# .999 = 1? **URL:** \ **Category:** Factual Questions\ **Created:** [July 31, 2000, 6:28am UTC](https://boards.straightdope.com/t/999-1/27517 "2000-07-31T06:28:03Z")\ **Posts on this page:** 20\ **Page:** 49
**Author:** ![The\_Hamster\_King](https://avatars.discourse-cdn.com/v4/letter/t/8edcca/32.png) [@The\_Hamster\_King](https://boards.straightdope.com/u/The_Hamster_King)\ **Post date:** [October 2, 2012, 12:23am UTC](https://boards.straightdope.com/t/999-1/27517/961 "2012-10-02T00:23:14Z")
> [@Valmont314](#): > > I was simply trying to point out why it is that the elements where colocated…not to embark on an effort to derive a new system. But, once you say that 0.9… \<\> 1 you ARE defining a new system, whether you intend to or not. Within the reals _as normally defined_ 0.9… = 1. It’s not a philosophical or metaphysical question, its simply a natural consequence of the axioms that define the reals. It’s the same number with two different ways of writing it … just like 1/3 and 0.3… and 0.1(ternary) are all the different ways of writing the same number. If you want to, you can come up with different axioms to define a different system of numbers behave differently than the reals do. You can set up the rules so that 0.9… denotes a different number than 1, or doesn’t denote anything whatsoever. But in general systems like that have other problems (like not being closed under multiplication) that make them less desirable as tools for working out problems with real-world consequences. ---
**Author:** ![The\_Hamster\_King](https://avatars.discourse-cdn.com/v4/letter/t/8edcca/32.png) [@The\_Hamster\_King](https://boards.straightdope.com/u/The_Hamster_King)\ **Post date:** [October 2, 2012, 12:30am UTC](https://boards.straightdope.com/t/999-1/27517/962 "2012-10-02T00:30:37Z")
> [@Valmont314](#): > > You do agree though that if I set the limits of precision to a Googleplex and take results out to 5 decimal places I am ok for 3 operations do you not? Nope. 5 \* (0.X1 / 5) \<\> (5 \* 0.X1) / 5 where the “X” denotes a googleplex of zeros. The system you describe is not closed under associativity. ---
**Author:** ![allotrope](https://avatars.discourse-cdn.com/v4/letter/a/edb3f5/32.png) [@allotrope](https://boards.straightdope.com/u/allotrope)\ **Post date:** [October 2, 2012, 12:32am UTC](https://boards.straightdope.com/t/999-1/27517/963 "2012-10-02T00:32:47Z")
> [@The\_Hamster\_King](#): > > Within the reals _as normally defined_ 0.9… = 1. It’s not a philosophical or metaphysical question, its simply a natural consequence of the axioms that define the reals. It’s the same number with two different ways of writing it … just like 1/3 and 0.3… and 0.1(ternary) are all the different ways of writing the same number. Of course this is true, and it’s tempting to just write it off as an axiomatic issue, except it really isn’t - not that I’m suggesting you are saying it is. I would just prefer not to go down that road again. ---
**Author:** ![Valmont314](https://avatars.discourse-cdn.com/v4/letter/v/3da27b/32.png) [@Valmont314](https://boards.straightdope.com/u/Valmont314)\ **Post date:** [October 2, 2012, 12:57am UTC](https://boards.straightdope.com/t/999-1/27517/964 "2012-10-02T00:57:25Z")
> [@The\_Hamster\_King](#): > > Nope. > > 5 \* (0.X1 / 5) \<\> (5 \* 0.X1) / 5 where the “X” denotes a googleplex of zeros. > > The system you describe is not closed under associativity. Incorrect The system is not closed under associativity at the nth digit where n represents the limits of precision. It is closed under associativity at the defined limits of measurement which in this case is the 5th decimal place and for some finite number of operations 5\*(.00001(X-5- ZEROS)/5 = 5\*(.00001(X-5-ZEROS)/5 ---
**Author:** ![Valmont314](https://avatars.discourse-cdn.com/v4/letter/v/3da27b/32.png) [@Valmont314](https://boards.straightdope.com/u/Valmont314)\ **Post date:** [October 2, 2012, 12:59am UTC](https://boards.straightdope.com/t/999-1/27517/965 "2012-10-02T00:59:49Z")
You have to start thinking about the number of operations as finite and limited. Which they always are for some real word calculation. ---
**Author:** ![Valmont314](https://avatars.discourse-cdn.com/v4/letter/v/3da27b/32.png) [@Valmont314](https://boards.straightdope.com/u/Valmont314)\ **Post date:** [October 2, 2012, 1:37am UTC](https://boards.straightdope.com/t/999-1/27517/966 "2012-10-02T01:37:05Z")
Where did everybody go? **…** ---
**Author:** ![Valmont314](https://avatars.discourse-cdn.com/v4/letter/v/3da27b/32.png) [@Valmont314](https://boards.straightdope.com/u/Valmont314)\ **Post date:** [October 2, 2012, 2:07am UTC](https://boards.straightdope.com/t/999-1/27517/967 "2012-10-02T02:07:45Z")
> [@The\_Hamster\_King](#): > > But, once you say that 0.9… \<\> 1 you ARE defining a new system, whether you intend to or not. > > Within the reals _as normally defined_ 0.9… = 1. It’s not a philosophical or metaphysical question, its simply a natural consequence of the axioms that define the reals. It’s the same number with two different ways of writing it … just like 1/3 and 0.3… and 0.1(ternary) are all the different ways of writing the same number. > > If you want to, you can come up with different axioms to define a different system of numbers behave differently than the reals do. You can set up the rules so that 0.9… denotes a different number than 1, or doesn’t denote anything whatsoever. But in general systems like that have other problems (like not being closed under multiplication) that make them less desirable as tools for working out problems with real-world consequences. Can somebody answer this question? If the statement .999… = 1 is no different from the statement .001 = .0010 Why are there discussions all over Internet about the first statement and no such discussions about anything resembling the second statement. Please tell me what I am missing. Why are people referring to this thread itself as a metaphor for something? If there were a similar thread on internet about the second statement would people find metaphors for it? I see a huge difference in the two statements and I can’t see how they could ever be compared. Its apples and oranges. If it’s simple as that…than someone should have simply said so at the beginning of the thread. ---
**Author:** ![Trinopus](https://avatars.discourse-cdn.com/v4/letter/t/2bfe46/32.png) [@Trinopus](https://boards.straightdope.com/u/Trinopus)\ **Post date:** [October 2, 2012, 2:15am UTC](https://boards.straightdope.com/t/999-1/27517/968 "2012-10-02T02:15:02Z")
> [@Valmont314](#): > > You do agree though that if I set the limits of precision to a Googleplex and > take results out to 5 decimal places I am ok for 3 operations do you not? > The system has to be tailored precisely and rigorously…if you would like to do that go for it. If it doesn’t work for all numbers, then the operation is “broken.” A single counter-example explodes the property. Sure, a typical Excel spreadsheet works just fine for balancing a checkbook. It fails miserably if you’re trying to multiply two 1,000-digit numbers. Your ideas fail miserably under certain conditions; your ideas are ill-conditioned. ---
**Author:** ![Trinopus](https://avatars.discourse-cdn.com/v4/letter/t/2bfe46/32.png) [@Trinopus](https://boards.straightdope.com/u/Trinopus)\ **Post date:** [October 2, 2012, 2:20am UTC](https://boards.straightdope.com/t/999-1/27517/969 "2012-10-02T02:20:18Z")
> [@Valmont314](#): > > Incorrect > > The system is not closed under associativity at the nth digit where n represents the limits of precision. > > It is closed under associativity at the defined limits of measurement which in this case is the 5th decimal place and for some finite number of operations > > 5\*(.00001(X-5- ZEROS)/5 = 5\*(.00001(X-5-ZEROS)/5 It isn’t “The associative property over a certain range of numbers.” It is the associative property of numbers. You’ve having a lovely time re-defining things just all over the place, but that isn’t going to work very well among people who understand number theory, algebra, and, perhaps most importantly, the philosophy of numbers. Your new ideas destroy the distributive and associative property of numbers; by insisting these properties only apply over subsets of numbers, you are destroying the property of closure. Under your ideas, algebra is not possible. You’re like the friend of Abraham Lincoln who wanted to call a tail a leg. You can do that. You can happily function in that world. The rest of us are having none of it, and are losing more and more respect for you, the deeper you dig yourself into this solipsistic pit. ---
**Author:** ![Valmont314](https://avatars.discourse-cdn.com/v4/letter/v/3da27b/32.png) [@Valmont314](https://boards.straightdope.com/u/Valmont314)\ **Post date:** [October 2, 2012, 2:29am UTC](https://boards.straightdope.com/t/999-1/27517/970 "2012-10-02T02:29:52Z")
> [@Trinopus](#): > > If it doesn’t work for all numbers, then the operation is “broken.” A single counter-example explodes the property. > > Sure, a typical Excel spreadsheet works just fine for balancing a checkbook. It fails miserably if you’re trying to multiply two 1,000-digit numbers. > > Your ideas fail miserably under certain conditions; your ideas are ill-conditioned. There is no single counter example when the number is used at the limits of measurement and the limits of operational steps (both as specified in the axioms) When you over-rev your engine “miserably” do you then go back to the dealer and ask him to pay for the engine because it was “broken” Engines exist in the real world not in the mind. Nice melodrama but you have failed to make your case that it is not possible to define a workable system that does not have co-located elements. It doesn’t have to be a Ferrari, it just has to be a Car. The existence of a car supports my assertions as does your failure to demonstrate that it is not possible to design such a system. … ---
**Author:** ![Valmont314](https://avatars.discourse-cdn.com/v4/letter/v/3da27b/32.png) [@Valmont314](https://boards.straightdope.com/u/Valmont314)\ **Post date:** [October 2, 2012, 2:33am UTC](https://boards.straightdope.com/t/999-1/27517/971 "2012-10-02T02:33:28Z")
> [@Trinopus](#): > > It isn’t You’ve having a lovely time re-defining things just all over the place, but that isn’t going to work very well LOL How are you suppose to define a set via axioms without defining axioms? ---
**Author:** ![Francis\_Vaughan](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/francis_vaughan/32/3093_2.png) [@Francis\_Vaughan](https://boards.straightdope.com/u/Francis_Vaughan)\ **Post date:** [October 2, 2012, 2:37am UTC](https://boards.straightdope.com/t/999-1/27517/972 "2012-10-02T02:37:25Z")
> [@Valmont314](#): > > Where did everybody go? > > **…** Sitting back with a bag of popcorn and a beer watching the show. Seriously, your comments and ideas are all over the place. In one post you say that it is axiomatic that there are ten decimal places, and yet later you claim that you are not building a new system. Then you make the statement that unless we can post something that directly addresses your assertions you will refuse to reply but simply write “…” There is simply no point conversing with someone who tries to argue like this. You change your system’s rules on an almost post by post basis, don’t address most of the comments made about your assertions, but cherry pick those you do wish to address, and change your rules of discourse as if you own the discussion. The thread is in General Questions. Factual answers to questions that have them. Postmodern interpretation of mathematics as a narrative is not such a pursuit. You could try a thread in IMHO. There is a long worn saying. _Never try to teach a pig to sing. It is a waste of time and it annoys the pig_. If you want to rise above the status of an amusical pig, try to at least work within the rules of logic, mathematics and polite discourse. We have no requirement to answer, nor any inclination, unless it improves the silence. ---
**Author:** ![Valmont314](https://avatars.discourse-cdn.com/v4/letter/v/3da27b/32.png) [@Valmont314](https://boards.straightdope.com/u/Valmont314)\ **Post date:** [October 2, 2012, 2:50am UTC](https://boards.straightdope.com/t/999-1/27517/973 "2012-10-02T02:50:24Z")
> [@Francis\_Vaughan](#): > > Sitting back with a bag of popcorn and a beer watching the show. > > Seriously, your comments and ideas are all over the place. In one post you say that it is axiomatic that there are ten decimal places, and yet later you claim that you are not building a new system. Then you make the statement that unless we can post something that directly addresses your assertions you will refuse to reply but simply write “…” > > There is simply no point conversing with someone who tries to argue like this. You change your system’s rules on an almost post by post basis, don’t address most of the comments made about your assertions, but cherry pick those you do wish to address, and change your rules of discourse as if you own the discussion. The thread is in General Questions. Factual answers to questions that have them. Postmodern interpretation of mathematics as a narrative is not such a pursuit. You could try a thread in IMHO. > > There is a long worn saying. _Never try to teach a pig to sing. It is a waste of time and it annoys the pig_. If you want to rise above the status of an amusical pig, try to at least work within the rules of logic, mathematics and polite discourse. We have no requirement to answer, nor any inclination, unless it improves the silence. Very constructive…You apparently think that you can make your arguments via ad hominem attacks as opposed to focusing on what is being discussed. Believe me I am not annoyed, on the contrary I find it quite befuddling to see people taking this topic so personally (as you are)…as if they were defending some sacred secret or something…sheesh, this is a free forum on Internet…lighten up man. ---
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [October 2, 2012, 2:57am UTC](https://boards.straightdope.com/t/999-1/27517/974 "2012-10-02T02:57:47Z")
> [@Francis\_Vaughan](#): > > There is simply no point conversing with someone who tries to argue like this. You change your system’s rules on an almost post by post basis, don’t address most of the comments made about your assertions, but cherry pick those you do wish to address, and change your rules of discourse as if you own the discussion. … It would be moderately interesting if **Valmont314** could give us a list of the axioms and definitions that he is currently using in his idiosyncratic number system. However, by now it is clear that he is no longer talking about real numbers as understood by mathematicians generally. ---
**Author:** ![Senegoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/senegoid/32/6606_2.png) [@Senegoid](https://boards.straightdope.com/u/Senegoid)\ **Post date:** [October 2, 2012, 3:22am UTC](https://boards.straightdope.com/t/999-1/27517/975 "2012-10-02T03:22:44Z")
Hey all youse guys, not to hijack the direction of this thread or anything, but I’ve got an actual question about .999… being = 1. (Remember that question?) When I took 1st semester Calculus (circa 1983), they were still teaching epsilon-delta proofs in first semester. (I’ve heard it said that ε-δ is no longer taught in first semester, but is now covered in more advanced upper-division classes. Is that true?) To be sure, we only got a superficial taste of how to do ε-δ limit proofs, nothing in depth. And I only barely remember even that after all these years. Okay, guys, we get .999 = 1 by defining .999… as the limit of a sequence of partial sums. And we’ve seen all the algebraic proofs ( the 1/3 = .333… times 3 proof, and the 10x - x = 9.999… - .999… = 9 proof). Now, how do you prove that .999… = 1 with an ε-δ limit proof? That is, how to prove that the sequence of partial sums .9, .99, .999, .9999 etc. approaches 1 as a limit, using ε-δ ? ---
**Author:** ![Valmont314](https://avatars.discourse-cdn.com/v4/letter/v/3da27b/32.png) [@Valmont314](https://boards.straightdope.com/u/Valmont314)\ **Post date:** [October 2, 2012, 3:26am UTC](https://boards.straightdope.com/t/999-1/27517/976 "2012-10-02T03:26:08Z")
For anyone reading this BS here is what I said regarding my final conclusions and assertions Regarding the reals I have gone as far as I feel that I can to go to understand what is being discussed here. The final conclusion for me is this: The lack of a definition of a least element (which is also the fundamental representation of interval in the Reals) for the Reals results in the inability to define a relationship between a least element and the limits of precision for the Reals. The definition of this relationship is necessary to balance the limits of precision for the Reals against the magnitude of the least element . The balancing is necessary to preclude the creation of co-located elements. The inability to define such a relationship results in co-located elements. The creation of co-located elements results in the following situation and two possibilities: The two symbolic representations of the co-located elements represent distinct elements of the set. The two symbolic representation of the co-located elements represent the same element. At this point, assuming nobody can demonstrate an error in the logic above, I would say that any further discussion of this topic can be viewed as a discussion of whether or not a Necker cube can be viewed as a single object with two representations or two objects with a single representation. Either answer is correct. * * * Now, how does that translate to my being obligated to construct a new number system to support my assertions? If I point out a manufacturer’s defect on a car am I then obligated to redesign the car to prove that there was defect? That’s not how the scientific method works. So enough is enough, way too much hostility around here lately. I tried to lighten things up a bit but that appears to be a lost cause. Sorry if I offended anybody in the course of trying to have a bit of fun as that surely was not my purpose here. All I can say is wow, there sure seems to be a lot of people around here who take this topic personally and I can’t figure out why, if that is the case, people refuse to discuss the topic beyond the stark, cold, logic of it all. Is that the only way to approach it? Well, I guess I won’t find the answer to that here because I am done. Have fun, you aren’t far from message 999. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [October 2, 2012, 3:30am UTC](https://boards.straightdope.com/t/999-1/27517/977 "2012-10-02T03:30:13Z")
> [@Senegoid](#): > > Now, how do you prove that .999 = 1 with an ε-δ limit proof? That is, how to prove that the sequence of partial sums .9, .99, .999, .9999 etc. approaches 1 as a limit, using ε-δ ? The goal is to prove that, for every rational ε \> 0, the absolute differences between that sequence and 1 eventually drop and stay below ε. In other words, for every natural N, the reciprocal absolute differences eventually surpass and stay above N. Well, the absolute differences between 1 and 0.9, 0.99, 0.999, etc., are 0.1, 0.01, 0.001, etc., whose reciprocals are 10, 100, 1000, etc. So you just have to know that every natural number is below some power of 10, which is basic enough. Note: This proof only works because I’ve said “for every rational ε \> 0”; it is part of the design of the system I am interested in (the Archimedean reals) that rationals suffice to differentiate any two distinct values. If I had said simply “for every number ε \> 0”, we could devise a system where this proof failed. ---
**Author:** ![Trinopus](https://avatars.discourse-cdn.com/v4/letter/t/2bfe46/32.png) [@Trinopus](https://boards.straightdope.com/u/Trinopus)\ **Post date:** [October 2, 2012, 3:33am UTC](https://boards.straightdope.com/t/999-1/27517/978 "2012-10-02T03:33:54Z")
> [@Valmont314](#): > > There is no single counter example when the number is used at the limits of measurement and the limits of operational steps (both as specified in the axioms) > > When you over-rev your engine “miserably” do you then go back to the dealer and ask him to pay for the engine because it was “broken” > > Engines exist in the real world not in the mind. > > Nice melodrama but you have failed to make your case that it is not possible to define a workable system that does not have co-located elements. > > It doesn’t have to be a Ferrari, it just has to be a Car. The existence of a car supports my assertions as does your failure to demonstrate that it is not possible to design such a system. > > … You’re missing the point: your system can get you across town. It can balance a checkbook. But it isn’t universal, and the standard systems of numbers are. The standard definition of number is closed under multiplication. Yours isn’t. The standard obeys the distributive and associative properties. Yours doesn’t. You’ve hog-tied a two-stroke lawnmower motor to a skateboard and are calling it a car. Yeah, it’ll get you to work and back… But it doesn’t have headlights, a horn, or a muffler. We, sir, don’t call it a car. Come back when you devise a number system that has the properties of – well, a number system! ---
**Author:** ![Senegoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/senegoid/32/6606_2.png) [@Senegoid](https://boards.straightdope.com/u/Senegoid)\ **Post date:** [October 2, 2012, 3:35am UTC](https://boards.straightdope.com/t/999-1/27517/979 "2012-10-02T03:35:56Z")
> [@Valmont314](#): > > \ > > The lack of a definition of a least element (which is also the fundamental representation of interval in the Reals) for the Reals results in the inability to define a relationship between a least element and the limits of precision for the Reals. The definition of this relationship is necessary to balance the limits of precision for the Reals against the magnitude of the least element . The balancing is necessary to preclude the creation of co-located elements. The inability to define such a relationship results in co-located elements. The creation of co-located elements results in the following situation and two possibilities: > > \ > > Now, how does that translate to my being obligated to construct a new number system to > support my assertions? If I point out a manufacturer’s defect on a car am I then obligated to redesign the car to prove that there was defect? That’s not how the scientific method works. > **So enough is enough, way too much hostility around here lately. I tried to lighten things up a bit but that appears to be a lost cause. Sorry if I offended anybody in the course of trying to have a bit of fun as that surely was not my purpose here. All I can say is wow, there sure seems to be a lot of people around here who take this topic personally and I can’t figure out why,** if that is the case, people refuse to discuss the topic beyond the stark, cold, logic of it all. > Is that the only way to approach it? Well, I guess I won’t find the answer to that here because I am done. Have fun, you aren’t far from message 999. **Valmont314** , **Valmont314**! We’ve misunderstood you all this time. Here you are bringing some lighthearted levity to this over-extended thread, and here are all the rest of us vainly trying to take you seriously! Well no wonder it gets tense. If only we had known. If only you had made this clear to us earlier. We all could have taken it in the entertaining manner you meant, instead of attacking you with pit bulls, venomous snakes, and vice grips! Wait a minute . . . Didn’t **Francis Vaughan** just say, a few posts up, that he’s enjoying the show with some popcorn and a beer? So there, at least one participant here correctly understood your intentions! Good show, good show! I did wonder a bit about your complaint that we don’t have a defined minimal real number (if I’m understanding what you’re on about there). A defined “infinitesimal” value or something? Do you envision a number system with some minimum quantum value, or a minimum pixel size, or a “Planck real” or something like that? Can you do ε-δ proofs too? ---
**Author:** ![Trinopus](https://avatars.discourse-cdn.com/v4/letter/t/2bfe46/32.png) [@Trinopus](https://boards.straightdope.com/u/Trinopus)\ **Post date:** [October 2, 2012, 3:36am UTC](https://boards.straightdope.com/t/999-1/27517/980 "2012-10-02T03:36:53Z")
> [@Valmont314](#): > > . . . How are you suppose to define a set via axioms without defining axioms? Well…mine are the standard Peano axioms. Can we see yours? You’re the big inventive genius whose ideas are supposed to challenge the established order, but, so far, you’ve produced…nothin’. [Previous page](https://boards.straightdope.com/t/999-1/27517.md?page=48) [Next page](https://boards.straightdope.com/t/999-1/27517.md?page=50)