0.99999~ = 1 ?

Actually, I have chosen not to recite gibberish of your own invention. As, for example, the system of fractions, in which 1/2 = 2/4. I have noted that no-one gets their knickers in a twist over that. The difference is that any systematic way of mapping decimal representations to real numbers must necessarily identify 0.9999~ as 1, whereas there is no equivalent necessity that, for example, the sequence of letters l-a-r-g-e should map to the same concept as the sequence b-i-g.In the case of 0.9999~ it is a matter of logical necessity. If we want our decimal notation and the associated algorithms to represent the structure of the real numbers, then 0.9999~ must be equal to 1. If it isn't, then our notation does not represent the real numbers. And now you're off on your own again. I recommend the English language to you as a medium of communication.Edited by Dr Adequate, : No reason given.

LOL. Whatever. When you have more to say other than "I don't get it", I will be ready to continue with you.Jon

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

Jon writes:

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So, does the 'real' in 'real numbers' have any relation to the 'real' in 'reality'?

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That is a philosophical question, not a mathematical one. You are playing to the distinction between Platonists and non-Platonists.Most mathematicians are Platonists, meaning that the objects that mathematics studies are real. "Three" is a real property just as much as color and texture are.And thus, the "Real" numbers are, indeed, real. They form the basis of the "continuum," which can be represented by the number line.There are some people, however, who do not share this idea. They think the concepts of mathematics are just that: Concepts. Abstractions of thought with no more reality than any other passing thought: "Ceci n'est pas un pipe." A picture of a pipe is not a pipe.But to most mathematicians, Real numbers truly are real. They are the basis of actual properties in the world. There is a precise distance between objects and that distance is part of the Real numbers.While it isn't nearly rigorous enough, a good shortcut description of the Reals is that it is the set of the Rationals and Irrationals. Numbers like pi and e, they are all Irrational. If we go with the idea of numbers like 1 and 12 being things that we can see, then we can certainly see other numbers like those and thus, the "Reals" have an effect upon our lives.

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namely, that the equality of 0.9999| and 1 is more of a result of the way in which you have defined your system

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You act like this is a bad thing. All proof functions this way. You define a systematic set of axioms and then derive proofs based upon the assumption that those things are true. Unless you are trying to pull a Bertrand Russell and question the very foundations of mathematics, then your argument makes no sense.If you are going to question that .999... = 1, then you will also necessarily have to question things like the Pythagorean Theorem, the quadrature of the lune, and all other mathematical proof.Is that what you're doing?

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defining it differently gives us different 'equalities' of infinites to 1.

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No, defining it differently leads to nonsensical statements. This isn't like non-Euclidean geometry where we can just tweak the Fifth Postulate and come up with something new and functional. You are trying to say that number really isn't number.0.999... = X
10 * 0.999... = 10X
9.999... = 10X
9.999... - 0.999... = 10X - X
9 = 9X
1 = X
Therefore, 0.999... = 1That isn't a question of "definition" the way you are using the term.

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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'.

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Why do you think that is a requirement? Surely your argument doesn't reduce to you not liking the use of the term "Real" to describe these numbers, does it?If the sum totality of your posts is that you simply wish to declare yourself to be a non-Platonist, then simply say it and be done with it.

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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.

When you are willing to communicate in the English language, please feel free to continue.As it is, you appear to be expressing your own mental confusion in a language of your own invention, the terms of which you have not troubled to define.I cannot correct mistakes that you cannot even adequately express.

RAZD writes:

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Sure (1+1)/2 = 2/2 = 1 but do we have that?

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Yes.Are you seriously questioning arithmetic? Don't get me wrong. There is something to be said about trying to get at the foundations of mathematics that allows us to do arithmetic. Bertrand Russell tried it with his Principia Mathematica, but you will note that it took him more than 62,000 steps to prove that 1 + 1 = 2.Now, if you really want to go that deeply into the system, then by all means do so. Just be up front that that's what you're doing and don't pretend that you are onto some fantastic secret that nobody has ever thought about.Very brilliant minds have already walked down this path. They have written lots of books about. While the concepts can be summarized here, the actual methodology cannot be because not only is it too complicated, we don't have any way of formatting the symbology required here in this forum.So a simple question: Are you seriously questioning the chain that (1 + 1)/2 = 2/2 = 1?This isn't begging the question. It's an actual proof that (1 + 1)/2 = 1.

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Is the 0.999~ in (3) the same as the 0.999~ in (4)?

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(*blink!*)You did not just say that, did you?Of course it is. Are you seriously claiming that every time a mathematician writes "1," we need to prove that it has the same properties as the other "1" used elsewhere in the proof?

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Every time you do it out to the same number of decimals you get different numbers with different remainders

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Incorrect.You have an infinite number of decimals and thus, you wind up with the same numbers and the same remainders.Or have you forgotten that this is an infinite expansion and thus inherits all the qualities of infinity?

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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.

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No, it's the exact same proof except you're using a different arithmetic operator. Dividing 1.999... by 2 is the exact same concept as subtracting 0.999... from 1. Both leverage the fact that it is an infinite decimal expansion and thus, every single "9" in the decimal is canceled out.You're trying to add woo-woo where there isn't any to be found.

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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.

Jon writes:

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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

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You're trying to add woo-woo where none exists. You're arguing philosophy, not mathematics.

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So, is there any reason to assert the existence of 0.9999| other than to dazzle the Kindergartners?

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I already showed you the answer to this question (Message 80): Yes. It is required as part of the proof that the Reals are uncountable.

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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.

Jon writes:

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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.

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That would require me to lie to you and I'm not going to do that.0.999... has real-world significance, is not just a "fancy mathematical parlor trick," or any other woo-woo you're trying to inject.It's simply another way to write the number "1" which is very real, indeed.If your entire thesis is that you are not a Platonist, then just say it and be done with it.

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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.

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Incrrect in every single option.0.999... is a real number. 0.999... is not distinct from 1 but rather is identical to it (and mathematically, identical is a stronger relationship than equality). And the objects of mathematics are real. Number is just as much a physical quality of the universe as color is.Now, if you're not a Platonist, we can understand why you might disagree. But rather than beat around the bush, just come right on out and say it.

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In length, if one believes the MATHSYSTEM to have a relationship to reality, then looking for that link will prove M=False

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Incorrect. Mathematics is real.

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which allows 0.9999| to not be REAL

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Incorrect. 0.999... is real. It is also Real.

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which fails to prove 0.9999| = 1.

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Incorrect. You have been shown numerous proofs that 0.999... is identical to 1 (which is an even stronger connection than equality).You're trying to add woo-woo where none exists.

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However, once we accept that MATHSYSTEM has no necessary relationship to reality

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But that would require me to lie to you and I won't do that.If all you're trying to say is that you're not a Platonist, then just say it and be done with it.

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then we can make M=True without second thought

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Huh? Of course mathematics is true. That's the reason why we study it.

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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.

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But that's what the proofs already told you.You're trying to add woo-woo where none exists.If all you're trying to say is that you're not a Platonist, then just say it and be done with it.

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In other words, our proofs or 0.9999| = 1 will not work unless we accept the MATHSYSTEM-REALITY link to be arbitrary

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Incorrect. We must accept that the conclusions of mathematics are necessary and required.After all, that's what proof is.If all you're trying to say is that you are not a Platonist, then say it and be done with it.

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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.

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But just as the word for an object refers to a real object, a symbol for a number refers to a real number.If all you're trying to say is that you're not a Platonist, then just say it and be done with it.

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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.

My dear Rrhain, Yes, I most certainly did, because it is not proven to be the same in the original, and we are just asked to take it on faith. That is not how I do mathematical proofs. And just asserting it doesn't make it so.I'm well aware of the reality here, all I've pointed out is that the proof offered was incomplete.Enjoy.

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Oh good grief ...Look, the equivalence relation on decimal representations definitely puts every decimal representation in the same equivalence class as itself. 1 = 1. 4.9 = 4.9. The square root of two equals the square root of two.

Hi Dr ACuriously you have just proven that you have not read my post in context.Enjoy.

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Jon writes:

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So you have chosen to assert that there is a disconnect from MATHSYSTEM and reality

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I can't speak for anybody else, but I must say that I thoroughly disagree with this. Mathematics is the very nature of reality.

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This is exactly what one would expect in a system which represents two identical real-world values (values from REALWORLDSYSTEM) with two different symbologies.

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By this logic, you have just said that the sky isn't really colored because we use the symbol "blue" to describe it in English and "azul" to describe it in Spanish.

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In fact, it is this disconnect that is primarily at the heart of the necessity of the DISTINCTness criteria

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Incorrect. Not only is 0.999... equal to 1, it is identical to it.

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1 = 3/3 = 1/3 + 2/3 = 0.3333| + 0.6666| = 0.9999| 1

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Do you truly not see that the last statement is a contradiction to the first statement and thus cannot be true? Since you have done a complete logical sequence from 1 to 0.999..., then you have just proven that 1 = 0.999....

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Afterall, any system that represented necessarily another system would not need a function to close paradoxical loopholes

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What "paradoxical loophole" are you talking about? There is no paradox. 0.999... is not mere equal to 1, it is identical to it.

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But all these things arise, of course, because MATHSYSTEM is not in a necessary relationship with REALWORLDSYSTEM.

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Incorrect. All that you have stated is false for mathematics is reality. It is the very nature of reality.If all you're trying to say is that you're a non-Platonist, then say it and be done with it.

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Afterall, in REALWORLDSYSTEM 0.9999| v. 1 is a formality which is meaningless

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Incorrect. It is a requirement that allows us to show that the Reals are uncountable.By your logic, the sky has no color because we can call it "blue" as well as "azul."

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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.

RAZD responds to me:

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Yes, I most certainly did, because it is not proven to be the same in the original, and we are just asked to take it on faith. That is not how I do mathematical proofs.

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Now, that isn't really true, now is it? When you were doing proofs in school, you didn't turn in a 300-page treatise that derived the existence of 1, did you?No, you simply wrote "1" and inherited all of the mathematical work that was done previously that showed that 1 existed and had the value that it did.Let's not play dumb.By your logic, 2 + 2 really can equal 5 because those two "2s" aren't necessarily the same.Please, let us not pretend that you are talking about limits.

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And just asserting it doesn't make it so.

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Indeed, which is why you need to prove your claim. Do you have evidence against the Principia? Since you clearly do not wonder whether or not the 15% tip given today is different from the 15% tip given tomorrow, then your claims that there is some need to show that two 1s are identical is disingenuous at best.Let us not play dumb.

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I'm well aware of the reality here, all I've pointed out is that the proof offered was incomplete.

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Except it wasn't. If you think it is, you need to explain why.0.999... is not only equal to 1, it is identical to it.And that's an even stronger relationship.

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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.

I haveEnjoy.

Hi RAZDAre you questioning whether 0.999~ is the same as 0.9999~?

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

Thank you Rhain for so forcefully showing how 0.99999999999999999999999999999999999999999999999999999999999999999999999999999999... is the same as 1.0.I just might want to present a useful way to bring 0.999~ into mathematical discussion. Surely you are acquainted with the concept of open and closed intervals on the Real Number axis.you have seen such things as [0.0, 1.0] and (0.0, 1.0). the former means we INCLUDE 0 and 1 and the latter means we do not, but everything in between. we even have such things as [0.0, 1.0) or (0.0, 1.0] which will include 0 or 1 respectively.someone could argue that 0.999~ is a beautiful way to represent the upper limit of the open interval (0.0, 1.0).give me an epsilon and i can get a delta (etc.).another topic:
the Heine-Borel Theorem is one of the coolest things ever.

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- xongsmith, 5.7d