0.99999~ = 1 ?

Do you really not understand? I assumed you were playing riverrat for this thread. I'm still not convinced, but whatever, anything for doubt.The definitions are made up after the fact and rely on some work by Cantor (set theory.) Initially, real numbers are a set devised to include both rational and irrational numbers. What they do not include are things like imaginary and undefined numbers. These are products of the square root of -1 and of 1/0 respectively. To the extent that new formulations are made up that do not seem to constitute rational or irrational numbers in the same way, the definitions will be expected to show why. For example, the square root of negative 2 could be considered irrational, but it isn't, specifically because it is imaginary.

Jon responds to me:

quote:

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What does it mean for numbers to converge upon e?

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On the face of it, this is either a trivial question or a very deep question. I don't know which way you mean. It depends on if you understand what the meaning of "converge" is in a mathematical sense. This is the stuff of Real Analysis which is generally the first course for aspiring mathematicians. Arithmetic, Algebra, Geometry, Trigonometry, Calculus, all of that is simple calculation. With Analysis, you finally start getting into the fundamental concepts of mathematics regarding why it even works in the first place.So to help me understand what sort of answer you are requesting, I'm going to have answer your question with a question:What do you think "converge upon e" means? Is this a question of what "converge" means? Is it a question of what e is? If you can let me know more details about what you are trying to work your way through, I can better respond.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.

Well, I could hardly do better then either Rrhain or Dr. Adequate, both honest-to-goodness mathematicians. Frankly, I'm not even qualified to have the debate; I only know what a real number is because I had read and understood the proof composed by honest-to-goodness mathematicians. And that you don't understand what "converge upon e" means means that you aren't either.So, if I'm not qualified to have the debate, why did I respond? That's because I recognized that you're not trying to debate, you're trying to be clever. (1.999 cleaver by 1/1.999) One doesn't debate established realities. One looks them up. One might debate whether something is an established reality; but, as per my admission and your demonstration, neither of us is qualified to enter that debate regarding real numbers. We are only in the position the sit back and admire our superiors, Rrhain and Dr. Adequate. (Or mine them for information if we can think of a question not so stupid as to embarrass ourselves.)And to touch upon the topic: 1- 0.000= 0.999Edited by lyx2no, : Tried some cleaver formatting that didn't work out as I thought it would.

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You are now a million miles away from where you were in space-time when you started reading this sentence.

And I await your ability to understand SG's argument, but I guess I'll be waiting a long long time... If this is what you took from SG's argument, then you understand neither mathematics nor logic...

So, does the 'real' in 'real numbers' have any relation to the 'real' in 'reality'?I think the confusion has come from me assuming that it in some way does, but the more folk describe this concept mathematically, it appears that I was on the right track with my first post: Which seems in line with what (I think) Catholic Scientist and RAZD have mentioned earlier; namely, that the equality of 0.9999| and 1 is more of a result of the way in which you have defined your system; defining it differently gives us different 'equalities' of infinites to 1. In other words, you have not given a definition (as far as I can tell) of 'real number' that defines it concretely in terms of 'reality', but merely the one you've given defines it in terms of the 'established mathematical system'. (RAZD (Message 12) gave the lovely example of the equivalence of .8888| and 1 in a nonary system.)Let us plug what you've introduced back into the proofsas I understand it(here, property Z is the properties you have listed that define 0.9999|too many to list separately, we variablize them):P "All #s with property Z are REAL within MATHSYSTEM (i.e., ((R/Z)/M) (M=MATHSYSTEM)"
P "0.9999| has property Z"
P "(The REAL of 0.9999| is True) given MATHSYSTEM is True (i.e., ((R/0.9999|)/M))"
P "[proof of M=True]"
C "(R (the REAL of 0.9999|) is True... period"Proof that .9999| is not DISTINCT (from 1):
P "It is false that 0.9999| is both REAL and DISTINCT (from 1) (~(RD))"
P "(R (the REAL of 0.9999|) is True... period"
C "0.9999| is not DISTINCT (from 1)"In essence, there has been little done in regards the explanations given for why 0.9999| is REAL other than to introduce the variable, in form of caveat, MATHSYSTEM, the veracity of which (i.e., its conformity to reality, last premise in first proof) must now be shown to support the proof for 0.9999| being REAL and making the conclusion that it is not DISTINCT (from 1) unavoidable (within the world of reality, in which we all strive to live; if it is the case that the only place where such a proof has relevance is within the world of number manipulation (MATHSYSTEM), and the MATHSYSTEM is not necessarily related to the real world, then I've little interest in continuing to understand the equality of 0.9999| and 1, since it would appear little more than a novelty of an imperfect system which allows functions and manipulations that do not have reality as their basis). Of course, it may be that our conditions for DISTINCT (from 1) are also only definable given MATHSYSTEM (D/M)i.e., that all parts of the proof's conformity to reality hinge on the veracity of the MATHSYSTEM, the disproval of which bringing destruction to the proof, but for now it will be enough to focus on just the REAL of 0.9999|.So, I look forward to your proof of M=true, or the admittance that the MATHSYSTEM is not necessarily related to reality.²Jon
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¹ It should probably be mentioned now that using Unicode (UTF-8) would result in the best display with all these symbols.
² If this is going too far off-topic, we can start another thread; just give the word, Admins.

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[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

Hi Jon,I had some trouble with Son Goku's proof as well. As this seems to be begging the question. Sure (1+1)/2 = 2/2 = 1 but do we have that?Is the 0.999~ in (3) the same as the 0.999~ in (4)? Every time you do it out to the same number of decimals you get different numbers with different remainders, and the average number (4) is always between (3) and 1, where you would expect it to be.I would think that a stronger proof would be to subtract 0.999~ from 1, or 1 from 0.999~, and what you get is a string of 0's, no matter where you stop. Or the frame shift proof: Note that this later method works for any repeating decimal: Enjoy.Edited by RAZD, : added

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If I've erred (possible), then please rephrase it in the form of a proof which can point out my errors; i.e., show which aspects of the following are inaccurate summations of SG's argument, and please correct them:P "All real numbers have properties X"
P ".9999| does not have properties X"
C ".9999| is = 1 and it is false that .9999| ≠ 1"So far, you folk have just been calling me foolish and telling me my logic is faulty; no one has yet to show me why or where. Is this to say that the argument he has given for ((0.9999| = 1) and ~(0.9999| ≠ 1)) is only relevant in the world of mathematics?I look forward to your active participation. Jon

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[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

I was under the impression that infinite numbers (which is what you get if you subtract 0.9999| from 1, no?) never have to stop. When we stop it, we are admittedly not working the the number proposed, which was infinite, and if we leave it be infinite, then we can never assess whether it is equivalent or not (can we?). In other words, there never seems to be a way to link the mathematical 0.9999| with anything in reality that would make it meaningful, such that asserting that a number like 0.9999| exists always appears to be a useless triviality.So, is there any reason to assert the existence of 0.9999| other than to dazzle the Kindergartners?

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[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

No. You get 0. 0 is finite. No. It's equal to 1. 1 is finite. Your meaning is obscure. Yes. Our system of decimal numbers allows us to write it.Would you agree that there is a use for the expression 0.3333|, namely to write 1/3 in decimal notation? And a use for 0.6666|, namely to write 2/3? Very well, what do you get if you add them together using the standard algorithm for the addition of decimal numbers?Oh yes, you get 0.9999|.If we declare that expression meaningless, we have a problem.

I have done so. This is, of course, not true.

Perhaps - we don't know. Next question?

Hi Jon, Well, you are starting with a mathematical concept, rather than objective evidence, and math does not need to conform to reality, just be internally consistent.For instance try to imagine a precisely equivalent 2nd of anything. What we get is 1 + ~1 = ~2 at best, and try to ignore the ~ parts.Second, the 0.999~ repeating decimal is purely an artifact of using the decimal system, an intellectual concept in itself.Enjoy.

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This has been my suspicion from early on, as I said in Message 87: I wish folk would have just agreed with me then that 0.9999| had no real-world significance and was instead just a fancy mathematical parlor trick.Interestingly, if the other folk would follow my proofs through (instead of dismissing them without reason), particularly the ones in Message 110, they would see that for 0.9999| to be a REAL number, and for 0.9999| to not be DISTINCT (from 1), that MATHSYSTEM would have to be such that it is not necessarily related to reality, which is a conclusion of my proofs summed up in your statement: In short, one must accept that either: 0.9999| is not a REAL number; 0.9999| is DISTINCT (from 1); or that the MATHSYSTEM has no necessary relationship to reality. In length, if one believes the MATHSYSTEM to have a relationship to reality, then looking for that link will prove M=False, which allows 0.9999| to not be REAL, which fails to prove 0.9999| = 1. However, once we accept that MATHSYSTEM has no necessary relationship to reality, then we can make M=True without second thought and 0.9999| will have to be REAL, which means that 0.9999| being DISTINCT (from 1) will have to be false, upholding our claim that 0.9999| and 1 are the same. In other words, our proofs or 0.9999| = 1 will not work unless we accept the MATHSYSTEM-REALITY link to be arbitrary; as much as the link between a word and its meaning, there is no necessary link between a number and the reality it attempts to describe. I expect the mathematicians to disagree. I did; thank you Jon

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[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

This is gibberish.Let me try to explain the situation.If we restricted ourselves to the rational numbers, then various perfectly nice and useful quantities such as pi and the square root of two would simply not exist in our number system.Therefore, we construct a system, the real numbers, in which these and other such irrational quantities are numbers (indeed, so that every Cauchy sequence has a limit).We then require some convenient form of representation whereby we can write the real numbers and perform arithmetical operations such as addition and multiplication on them. It is hardly convenient to work with Cauchy sequences for practical purposes.One representation of the real numbers commonly used for this purpose is decimal notation. One of the things which we are allowed to do in decimal notation is to write that a sequence of digits after the decimal point recurs indefinitely. This allows our notation to express precisely all rational numbers, which is nice.Since we are allowed to represent such infinite string of digits in our notation, we can write 0.9999~.If we declare this to be not equal to 1, we involve ourselves in paradox, for then we could write 1 = 3/3 = 1/3 + 2/3 = 0.3333~ + 0.6666~ = 0.9999~ ≠ 1.We could simply place a taboo on writing 0.9999~ (and any other expression ending in recurring nines), but in the first place, why should we? --- and in the second place, such expressions will arise naturally from our algorithms for addition, as when we add 0.3333~ and 0.6666~.

Excellent. So you have chosen to assert that there is a disconnect from MATHSYSTEM and reality, and without such a necessary link, you are free to claim MATHSYSTEM = True. This is exactly what one would expect in a system which represents two identical real-world values (values from REALWORLDSYSTEM) with two different symbologies. (That is, one would expect there to be a necessary disconnectand not a necessary relationbetween any two systems if the same thing in one system had altering representations in the other, cf. Language.)In fact, it is this disconnect that is primarily at the heart of the necessity of the DISTINCTness criteria, which says that "for any numbers A and B with the following properties X, they are not formally DISTINCT in the REALWORLDSYSTEM, though being formally distinct in the MATHSYSTEM". This is necessary, because as you point out, without it, we could get ourselves into a paradoxical mess in which the MATHSYSTEM would start out representing the REALWORLDSYSTEM and then end up not doing so (which is fine if we do not wish the system to work, but we do wish it to, so it is far from fine).1 = 3/3 = 1/3 + 2/3 = 0.3333| + 0.6666| = 0.9999| ≠ 1So, the MATHSYSTEM introduces a function that equates 0.9999| with 1 and thereby closes the paradoxical loophole, such that we may continue using the MATHSYSTEM as a representation of the REALWORLDSYSTEM. Of course, just because we can mend one system so that it will represent another, does not indicate a necessary representative property of the former system in regards the latter, i.e., it does not show that the former system represents by necessity the latter system, but merely shows that it represents it, not necessarily by necessity.Afterall, any system that represented necessarily another system would not need a function to close paradoxical loopholes, as such loopholes would not exist. The fact that the MATHSYSTEM has introduced such a function in an attempt to represent the REALWORLDSYSTEM, shows that the MATHSYSTEM is not necessarily representative of the REALWORLDSYSTEM.But all these things arise, of course, because MATHSYSTEM is not in a necessary relationship with REALWORLDSYSTEM. Afterall, in REALWORLDSYSTEM 0.9999| v. 1 is a formality which is meaningless... how d'you say... gibberish.JonEdited by Jon, : A is for: Indefinite article

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[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin