# .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:** 23
**Author:** ![allotrope](https://avatars.discourse-cdn.com/v4/letter/a/edb3f5/32.png) [@allotrope](https://boards.straightdope.com/u/allotrope)\ **Post date:** [August 7, 2012, 3:12am UTC](https://boards.straightdope.com/t/999-1/27517/441 "2012-08-07T03:12:38Z")
> [@Indistinguishable](#): > > Of course you can prove it… _from other rules_. > > A typical proof is along the lines of what I had written in [this post](http://boards.straightdope.com/sdmb/showpost.php?p=15355286&postcount=399), which [you dismissed](http://boards.straightdope.com/sdmb/showpost.php?p=15355319&postcount=400), but which, I assure you, is what your calculus books will expect. > > That argument is based on certain rules as to how limits at infinity are to be calculated. Different rules would lead to different conclusions. > > Every proof is relative to the rules/definitions/axioms/whatever-you-call-them being employed within that proof. And yet for some reason, you call proofs, assumptions. Care to explain that? ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [August 7, 2012, 3:15am UTC](https://boards.straightdope.com/t/999-1/27517/442 "2012-08-07T03:15:05Z")
> [@Jragon](#): > > I actually **am** curious what the exact axiomatic construction of the Reals is (whichever is simplest). As far as I can tell, the only axiom used to construct the real numbers that even involves infinity is that the set of all reals have a least upper bound/supremum of aleph-0. > > The other stuff is obvious (definition of addition, multiplication, etc), I’m curious if there’s a comprehensive list of all of the axioms that define the reals. All the ones I’ve found online so far omit anything about infinitesimals or infinities (aside from least upper bound). > > I’m not asking you to construct such a thing yourself, I just can’t find it, it HAS to exist SOMEWHERE. There are in fact many different possible formal constructions, which are equivalent in the classical context but which come apart in other contexts (most notably, within “constructive”/“intuitionistic” mathematics, as in topos theory). One quite natural and useful way is by saying that the real numbers are Dedekind cuts of rationals, where a Dedekind cut consists of a partition of the rationals into a downwards closed set L with no maximum element, a set with at most one element M, and an upwards closed set R with no minimum element. The idea is that this represents how the real number is in comparison to every rational: it’s above those in L, equal to those in M, and below those in R. The “no maximum/no minimum” element rules ensure it is not infinitesimally close to any rational without being equal to it. (One then defines arithmetic operations on the reals accordingly.) Note that this has, as a consequence, that any two reals that are above and below the same rationals are equal. And, accordingly, any two reals that are infinitesimally close are equal. We could present the same system via different axioms, taking that more explicitly as one of the defining rules. It’s just semantics what you call the foundational axioms and what you call the consequences. ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [August 7, 2012, 3:15am UTC](https://boards.straightdope.com/t/999-1/27517/443 "2012-08-07T03:15:35Z")
> [@allotrope](#): > > I could be mistaken, but I’m under the impression that Indistinguishable claims to be a maths expert, so I’m sure he can help you out. He **is** a math expert. Despite the fact that I can’t find specifics I’m sure what he’s saying is right. Not just because he’s answered other questions in the past in great detail, but because of what I know from various mathematics courses. You can define a hell of a lot of stuff that is incredibly bizarre. Such as number systems with non-transitive or start-point dependent inequalities. Sorting the numbers on a clock in clockwise order starting at an arbitrary point is one such example where weird mathematical systems come into play. Trust me, if 1 \> 12 can be true depending on your rule system, there’s a **lot** that depends on your rule system. > [@Indistinguishable](#): > > There are in fact many different possible formal constructions, which are equivalent in the classical context but which come apart in other contexts (most notably, within “constructive”/“intuitionistic” mathematics, as in topos theory). > > One way is by saying that the real numbers are Dedekind cuts of rationals, where a Dedekind cut consists of a partition of the rationals into a downwards closed set L, a set with at most one element M, and an upwards closed set R. > > The idea is that this represents how the real number is in comparison to every rational: it’s above those in L, equal to those in M, and below those in R. > > One then defines arithmetic operations on the reals accordingly. I thought it might have to do with the Dedekind cut construction, but I didn’t have enough time as of that post to really digest the set construction notation fully. Thanks. ---
**Author:** ![allotrope](https://avatars.discourse-cdn.com/v4/letter/a/edb3f5/32.png) [@allotrope](https://boards.straightdope.com/u/allotrope)\ **Post date:** [August 7, 2012, 3:18am UTC](https://boards.straightdope.com/t/999-1/27517/444 "2012-08-07T03:18:44Z")
> [@Jragon](#): > > He **is** a math expert. Despite the fact that I can’t find specifics I’m sure what he’s saying is right. Not just because he’s answered other questions in the past in great detail, but because of what I know from various mathematics courses. You can define a hell of a lot of stuff that is incredibly bizarre. Such as number systems with non-transitive or start-point dependent inequalities. Sorting the numbers on a clock in clockwise order starting at an arbitrary point is one such example where weird mathematical systems come into play. > > Trust me, if 1 \> 12 can be true depending on your rule system, there’s a **lot** that depends on your rule system. > > I thought it might have to do with the Dedekind cut construction, but I didn’t have enough time as of that post to really digest the set construction notation fully. Thanks. Based on what I know, he’s been flunking out. So knowing some set theory doesn’t impress me. And I’m still waiting for an answer to my last question. ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [August 7, 2012, 3:21am UTC](https://boards.straightdope.com/t/999-1/27517/445 "2012-08-07T03:21:48Z")
> [@allotrope](#): > > Based on what I know, he’s been flunking out. So knowing some set theory doesn’t impress me. > > And I’m still waiting for an answer to my last question. Did you miss his post a few hundred posts back demonstrating how in the hyperreals .999… doesn’t necessarily equal 1? If not, I suggest you go back and find it. I doesn’t answer your question directly, but it certainly shows he’s correct. ---
**Author:** ![allotrope](https://avatars.discourse-cdn.com/v4/letter/a/edb3f5/32.png) [@allotrope](https://boards.straightdope.com/u/allotrope)\ **Post date:** [August 7, 2012, 3:23am UTC](https://boards.straightdope.com/t/999-1/27517/446 "2012-08-07T03:23:09Z")
> [@Jragon](#): > > Did you miss his post a few hundred posts back demonstrating how in the hyperreals .999… doesn’t necessarily equal 1? If not, I suggest you go back and find it. I doesn’t answer your question directly, but it certainly shows he’s correct. He’s certainly NOT correct about the proof I presented being based on assumptions. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 7, 2012, 3:27am UTC](https://boards.straightdope.com/t/999-1/27517/447 "2012-08-07T03:27:13Z")
> [@Trinopus](#): > > Holy Mother of God, NO! > > (Wow, that’s awkward, as ". . . " means “snip” and is not the same as “…” which means “repeated indefinitely.” But…moving on…) > > Neither of the two strings you just used is defined. You can’t put a digit after “…” > > “.000…0” has no meaning. I would like to ask you to define it, please; I’m pretty sure you cannot. > > This is all precedented; according to Martin Gardner, Charles Dodgson also didn’t understand limits. He complained that the infinitesimal calculus didn’t “eliminate” the difference between a limit and the actual number, but only reduced it to a very small, yet positive difference. When even a maths professor couldn’t get it, I think it is forgiveable for one of hoi polloi (the rest of us!) not to get it. > > Also worth quoting is Norton Juster’s quip from “The Phantom Tollbooth:” “‘Just follow that line forever,’ said the Mathemagician, ‘and when you reach the end, turn left.’” Fair observations… and really inconsequestial to what I am now arguing, just for fun. It has been stated that there is absolutely no end or that at least you could never get to the end of an infinite decimal expansion. I blieve i have refuted that. Have way at it. I am inviting it. ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [August 7, 2012, 3:29am UTC](https://boards.straightdope.com/t/999-1/27517/448 "2012-08-07T03:29:03Z")
> [@allotrope](#): > > He’s certainly NOT correct about the proof I presented being based on assumptions. Okay, there are multiple number systems. The Reals and the Hyperreals are two of them, starting with different axioms. If he shows that the hyperreals, a system with different axioms, can have .999=/=1, then it follows, logically, that .999=1 is dependent on the axioms you can use. Again, if I can construct a system (a system you use every day) where 12+1 = 1\*, I’d say it’s pretty much a given that anything you care to present is based on axioms. Hell, the .999… = 1 vs .999 =/= 1 problem doesn’t even **exist** in the rationals, integers, or naturals. So you could say that some of the axioms in question are the very properties that make the reals and the rationals different sets. - Or 23+1=0, for you 24-hour clock users. ---
**Author:** ![allotrope](https://avatars.discourse-cdn.com/v4/letter/a/edb3f5/32.png) [@allotrope](https://boards.straightdope.com/u/allotrope)\ **Post date:** [August 7, 2012, 3:35am UTC](https://boards.straightdope.com/t/999-1/27517/449 "2012-08-07T03:35:02Z")
> [@Jragon](#): > > Okay, there are multiple number systems. The Reals and the Hyperreals are two of them, starting with different axioms. If he shows that the hyperreals, a system with different axioms, can have .999=/=1, then it follows, logically, that .999=1 is dependent on the axioms you can use. > > Again, if I can construct a system (a system you use every day) where 12+1 = 1\*, I’d say it’s pretty much a given that anything you care to present is based on axioms. Hell, the .999… = 1 vs .999 =/= 1 problem doesn’t even **exist** in the rationals, integers, or naturals. So you could say that some of the axioms in question are the very properties that make the reals and the rationals different sets. > > - Or 23+1=0, for you 24-hour clock users. And everything you just said is completely irrelevant to the issue at hand. ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [August 7, 2012, 3:37am UTC](https://boards.straightdope.com/t/999-1/27517/450 "2012-08-07T03:37:28Z")
> [@allotrope](#): > > And everything you just said is completely irrelevant to the issue at hand. No, it’s not. You asked how the proof was based off assumptions. I pointed out that the assumptions are tied to the fact that you’re using real numbers. If there are number systems **other than reals** , and more importantly, systems **other than reals where the proof fails** then it follows that assumptions are key to the proof working, because they fail in cases where those assumptions aren’t present (i.e. not using Reals). ---
**Author:** ![allotrope](https://avatars.discourse-cdn.com/v4/letter/a/edb3f5/32.png) [@allotrope](https://boards.straightdope.com/u/allotrope)\ **Post date:** [August 7, 2012, 3:40am UTC](https://boards.straightdope.com/t/999-1/27517/451 "2012-08-07T03:40:14Z")
> [@Jragon](#): > > No, it’s not. You asked how the proof was based off assumptions. I pointed out that the assumptions are tied to the fact that you’re using real numbers. If there are number systems **other than reals** , and more importantly, systems **other than reals where the proof fails** than it follows that assumptions are key to the proof working, because they fail in cases where those assumptions aren’t present (i.e. not using Reals). No, I know how proofs are based on axioms. I want to know why Indistinguishable is calling proofs assumptions. I think that’s a pretty egregious error and if you don’t, then I don’t know what to say to you. ---
**Author:** ![Half\_Man\_Half\_Wit](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/half_man_half_wit/32/21766_2.png) [@Half\_Man\_Half\_Wit](https://boards.straightdope.com/u/Half_Man_Half_Wit)\ **Post date:** [August 7, 2012, 3:45am UTC](https://boards.straightdope.com/t/999-1/27517/452 "2012-08-07T03:45:56Z")
> [@erik150x](#): > > It has been brought up and a valid argument to make. But the diefinition is imposed artifially if you ask by the definition we place in reals using limits. It doesn;t have to be that way. Look at the natural numbers. You don’t say because there are no numbers between 2 and 3, that they are the same number do you? Why must this be for reals… except that we impose this via the limit definiton of real numbers. Well, of course the definition is arbitrary, but it’s the agreed upon definition of the real numbers. If you want to impose your own definitions, you can make any relation between the symbols ‘0.999…’ and ‘1’ true: that 0.999… is smaller, larger, greener, more to the left, whatever you can think of; in this way, the discussion is pointless. And there’s an important difference between the reals and the natural numbers (though, of course, only if we use ‘reals’ and ‘natural numbers’ to mean what mathematicians typically take them to mean; again, you may redefine according to your tastes, which however makes meaningful discourse impossible), which is that there are more reals than natural numbers. ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [August 7, 2012, 3:49am UTC](https://boards.straightdope.com/t/999-1/27517/453 "2012-08-07T03:49:08Z")
> [@allotrope](#): > > No, I know how proofs are based on axioms. I want to know why Indistinguishable is calling proofs assumptions. I think that’s a pretty egregious error and if you don’t, then I don’t know what to say to you. He never called proofs assumptions. He said that, specifically, in that proof, lim(n\>inf) { 1/10[sup]n[/sup] } = 0 (or simply 1/10^{infinity} = 0) is an assumption. That’s a completely different statement. It doesn’t matter if that assumption is **also** proven somewhere else. Within that proof itself that particular premise acts as an assumption of the way limits work. To use a parallel, using the quadratic formula to find the roots of a parabola is an assumption of the quadratic formula. It doesn’t matter if it works, it doesn’t matter if you **can** prove it by completing the square on an arbitrary quadratic equation. If you do not prove it within the problem itself, it’s an assumption. That’s what he was saying, that limits and the properties of infinity – regardless of their provability, axiomatic basis, whatever in the reals – were assumed within the proof you gave. The only thing that proof proves is that .999…=1, everything other bit of math used in that proof to reach that conclusion is an assumption. ---
**Author:** ![Trinopus](https://avatars.discourse-cdn.com/v4/letter/t/2bfe46/32.png) [@Trinopus](https://boards.straightdope.com/u/Trinopus)\ **Post date:** [August 7, 2012, 3:52am UTC](https://boards.straightdope.com/t/999-1/27517/454 "2012-08-07T03:52:01Z")
> [@erik150x](#): > > Fair observations… and really inconsequestial to what I am now arguing, just for fun. > > It has been stated that there is absolutely no end or that at least you could never get to the end of an infinite decimal expansion. I blieve i have refuted that. Have way at it. I am inviting it. Please define the term “.000…1” You’re making some lively assertions, but you haven’t given us any definitions. ---
**Author:** ![allotrope](https://avatars.discourse-cdn.com/v4/letter/a/edb3f5/32.png) [@allotrope](https://boards.straightdope.com/u/allotrope)\ **Post date:** [August 7, 2012, 3:53am UTC](https://boards.straightdope.com/t/999-1/27517/455 "2012-08-07T03:53:35Z")
> [@Jragon](#): > > He never called proofs assumptions. He said that, specifically, in that proof, lim(n\>inf) { 1/10[sup]n[/sup] } = 0 (or simply 1/10^{infinity} = 0) is an assumption. That’s a completely different statement. > > It doesn’t matter if that assumption is **also** proven somewhere else. Within that proof itself that particular premise acts as an assumption of the way limits work. > > To use a parallel, using the quadratic formula to find the roots of a parabola is an assumption of the quadratic formula. It doesn’t matter if it works, it doesn’t matter if you **can** prove it by completing the square on an arbitrary quadratic equation. If you do not prove it within the problem itself, it’s an assumption. > > That’s what he was saying, that limits and the properties of infinity – regardless of their provability, axiomatic basis, whatever in the reals – were assumed within the proof you gave. The only thing that proof proves is that .999…=1, everything other bit of math used in that proof to reach that conclusion is an assumption. It is in no way an assumption. It flows directly from the definition of the infinitesimal calculus. To even attempt to call it an assumption is a gross insult to intelligence of your audience. I suppose you’ve been getting away with it up until now - which is surprising. I though that we had some actual scientists and mathematicians here. ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [August 7, 2012, 3:55am UTC](https://boards.straightdope.com/t/999-1/27517/456 "2012-08-07T03:55:44Z")
> [@allotrope](#): > > It is in no way an assumption. It flows directly from the definition of the infinitesimal calculus. To even attempt to call it an assumption is a gross insult to intelligence of your audience. I suppose you’ve been getting away with it up until now - which is surprising. I though that we had some actual scientists and mathematicians here. You do realize that even 1+1=2 is an assumption about the properties of the “+” and “=” operators, right? ---
**Author:** ![allotrope](https://avatars.discourse-cdn.com/v4/letter/a/edb3f5/32.png) [@allotrope](https://boards.straightdope.com/u/allotrope)\ **Post date:** [August 7, 2012, 3:58am UTC](https://boards.straightdope.com/t/999-1/27517/457 "2012-08-07T03:58:29Z")
> [@Jragon](#): > > You do realize that even 1+1=2 is an assumption about the properties of the “+” and “=” operators, right? Those are axioms. Please use the correct terminology. The behavior of limits is not axiomatic. It is only based on axioms. But I’ll tell you a secret. In deductive systems, EVERYTHING is based on an axiom. Shhhhhh ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 7, 2012, 4:00am UTC](https://boards.straightdope.com/t/999-1/27517/458 "2012-08-07T04:00:33Z")
> [@Half\_Man\_Half\_Wit](#): > > Well, of course the definition is arbitrary, but it’s the agreed upon definition of the real numbers. If you want to impose your own definitions, you can make any relation between the symbols ‘0.999…’ and ‘1’ true: that 0.999… is smaller, larger, greener, more to the left, whatever you can think of; in this way, the discussion is pointless. And there’s an important difference between the reals and the natural numbers (though, of course, only if we use ‘reals’ and ‘natural numbers’ to mean what mathematicians typically take them to mean; again, you may redefine according to your tastes, which however makes meaningful discourse impossible), which is that there are more reals than natural numbers. Your points are valid. I guess I was merely trying to express that perhaps, we impose this restrisction unecccessarly. I mean if you are using limits to define your numebrs, then there is no other way around it. There would have to be a real number between the 2, agreed. But I at the time I was trying to say we don’t have define numebrs this way. Anyway… we make rules, but they are not (or at least should not) be completely arbitrary in math. I have a nagginng feeling limit definitons are imposed on numbers because people don’t like infinitesimals. Whish is ok, probaably makes sense too. But it also to me sweeps some issue under the rug. Unless you just want to say they plain don’t exist in any real way to begin with. If you want to say numbers don’t exist in any real way but that which we imagine, I am sorry but I am not in that camp. I take a platonic view there. I am more concerned now with whther we can or cannot “reach” or “find” the end of an infinite sequnece of 9s or whatever. Which I have been arguing for some pages now… With various bewlidering resonses in mind me anyway 😉 ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 7, 2012, 4:02am UTC](https://boards.straightdope.com/t/999-1/27517/459 "2012-08-07T04:02:35Z")
> [@Trinopus](#): > > Please define the term “.000…1” > > You’re making some lively assertions, but you haven’t given us any definitions. an infinite string of zero’s except for the 1/infinity decimal place. It seems a fariously obvious definiton [given my context], but i guess there could be some confusion. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [August 7, 2012, 4:06am UTC](https://boards.straightdope.com/t/999-1/27517/460 "2012-08-07T04:06:40Z")
> [@allotrope](#): > > It is in no way an assumption. It flows directly from the definition of the infinitesimal calculus. To even attempt to call it an assumption is a gross insult to intelligence of your audience. I suppose you’ve been getting away with it up until now - which is surprising. I though that we had some actual scientists and mathematicians here. The whole point of my “Everything in math is just rules” thing was to say “Everything depends on the definitions; with different interpretations, all this ‘0.999… has infinitely many 9s and is just less than 1, by an infinitesimal amount’ stuff is perfectly sensible”. So, yes. Certain things flow directly from the standard definitions of the infinitesimal calculus. And other things would flow with different definitions, or different interpretations of the terminology, or different axioms or different rules (these all amount to the same thing). That’s what I was saying. The question “Does 0.999… = 1?” is dependent on your interpretation of the notation. I’ve noted the standard interpretation, and also noted other, reasonable interpretations one might have used instead. For example, you will find that [Keisler’s textbook on calculus](http://www.math.wisc.edu/~keisler/calc.html) discusses the infinitesimal calculus in the context of different definitions (or different interpretations of the terminology, or different rules, or whatever you want to call it), such that 1/10[sup]n[/sup] is positive rather than zero when n is infinite. And why not? There’s nothing wrong with that. Nothing in math is a brute external fact independent of our made-up rules, any more than bishops moving diagonally is a brute external fact independent of our say-so. That was my only point. > [@allotrope](#): > > The behavior of limits is not axiomatic. It is only based on axioms. But I’ll tell you a secret. In deductive systems, EVERYTHING is based on an axiom. Well, that’s my point… When I said “Everything in math is just rules”, you went off on me. But now you are claiming essentially the same thing, yet taking it as somehow against me? [One might terminologically quibble about some rules being called “axioms” and others being called “rules of inference” or “rules of logic”, but this is only a difference in our attitude; it’s all the same thing, whether we call them “rules”, “axioms”, “laws”, “definitions”, or what have you… The “rules of logic” are as malleable as any other; see classical vs. intuitionistic (vs. paraconsistent vs. linear vs. quantum vs. noncommutative vs…) logic] You claimed > [@allotrope](#): > > And as the equation we just looked at shows, when the decimal recurs ad infinitum and n = ∞, then it is not a rule but in actual fact there is no residual infinitesimal. There’s no sense in saying “it’s not a rule but in actual fact…” about anything in mathematics, and particularly not about something which has been demonstrated to be quite dependent on the ambient rule-system, in the context of an argument about that choice of rule system. [Previous page](https://boards.straightdope.com/t/999-1/27517.md?page=22) [Next page](https://boards.straightdope.com/t/999-1/27517.md?page=24)