0.99999~ = 1 ?

Yes, as many have pointed out, .999... = 1 exactly. There is no variation between the two of any kind.This is not a problem for mathematics and, in fact, an extremely significant proof relies upon this notation: That the set of Reals is uncountable.Suppose the Reals are countable. If so, then you can list them since they would be in one-to-one correspondance with the set of Countable numbers (1, 2, 3, ...). Suppose you had a list of all the Real numbers between 0 and 1, inclusive. Write them out using the repeated-decimal formation where we write .999... for 1. This also means that for any decimal that stops, we use a repeating 9 decimal so that instead of 0.5, we should write 0.4999....The reason why we do this is that it guarantees that we have a unique representation of each number. No number terminates but instead continues on, even if it is with a bunch of 9s. Thus, we can ensure that two numbers are identical if and only if they are exactly the same at every single decimal place.So, we'll have a list of decimals:a₁ = 0.a₁₁a₁₂a₁₃...
a₂ = 0.a₂₁a₂₂a₂₃...
a₃ = 0.a₃₁a₃₂a₃₃...
.
.
.Now, construct b:b = 0.b₁b₂b₃...Where b_n = a_nn such that:If a_nn = 1, then b_n = 2.
If a_nn <> 1, then b_n = 1.Thus, it is clear that b <> a_n for all n. They differ precisely at the nth decimal place.And yet, b is a number between 0 and 1, inclusive. Therefore, the number of numbers between 0 and 1 is uncountable and thus, the size of the Reals is also 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.

Mr Jack responds to me:

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It's not necessary to write them as 0.9999~ for that proof to work, concluding with an infinite series of 0s also works.

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Then how do you distinguish between "0.5000..." and "0.4999..."? The proof rests upon having a decimal expansion and we need to make sure that we don't skip anything or introduce paradox where a number written one way triggers something that it wouldn't have had it been written the other way.I admit, it's been a while since my Real Analysis class where this was brought up, but that's how I remember the proof: By enforcing a standardized notational format, we can ensure that the process looks at each number individually and we don't overlook something simply because it wasn't notated correctly.

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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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Can you provide a definition of 'real number' so that I do not have to look for one myself?

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Well, the problem is that a "Real" number isn't nearly as easily defined as other kinds of numbers. For example, there are the "Natural" numbers: 1, 2, 3, .... Notice that "0" is not among them. If you add 0 to it, you get "Whole" numbers. Add the negatives, and you have the "Integers."A "Rational" number is any number that can be expressed as a/b where a and b are Integers.Real numbers, though, start to get complicated. One of the simplest way of constructing the "Reals" is through decimal expansion. For any number, you can converge upon its decimal representation: 2, 2.7, 2.71, 2.718, 2.7182, etc. will converge upon e, a Real number.

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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 responds to me:

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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.

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.

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.

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.

RAZD responds to me:

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I have

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Incorrect.Are you denying that division of 1.999... by 2 results in a pair-wise identity in decimal place with 0.999...? For that is what was given as justification (Message 4): Are you saying that this statement isn't true? That 1.999.../2 <> 0.999...?You don't just get to whine about your doubt. You need to provide your evidence as to why the equality isn't true or why it is that 0.999... <> 0.999.... Is there something specific that is bothering you? If so, what is it?Be specific.

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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 writes:

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In the process it uses another version of 0.999~

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Incorrect.In the process, it calculates that (1 + 0.999...)/2 = 0.999.... This can only be true if 0.999... = 1.Are you saying that (1 + 0.999...)/2 does not result in a pair-wise identity in each decimal place with 0.999...?Let us not play dumb.

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One can't use the conclusion as part of the proof eh?

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And if that were the case, you might have a point. But instead, we are finding that a calculated value (the quotient) is identical to the constant we used. And because of that, in the context of the proof, this means that 0.999... = 1.(x + x)/2 = xThat second "x" is a calculated x, yes, but that's the point of the proof. Because if (y + x)/2 = x, then y = x.So we start with y = 1 and x = 0.999... and then do a calculation on it. If we wind up with 0.999..., then we know that y = x and 1 = 0.999....Are you denying that (1 + 0.999...)/2 = 0.999...?

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A much simpler process is take 0.999~, multiply it by two (=1.999~), where the 9's are exactly aligned from the decimal, and subtract the original ( 1), QED

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It isn't "simpler."It's backwards.Your method also works since identities are isomorphic and thus the implications go both ways [(A=>B) => (B=>A)], but it is hardly simple for it requires the exact same process: Performing a mathematical operation upon an infinite decimal.You have the exact same problem you are complaining about: What makes you think the 9s in 1.999... are the same as the 9s in 0.999...?Easy: The mathematical operation is correct and the notation shows it to be so.Whether we add and then divide or we multiply and then subtract is immaterial.

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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.

Jazzns writes:

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Yea transcendentals are contained in the irrationals.

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Yeah. The Rationals are simply defined as the set of numbers that can be expressed as a/b where a and b are Integers. The Irrationals are those numbers that cannot be expressed that way.Thus pi, e, and the square root of 2 are all irrational.Transcendental numbers are those that aren't algebraic, meaning roots of a polynomial equation with rational coefficients. The square root of 2 is algebraic, but pi and e aren't.And there is the constructible dichotomy: Those numbers which can be constructed via straightedge and compass are "constructible" while those that can't aren't. The square root of 2 is constructible, pi is not. And that's why it is impossible to square the circle: The square root of pi is not constructible.All constructible numbers are algebraic, but not all algebraic numbers are constructible. All rational numbers are constructible, but not all constructible numbers are rational.There are all sorts of ways to divide the real numbers down. That's part of the reason that Real Analysis is so fascinating: There are just so many numbers and with different properties to them that you can divvy them up any which way and learn all sorts of interesting things about the way numbers can behave.

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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 writes:

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I'm aware that functionally they are similar

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It is more than just "similar." They are identical. This is part of the reason why in the proof that the Reals are uncountable that you have to choose the notation of the repeated decimal format.

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I just find them conceptually different, as one seems (unnecessarily) more manipulated than the other, and it seems that the main argument is about how you phrase the problem, rather than the actual solutions.

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And yet, as I have pointed out more than once, there is at least one problem that hinges upon this equivalency.There are plenty of things in physics that make absolutely no sense intuitively and yet we know they are true. We run the experiments and we get results that only make sense if the world works in ways completely against what we would normally expect. The very idea that no, you can't go faster than the speed of light is one of them. A car going at 40 miles an hour crashes headlong into a car coming toward it at 50 miles an hour. Their combined force impact is NOT equivalent to the single mass of both cars crashing into a stationary object at 90 miles an hour. Relativistic effects are present at all levels. The only reason we didn't figure it out sooner is because the effect is so small that we didn't notice it. Now that we can mover faster and farther, those effects become more noticeable.Mathematics is the same way: Some things don't seem right but they are and unlike physics, you can actually prove it forever to be true.

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My dad makes a comment about mathematicians understanding the problems very well, but not understanding how to communicate their understanding to those who do not understand maths, so I try for the simplest paths possible.

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The thing is that just like physics and biology, the really interesting questions don't have simple answers. There's a reason that it takes over 60,000 steps to prove that 1 + 1 = 2 from first principles. It took Russell and Whitehead years to develop the proof and Russell nearly went mad over it. There are plenty of pithy phrases we can say, but to be rigorous about it requires that we abandon pretenses of simplicity and admit that it's hard work to understand this stuff.

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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.

Huntard writes:

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What is it about math that makes people go mad, anyway? Or is it simply mad people that practice math?

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That's a question many have asked. You might want to pick up a copy of Logicomix which is about Russell, the development of the Principia, and an interesting study about the nature of looking into the void and having it look back at you.Russell's reaction to the foundations of mathematics was a bit different from Cantor, but Cantor was most likely bipolar.The interesting bit of Russell's background is that his parents were involved in a menage a trois and in the end, Russell seems to at the very least not minded his wife's affairs...until she had had two children by another man.

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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.