutable? 0.333...*3 has never equaled 0.999... according to my knowledge of simple algebra. That would be an incorrect multiplication of repetends, and go against the multiplicative property. Likewise with 10*0.999...=9.999... — Preceding unsigned comment added by 207.118.6.122 (talk • contribs)

No, it's correct. How exactly does is the multiplication of repetends incorrect? In 0.333...*3, 3 is multiplied by each digit in 0.333..., or 1/3, yielding the infinite series 0.999..., or 1. The same goes for 10*0.999..., which equals 10. Supadawg (talk • contribs) 01:21, 2 September 2006 (UTC)

Well, 0.333...*3=0.333...+0.333...+0.333..., and when placed that way, they do not add up to 0.999... The addition of the repetends makes a difference, how does the multiplication not?

What is 0.333...+0.333...+0.333... then? Melchoir 01:36, 2 September 2006 (UTC)

1? — Preceding unsigned comment added by 207.118.6.122 (talk • contribs)

Yes, they add up to 1, but they also add up to 0.999.... This is because 0.999... equals 1. Please explain how they do not add up that way, because you can proceed in adding them up the same way as with multiplication: .3 + .3 + .3 = .9, .03 + .03 + .03 = .09, into infinity. And please sign your comments, using four tildes (~~~~) so we can keep track of who said what. Supadawg (talk • contribs) 01:48, 2 September 2006 (UTC)

Loclity, Non-locality and the Real-line

The standard Set/Member relation is based on a xor connective.

For example: Any given x is (a member) xor (not a member) of set A and there is no intermediate state.

Fuzzy logic expands standard membership (0 xor 1) by using x, which defines the degree of a membership between 0 and 1 (0 or 1 are included too).

If 0<x<1, and [0,x] belongs to set A then (x,1] does not belong to set A ( [0,x] xor (x,1] )

So, in both cases a xor connective is used as the logical basis of the Set/Member relations.

0, 1, [0,x] or (x,1] (where 0<x<1) are all local mathematical objects because we can clearly define their locality (they are "in" xor "out" of some mathematical object).

An object that is not a set but can be a member of a set, is called an urelemnt ( http://72.14.221.104/search?q=cache:JRO16vlJWNQJ:en.wikipedia.org/wiki/Urelement+urelement&hl=en&ct=clnk&cd=1 ).

A sub-object is a part of an object.

Since an urelement is not a set, it does include any sub-object as a part of it, or in other words, it is an atomic singleton.

x is a urelement.

If x is a member and not a member of A then x is a non-local mathematical object.

The best way to notate this is: _{_} , where __ is a urelement.

__ can be both a local ( {__} xor __{ } ) or a non-local ( _{_} ) urelement.

If we wish to find the best way to notate a local urelement (a local atomic singleton) than we use . (a point).

An example: {.} xor .{ }

In the standard Set/Member relations each member is a sub-object xor not a sub-object of a set, and both the empty and any a non-empty set are defined by the ways that these sub-objects belong xor don't belong to them.

Since . and __ are atomic singletons, they are not defined by each other (they are mutually independent exactly like two axioms) and we can expand the membership concept beyond the Set/Member dependency.

n>1

k=n-1

If we are using a membership among mutually independent objects, then 0.kkk...[base n] < 1

For more details please read http://www.geocities.com/complementarytheory/TOUM.pdf

DoronSahdmi 21:03, 11 September 2006 (UTC)

Perhaps. This may have made sense where it was originally placed, but it doesn't have anything to do with .999... = 1. — Arthur Rubin | (talk) 23:28, 11 September 2006 (UTC)

Please read the bold sentence that I wrote above , DoronSahdmi 11:18, 12 September 2006 (UTC)

"membership among mutually independent objects" means ...? I still assert that — even if it means something to you — it has nothing to do with any recognized set theory. (I started trying to read the above-mentioned paper. I got completely lost after the third or fourth definition where it wasn't clear whether variables or operations were being defined.) — Arthur Rubin | (talk) 07:14, 13 September 2006 (UTC)

Doron, a quick google search indicates that you seem to believe you have re-invented mathematics. At best, this is exteremely presumptuous. I wish you good luck in your quest, but that doesn't change the fact that Wikipedia is definitely not the place for your speculations. -- Meni Rosenfeld (talk) 08:19, 13 September 2006 (UTC)

My argument is vert simple: If a non-local urelement (a line for example) and a local urelement (a point of example) are two independent atoms, then no non-finite collection can be a non-local urelement.

In other words, any given collection is an incomplete mathematical object when it is compared to the completeness of a non-local urelement.

In that case 0____1 is a non-local urelement and no sequence (where a sequence is an ordered collection) of the form 0.kkk...[base n] (n>1 , k=n-1) can reach its completeness.

Instead of speculating abuot me please reply about my argument.

DoronSahdmi 20:38, 13 September 2006 (UTC)

A line is, by definition, not an urelement but a set. Furthermore, 0.9999... is not a sequence, but the limit of a sequence (or the sum of a series). Your use of non-standard mathematical language is rather irritating, and I agree with Arthur Rubin that your "definitions" are extremely unclear. Finally, you might want to consider that real numbers usually are defined as equivalence classes of Cauchy sequences of rationals; thus, not even 1 (considered as a real number) is an urelement, but a set... Yours, Huon 21:15, 13 September 2006 (UTC)

I would be more than happy to respond to your argument, if only I understood a word of it. Unfortunately, it doesn't contain even one mathematically meaningful statement. -- Meni Rosenfeld (talk) 07:58, 14 September 2006 (UTC)

Indeed by the standard notion, a line is a set.

I am talking about a mathematical framework, where a line is not a set. In this framework a line is a non-local urelement, and in this framework 0.999... is not a limit of a sequence but a sequence that exists on inifintly many scale levels that cannot reach the completeness of a non local urelement (in my new framework, the real-line is a non-local urelement).

I clearly showed above the logical basis that is used in the case of standard (0 xor 1) logic, Fuzzy-logic ( [0,x] xor (x,1] where 0<x<1 ) and my new compelementry-Logic, where x is "in" and "out" A is not a contradiction but the logical basis of non-locality.

I wonder what prevent from you to grasp the existence of a framework, where a local urelement (a point, for exmaple) and a non-local urelement (a line, for example) can be members of a set, but no one of tham is a set.

In this case a line and a point are two atoms (non-composed objects) and as a result, no set of infinitly many points is an atomic non-local urelement.

In this framework, 0.kkk...[base n] (n>1 , k=n-1) is smaller than 1. DoronSahdmi 15:56, 14 September 2006 (UTC)

Your "new framework" is precisely what Meni Rosenfeld meant by "reinventing mathematics". Now while that might be a great new idea, unless you publish it in a peer-reviewed journal or something like that, it is also original research, which is by policy unsuitable for Wikipedia. Thus, this article should reflect the standard framework of mathematics. --Huon 17:07, 14 September 2006 (UTC)

This work was presented as a "short communication" under the name "Complementary Mathematics" (the presentation can be seen in http://www.createforum.com/phpbb/viewtopic.php?t=60&mforum=geproject ) by my colleague Moshe Klein in http://icm2006.org/v_f/web_fr.php at 25.8 17:15 at room 403r and after the presentation, more than 40 mathematicians wished to get a copy of the work. Also I personally presented it to Benoit Mandelbrot and he was exited by the new mathematical possibilities of this work. DoronSahdmi 22:02, 14 September 2006 (UTC)

If your work has been received so well, I'm sure it will be published soon. We can wait. Melchoir 22:40, 14 September 2006 (UTC)

The Membership concept needs logical foundations in order to be defined rigorously.

Let in be "a member of ..."

Let out be "not a member of ..."

Definition 1: A system is any framework which at least enables to research the logical connectives between in and out.

Let a thing be nothing or something.

Let x be a placeholder of a thing.

Definition 2: x is called local if for any system A, x is in A xor x is out A implies true.

The true table of locality is:

in out

0 0 → 0

0 1 → 1

1 0 → 1

1 1 → 0

Let x be nothing

Definition 3: x is called non-local if for any system A, x is in A nor x is out A implies true.

The true table of non-locality when x is nothing is:

in out

0 0 → 1

0 1 → 0

1 0 → 0

1 1 → 0

Let x be something

Definition 4: x is called non-local if for any system A, x is in A and x is out A implies true.

The true table of non-locality when x is something is:

in out

0 0 → 0

0 1 → 0

1 0 → 0

1 1 → 1

Let system Z be the complementation between non-locality and locality.

The true table of Z is:

in out

0 0 → 1

0 1 → 1

1 0 → 1

1 1 → 1

Let us return once more to Fuzzy-Logic:

If x=0 then x is not a member.

If 0 < x < 0.5 then x is mostly not a member.

If x=0.5 then x is equaly a member and not a member.

If 0.5 < x < 1 then x is mostly a member.

If x=1 then x is a member.

All these memberships are based on the proportion between x and 1, in system Z.

If the system is limited only to R members, then the proportion between x and 1 is distinct.

If non-local numbers are included, then the proportion between x and 1 is non-distinct.

In system Z the non-local number 3.14...[base 10] < local number pi, ... etc.,..., etc.

DoronSahdmi 16:53, 17 September 2006 (UTC)