.999...=1?
Someone attempted to prove this...

 .999...=1? •John Creighton Forum Veteran Usergroup: Members Joined: Apr 22, 2012 Total Topics: 96 Total Posts: 766 #171 - Quote - Permalink Posted Jul 21, 2012 - 9:28 PM: t0mr wrote:Yes, I guess the equality could be an implicit function of epsilon. But if it is an implicit function that means the values the function produces do not necessarily have to be one to one with values of epsilon. In other words, more than one N could apply to a single epsilon value. I do not know if that is our case however.I agree there could be potential problems but since we are able to solve for N explicitly this is not an issue in this case.Also I dont get a very friendly result when I try to isolate the n for the reduced form of your inequality: |∑(n to ∞)9*10^(-n)| < ε.In:http://www.mymathforum.com/viewtop...8&t=32227&p=130328#p130328I use inequality (3) first to get equation (6).Now to do things more explicitly substitute:N=1-log_{10}(ε) for n on the right hand side of equation (6). After some algebra you should be able to get the value ε on the right side of the inequality. Also I'm using log base 10. Not the natural logarithm. My substitution is valid because n>N so the subsituation gives something larger then would be the case for any value n greater then N because the function decreases with increasing n.If it still isn't clear then when I get time I'll try to more explicitly clarify these steps. Edited by John Creighton on Jul 21, 2012 - 9:42 PM •DanLanglois Unmoderated Member Usergroup: Unmoderated Member Joined: Jul 16, 2012 Total Topics: 4 Total Posts: 190 #172 - Quote - Permalink Posted Jul 21, 2012 - 10:03 PM: You haven't used only proofs of ∀x[s(x) = t(x)]. Which you should be able to do. Equalities between numerals are true, if the numerals are the same or they have the same form. In discussing certain propositions (e.g. implications with universal antecedents), and contemplating functions of types ∀xF(x) =⇒ ∀xG(x) which take functions (or proofs) as arguments, the idea you are giving here is that the object of number theory is an iterative process, and this iterative process is representable in intuition, the numerals do not stand for anything. Perhaps, if we can start counting and both numerals tell us to stop at the same time, they are equal. In which case, you may have proven that if it's not equal to 1, it is less than 1. Is e real? Yes. Is e^2 real? Yes, but with a barely perceptible delay. e^3? Yes, but with more delay. Always yes. 2 to the 100th power? [performs quick 'n' dirty calculation in 3,000,000,000,000,000,000,000 years.] Yes. Edited by DanLanglois on Jul 22, 2012 - 1:37 AM •John Creighton Forum Veteran Usergroup: Members Joined: Apr 22, 2012 Total Topics: 96 Total Posts: 766 #173 - Quote - Permalink Posted Jul 21, 2012 - 10:17 PM: DanLanglois wrote:You haven't used only proofs of ∀x[s(x) = t(x)]. Which should be able to do. Equalities between numerals are true, if the numerals are the same or they have the same form. In discussing certain propositions (e.g. implications with universal antecedents), and contemplating functions of types ∀xF(x) =⇒ ∀xG(x) which take functions (or proofs) as arguments, the idea you are giving here is that the object of number theory is an iterative process, and this iterative process is representable in intuition, the numerals do not stand for anything. Perhaps, if we can start counting and both numeralstell us to stop at the same time, they are equal. In which case, you have perhaps proven that if it's not equal to 1, it is less than 1. It would be more helpful if you told me which specific step you disagree with in:http://www.mymathforum.com/viewtop...8&t=32227&p=130328#p130328The form of the proof you are asking for is not relevant because I am not trying to prove a general proposition but only a certain limit. •t0mr Unmoderated Member Usergroup: Unmoderated Member Joined: Jul 03, 2012 Total Topics: 1 Total Posts: 36 #174 - Quote - Permalink Posted Jul 21, 2012 - 10:31 PM: Dan Langlois wrote: As an analogy, I take his position to be applicable to multiplication tables. If you take a look at them, how is this easy? Translate it for me, is there a set of equations? Are they suitable? (somebody tries to oblige..) But, there are special reasoning methods, here. There are different kinds of definitions, say what is a definition? This 'proof' contains an algorithm for the construction of mathematical objects, this one contains subproofs. Is the universe unique, or a multitude of many universes?The universe is a unique multiverse in an average sided ultraverse. But that’s it. The strings end there. I prefer informal mathematical reasoning, appealing to intuition, which is an unusual position, and I strongly prefer it. The notion that mere intuition is not a sufficient guiding principle any more, plays right into t0mr's hands--seeing the way attempts to shoot him down are constructed makes me want to take his side, spin people's wheels. It's not gonna help, bringing this stuff in about properties of point-sets and continuous functions such as space-filling curves, nowhere differentiable continuous functions, paradoxical decompositions of the sphere (I'm anticipating how bad this might get).To me intuition is more like the feeling that something is not right when you hear it. For instance, when you see two different numbers that are endlessly distinct from one another like .999… and 1.000… and someone tells you they are the same thing. Intuition is the thing that puts the “That doesn’t seem right.” in your brain yet you have not completely found or formulated the reasoning behind why it doesn’t seem right. Now often intuition is right and if you have a “that doesn’t seem right” moment you should flush out the impulse to better understand why something seems the way it does. But what we are discussing here should be beyond the realm of intuition as I am defining it.I like the formal definitions and proofs we have been working with. But I agree that the near ‘endless’ mathematical doctrines that exist could really bog down any hope for a reasonable approach to our problem.I am proposing we finally create and/or agree upon a definition for (1) infinity and (2) our number system. From there we could all draw conclusions from the same two concise source definitions. (I tried to do this before in a long post but I think the message was lost in the details.) We need these two things (1) and (2) defined because without infinity we cannot have the number system and without the number system we cannot have .999… or 1. •DanLanglois Unmoderated Member Usergroup: Unmoderated Member Joined: Jul 16, 2012 Total Topics: 4 Total Posts: 190 #175 - Quote - Permalink Posted Jul 21, 2012 - 11:12 PM: John Creighton wrote:It would be more helpful if you told me which specific step you disagree with in:http://www.mymathforum.com/viewtop...8&t=32227&p=130328#p130328The form of the proof you are asking for is not relevant because I am not trying to prove a general proposition but only a certain limit. To be clear, you've got me convinced, totally. t0mr is another matter. Your successful (w/me, totally) effort amounts to "let’s compute a limit or two using 'these properties'". Me, I'd be too embarrassed to ask you to prove some of the basic properties and facts about limits. However, t0mr writes: I know that a repeating decimal can be represented by an infinite series and I know that series has the form of a geometric series. I also know that once in the form of a geometric series we can find the sum of that series if the series converges. But did you know that in deriving the formula for the sum of a geometric series the limit process is used.Under the circumstances, I dunno, somebody's gonna eventually try posting the precise definition of the limit, maybe, a mouthful, and I do I take it as being obvious that there's not any dispute here if we all have a fairly good feel for doing that kind of proof, are very comfortable using the definition of the limit to prove limits. I imagine that if some properties of limits get posted then that will turn into the proof of some of these properties being posted. Really? t0mr wrote:I am proposing we finally create and/or agree upon a definition for (1) infinity and (2) our number system. From there we could all draw conclusions from the same two concise source definitions. (I tried to do this before in a long post but I think the message was lost in the details.) We need these two things (1) and (2) defined because without infinity we cannot have the number system and without the number system we cannot have .999… or 1You mention having offered a 'long post', checking back for it, I'm not so sure what you are referring to. I did find this: The word infinite when used to describe a set is not defining a specific object or specific numerical value. I picked up on that sentiment already. Also, I suppose that you are right, that without infinity we cannot have the number system. I might rephrase this as, certain sets are of widespread interest, integers, rational numbers, etc. However, a 'definition' of infinity and our number system, concise source definitions, these things defined, well, what will you accept as a 'definition'? Everything must be defined in terms of something else. What is a line? A point? Are you going to run to Euclid's elements, cuz he gives a definition, a point has no parts, right, that's hokem. Then what is the definition? You don't need definitions, they wouldn't even give objective validity if they could be had. You need intuition, a word that I find I cannot use in a context like this, w/out getting accused of having much more enthusiasm for foundation of mathematics work of the 20th century than I actually have. But. Definitions will help you, then, fine, good luck w/that. Edited by DanLanglois on Jul 22, 2012 - 12:07 AM •t0mr Unmoderated Member Usergroup: Unmoderated Member Joined: Jul 03, 2012 Total Topics: 1 Total Posts: 36 #176 - Quote - Permalink Posted Jul 22, 2012 - 6:31 AM: I use inequality (3) first to get equation (6).Now to do things more explicitly substitute:N=1-log_{10}(ε) for n on the right hand side of equation (6). After some algebra you should be able to get the value ε on the right side of the inequality. Also I'm using log base 10. Not the natural logarithm. My substitution is valid because n>N so the subsituation gives something larger then would be the case for any value n greater then N because the function decreases with increasing n.If it still isn't clear then when I get time I'll try to more explicitly clarify these steps. It still isn't clear. But you agree the proof is better with an explicit function for N? Since epsilon can be any arbitrary value then and not just (10^-(N-1)) values. What I am stuck is how you can get an explicit function because I cannot. If you could start the proof out in the form |f(n) -L| < ε then manipulate it to the point you have above it would help me alot. Also could you do it with natural logarithms because the notation in the above N=1-log_{10}(ε) is confusing to me and I am more familiar with natural logs. I am sorry I still have questions. If this site had the capabilities for notation that your linked site did we would probably already have this settled. Can we post pictures here? Then we might be able to post these problems more clearly. •John Creighton Forum Veteran Usergroup: Members Joined: Apr 22, 2012 Total Topics: 96 Total Posts: 766 #177 - Quote - Permalink Posted Jul 22, 2012 - 7:01 AM: t0mr wrote: It still isn't clear. But you agree the proof is better with an explicit function for N? Since epsilon can be any arbitrary value then and not just (10^-(N-1)) values. What I am stuck is how you can get an explicit function because I cannot. If you could start the proof out in the form |f(n) -L| < ε then manipulate it to the point you have above it would help me alot. Also could you do it with natural logarithms because the notation in the above N=1-log_{10}(ε) is confusing to me and I am more familiar with natural logs. I am sorry I still have questions. If this site had the capabilities for notation that your linked site did we would probably already have this settled. Can we post pictures here? Then we might be able to post these problems more clearly. In: N=1-log_{10}(ε)The under score "_" denotes subscript. The squiggly brackets "{...}" bracket off what is being subscripted. There by log_{10} I mean log base 10. This notation comes form Latex. It will be handy to know if we et TeX capabilities on this forum to write math equations. For more info try one of these links:http://www.maths.tcd.ie/~dwilkins/LaTeXPrimer/http://ece.uprm.edu/~caceros/latex/introduction.pdfHere is a property of log base 10:10^{log_{10}(x))=XHere I'm using ^ to denote superscript (AKA exponentiation) and again the squiggly brackets bracket off what is being superscripted. For more info on log base 10 see:http://en.wikipedia.org/wiki/Common_logarithm On Jul 22, 2012 - 7:34 AM, jorndoe responded: How about $$N = 1 - log_{10}(\epsilon)$$ and $$\forall x \in \mathbb{R} > 0 [ 10^{log_{10}(x)} = x ]$$ ? •John Creighton Forum Veteran Usergroup: Members Joined: Apr 22, 2012 Total Topics: 96 Total Posts: 766 #178 - Quote - Permalink Posted Jul 22, 2012 - 7:31 AM: John Creighton wrote:In: N=1-log_{10}(ε)The under score "_" denotes subscript. The squiggly brackets "{...}" bracket off what is being subscripted. There by log_{10} I mean log base 10. This notation comes form Latex. It will be handy to know if we et TeX capabilities on this forum to write math equations. For more info try one of these links:http://www.maths.tcd.ie/~dwilkins/LaTeXPrimer/http://ece.uprm.edu/~caceros/latex/introduction.pdfHere is a property of log base 10:10^{log_{10}(x))=XHere I'm using ^ to denote superscript (AKA exponentiation) and again the squiggly brackets bracket off what is being superscripted. For more info on log base 10 see:http://en.wikipedia.org/wiki/Common_logarithmI'm testing my above expression with the math notation used in Paul's post #14 of the Latex topic.\begin{aligned} 10^{log_{10}(x)}=x \end{aligned}Seems to work. Maybe I can post more of the math stuff here. I'm not sure why it isn't set up to work with the BB code math tags though. Edited by John Creighton on Jul 22, 2012 - 7:38 AM •Kwalish Kid PF Addict Usergroup: Members Joined: Sep 26, 2004 Total Topics: 56 Total Posts: 1162 #179 - Quote - Permalink 1 of 1 people found this post helpful Posted Jul 22, 2012 - 7:52 AM: t0mr wrote:To me intuition is more like the feeling that something is not right when you hear it. For instance, when you see two different numbers that are endlessly distinct from one another like .999… and 1.000… and someone tells you they are the same thing.These are not numbers! They are representations of numbers. You have been told that again and again and again. You have claimed that you knew the difference. You have already been corrected at least once on your misunderstanding, why do you persist? I like the formal definitions and proofs we have been working with. But I agree that the near ‘endless’ mathematical doctrines that exist could really bog down any hope for a reasonable approach to our problem.Riiiiight. You like the math, except for all the math parts. •DanLanglois Unmoderated Member Usergroup: Unmoderated Member Joined: Jul 16, 2012 Total Topics: 4 Total Posts: 190 #180 - Quote - Permalink Posted Jul 22, 2012 - 1:27 PM: I take t0mr's position to be, in a nutshell, boiled right down to one sentence, that a series is not a number. So I don't think that 'these are not numbers' is going to settle anything, I myself am not sure how to parse this distinction between a number and a representation of a number. Sounds pretty deep, eh? Is there a proper isomorphism going on here? Now we have two debates. These numbers are not numbers!!! Edited by DanLanglois on Jul 22, 2012 - 1:37 PM

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