# .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:** 74
**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:** [July 28, 2014, 3:03pm UTC](https://boards.straightdope.com/t/999-1/27517/1466 "2014-07-28T15:03:27Z")
> [@Indistinguishable](#): > > Wait, what do you mean by this? What digits would you get instead if you wrote 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + … as a digit expansion? That depends on how you want to define your notation—you could, for instance, define the first place as the coefficient of ‘1’, the second as the coefficient of ‘-1/3’, the third as the coefficient of ‘1/5’, and so on, in which case it’d be 1111… Not sure that’s a sensible notation, or even which numbers you could represent using it, but nobody seems to be forcibly refraining me from using it. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 28, 2014, 3:26pm UTC](https://boards.straightdope.com/t/999-1/27517/1467 "2014-07-28T15:26:09Z")
Ah, ok. I must’ve read it too quickly to make the connection to the just-prior context of “The same number can have different digit expansions in different bases”. ---
**Author:** ![Budget\_Player\_Cadet](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/budget_player_cadet/32/205_2.png) [@Budget\_Player\_Cadet](https://boards.straightdope.com/u/Budget_Player_Cadet)\ **Post date:** [July 28, 2014, 3:33pm UTC](https://boards.straightdope.com/t/999-1/27517/1468 "2014-07-28T15:33:01Z")
> [@The\_Hamster\_King](#): > > I’m still hoping for a response to the question I asked 3 pages ago. What’s wrong with this proof? > > 1. x = 0.333… > 2. 10x = 3.333… > 3. 3 + x = 3.333… > 4. 10x = 3 + x > 5. 9x = 3 > 6. x = 3/9 > 7. x = 1/3 > 8. 0.333… = 1/3 I think he would object to step 1. ---
**Author:** ![7777777](https://avatars.discourse-cdn.com/v4/letter/7/9de0a6/32.png) [@7777777](https://boards.straightdope.com/u/7777777)\ **Post date:** [July 28, 2014, 4:49pm UTC](https://boards.straightdope.com/t/999-1/27517/1469 "2014-07-28T16:49:42Z")
> [@Budget\_Player\_Cadet](#): > > Here’s another one - can you name me the difference between 0.9999… and 1? Like, if two real numbers a and b are different, then in this equation: > > | a - b | \> e; e element of Real Numbers > > e will, by definition of the real numbers, always be larger than 0. So can you find e? It’s not 0.01. It’s not 0.0001. It’s not 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001. > > What is it? The problem with your question is the same as with writing down all the digits of the number pi, π . You cannot write down all the decimals of π, and you cannot write down the value of the infinitesimal value between 1 and 0.999999…Just because you cannot write down the exact value of π does not mean that π does not exist. Therefore you cannot write down the value of the infinitesimal, nevertheless it exists. The infinitesimal is a variable, and it can be described by x which can be then used in various formulas to deal with it. It is not a fairy tale or a unicorn. Another example: what it the largest amount or the largest number? You cannot know its value because if you do, you can always add number 1 to it and get still larger value. The largest number cannot be infinity because of the formula ∞ = ∞ + 1, its value does not anymore increase. The same goes with the smallest amount, the smallest number. It exists although you don’t know its value. I know that that you will deny the largest number or amount and the smallest number, or amount. You think there are no such numbers. I know, you deny everything that I say. You just need to think yourself now and don’t expect I tell you what they are. They are not secrets, but why should I tell you what they are because I know you will deny them? But think about physics, the most fundamental laws always deal with the principle of minimum, why there should not be a corresponding principle also in mathematics, something like a smallest amount of a known largest amount? ---
**Author:** ![Budget\_Player\_Cadet](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/budget_player_cadet/32/205_2.png) [@Budget\_Player\_Cadet](https://boards.straightdope.com/u/Budget_Player_Cadet)\ **Post date:** [July 28, 2014, 5:19pm UTC](https://boards.straightdope.com/t/999-1/27517/1470 "2014-07-28T17:19:38Z")
> [@7777777](#): > > The problem with your question is the same as with writing down all the digits of the > number pi, π . > You cannot write down all the decimals of π, and you cannot write down the value > of the infinitesimal value between 1 and 0.999999…Just because you cannot > write down the exact value of π does not mean that π does not exist. True, but you can mathematically determine the value of pi. You cannot write it all down, but you can [define it](http://en.wikipedia.org/wiki/Pi#Definition), mathematically, and then use that definition to proceed forward. So in that vein, how would you define the infinitesimal value between 1 and 0.9999…? I mean, the best way I can think of to define it is as 0.0000…0001. The problem with that though, is that there is an _infinite_ number of zeroes there. The 1 just simply never shows up. > [@](#): > > Therefore > you cannot write down the value of the infinitesimal, nevertheless it exists. Good news - I’m not asking you to write it down, I’m asking you to _define_ it. I could never write down Pi, but I can easily define it as “The circumference of a circle divided by its diameter”, which gives me Pi. > [@](#): > > Another example: what it the largest amount or the largest number? > You cannot know its value because if you do, you can always add number 1 > to it and get still larger value. The largest number cannot be infinity because of > the formula ∞ = ∞ + 1, its value does not anymore increase. The same goes with the smallest amount, the smallest number. It exists although you don’t know its value. Actually, this was one of the first things we learned in Analysis 1 last semester - no, the largest number and the smallest number _don’t_ exist within the set of the real numbers. The concept only makes sense within intervals, and even then only within _inclusive_ intervals. For example the interval [0,1], which is an interval containing 0, 1, and every real number between the two. It has a clearly defined largest and smallest number (1 and 0, respectively). But the interval (0,1), which is an interval containing every real number between 0 and 1 but _not_ 0 and 1 does _not_ have a largest or smallest number! And (-inf, inf), the interval containing the set of real numbers, just like every other exclusive interval, _does not have a maximum or a minimum_ - defined as the number which is larger/smaller than every other number in the set and contained within the set. Rather, they have a “supremum” and an “infimum” - the number which is larger/smaller than every other number in the set, but which must not be included in the set. It seems intuitive that there would be a largest number, but when you actually go looking for it… it’s _just not there_. Because, mathematically speaking, the term makes no sense. > [@](#): > > I know that that you will deny the largest number or amount and the smallest number, > or amount. You think there are no such numbers. I know, you deny everything that I say. I deny this because this is the very first subject covered in college-level Analysis courses. It’s a fundamental principle without which you cannot truly understand things like the limes of a row or function. If you have evidence that my professors were wrong, then _show it_! Don’t just act like you have the truth but “I’m not going to show it to you guys because I’m salty.” If you have evidence, _let’s hear it!_ > [@](#): > > They are not secrets, but why should I tell you what they are because I know > you will deny them? If you already know I will deny them, then you already know that they’re flawed and unconvincing. I would like to give you more credit than that. > [@](#): > > But think about physics, the most fundamental laws always > deal with the principle of minimum, why there should not be a corresponding > principle also in mathematics, something like a smallest amount of a known > largest amount? Because physics is _not_ mathematics! Mathematics is a completely theoretical construct, built from the ground up on axioms which by design of those axioms attempts at least to some degree to mirror the real world. Physics, on the other hand, is based on the real world. In the real world, there are certain clear limits. You cannot go faster than the speed of light. You cannot be smaller than the Planck length. Mathematics has no such limitation - which is part of why mathematics, when applied to physics or indeed any “real-world” science, must be done so with _extreme care_ - otherwise you end up with downright silly results (such as people extrapolating that because the human population grew at a speed X for the past 50 years, that it always grew that fast and therefore the world was 4000 years old or some crap like that). In mathematics, the concepts of minimum and maximum are not universally applicable. Not every set or interval has a minimum or a maximum. In fact, an exclusive interval, almost by definition, does not. If you believe the interval (0,1) has a maximal or minimal element, then please, _define_ that element. Not write it down, _define_. The set of real numbers is an exclusive interval. If you want, if it would help you, I could try to dredge up my notes and slides from that class. Would that help you at all? ---
**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:** [July 28, 2014, 6:12pm UTC](https://boards.straightdope.com/t/999-1/27517/1471 "2014-07-28T18:12:55Z")
> [@7777777](#): > > The problem with your question is the same as with writing down all the digits of the > number pi, π . > You cannot write down all the decimals of π, and you cannot write down the value > of the infinitesimal value between 1 and 0.999999…Just because you cannot > write down the exact value of π does not mean that π does not exist. Therefore > you cannot write down the value of the infinitesimal, nevertheless it exists. What you can and cannot write down is wholly a matter of your notational system. In base π, for example, π is exactly 1; however, numbers having a finite digit representation in the decimal system won’t have on in the base π system. Nevertheless, and independently of the number system, π has a definite value, as does 0.9999~ (it’s 1, by the way). Your ‘infinitesimal difference’ doesn’t. ---
**Author:** ![7777777](https://avatars.discourse-cdn.com/v4/letter/7/9de0a6/32.png) [@7777777](https://boards.straightdope.com/u/7777777)\ **Post date:** [July 28, 2014, 7:12pm UTC](https://boards.straightdope.com/t/999-1/27517/1472 "2014-07-28T19:12:09Z")
> [@Budget\_Player\_Cadet](#): > > Actually, this was one of the first things we learned in Analysis 1 last semester - no, the largest number and the smallest number _don’t_ exist within the set of the real numbers. The concept only makes sense within intervals, and even then only within _inclusive_ intervals. For example the interval [0,1], which is an interval containing 0, 1, and every real number between the two. It has a clearly defined largest and smallest number (1 and 0, respectively). But the interval (0,1), which is an interval containing every real number between 0 and 1 but _not_ 0 and 1 does _not_ have a largest or smallest number! And (-inf, inf), the interval containing the set of real numbers, just like every other exclusive interval, _does not have a maximum or a minimum_ - defined as the number which is larger/smaller than every other number in the set and contained within the set. Rather, they have a “supremum” and an “infimum” - the number which is larger/smaller than every other number in the set, but which must not be included in the set. It seems intuitive that there would be a largest number, but when you actually go looking for it… it’s _just not there_. Because, mathematically speaking, the term makes no sense. > > I deny this because this is the very first subject covered in college-level Analysis courses. It’s a fundamental principle without which you cannot truly understand things like the limes of a row or function. If you have evidence that my professors were wrong, then _show it_! Don’t just act like you have the truth but “I’m not going to show it to you guys because I’m salty.” If you have evidence, _let’s hear it!_ > > If you already know I will deny them, then you already know that they’re flawed and unconvincing. I would like to give you more credit than that. > > Because physics is _not_ mathematics! Mathematics is a completely theoretical construct, built from the ground up on axioms which by design of those axioms attempts at least to some degree to mirror the real world. Physics, on the other hand, is based on the real world. In the real world, there are certain clear limits. You cannot go faster than the speed of light. You cannot be smaller than the Planck length. Mathematics has no such limitation - which is part of why mathematics, when applied to physics or indeed any “real-world” science, must be done so with _extreme care_ - otherwise you end up with downright silly results (such as people extrapolating that because the human population grew at a speed X for the past 50 years, that it always grew that fast and therefore the world was 4000 years old or some crap like that). > > In mathematics, the concepts of minimum and maximum are not universally applicable. Not every set or interval has a minimum or a maximum. In fact, an exclusive interval, almost by definition, does not. If you believe the interval (0,1) has a maximal or minimal element, then please, _define_ that element. Not write it down, _define_. The set of real numbers is an exclusive interval. If you want, if it would help you, I could try to dredge up my notes and slides from that class. Would that help you at all? Regarding the largest number, if you want to know what it is, you should solve the equation: Z + 1 = ∞ Z is the variable representing the largest number, if you add number 1 to it you don’t arrive at still a larger number, instead you will arrive at infinity. Writing down the value of Z will lead to a problem, as if it did not exist because if you do it, for example you say it is 1000, and then add 1 , you will arrive at 1001 which is larger than 1000. What solution do you propose to this problem? A wrong answer is that there is no Z. The infinitesimal x can be also derived from the same formula: Z = -log(x) , log is base 10 logarithm you can easily make sure that the smaller the value of x is, the larger will be the corresponding value of Z ---
**Author:** ![Budget\_Player\_Cadet](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/budget_player_cadet/32/205_2.png) [@Budget\_Player\_Cadet](https://boards.straightdope.com/u/Budget_Player_Cadet)\ **Post date:** [July 28, 2014, 8:37pm UTC](https://boards.straightdope.com/t/999-1/27517/1473 "2014-07-28T20:37:14Z")
> [@7777777](#): > > Regarding the largest number, if you want to know what it is, you should > solve the equation: > > Z + 1 = ∞ > > Z is the variable representing the largest number, if you add number 1 to it > you don’t arrive at still a larger number, instead you will arrive at infinity. > Writing down the value of Z will lead to a problem, as if it did not exist because > if you do it, for example you say it is 1000, and then add 1 , you will arrive at > 1001 which is larger than 1000. What solution do you propose to this problem? > A wrong answer is that there is no Z. No, a _right_ answer. The equation you posted above is meaningless. If Z is within the realm of the real, natural, or complex numbers, there is _no number_ for which that equation is true. Infinity simply doesn’t work that way. Indeed, the only _thing_ for which the above equation could possibly be true is infinity. Like, to simplify this back to the realm of the natural numbers, one way of defining the natural numbers is to first define “1” and “0”, and then recursively add +1 to get 2, 3, 4, 5, etc. The thing about this, though, is that at no point does this _end_. For _any_ natural number you could possibly imagine, it’s possible to, again, apply “+1” and get another natural number. Not infinity, _n element of N_. This **never, ever stops**. And because the natural numbers are a subset of the real numbers, the same rule applies there. There is not a single real number Z for which Z+1 does not equal another real number. You can just keep iterating upwards. In order for this not to be true, Z would have to _not_ be a real number. This is simply a property of the set of real numbers (and natural numbers, and rational numbers, and complex numbers) - it keeps going _forever_. The concept of infinity is simply a way to stratify this, and explain the behavior of certain constructs - for example, when we say lim[x-\>0] (1/x) = ∞, what we mean is that the closer we get to zero (which we’re not allowed to enter into the equation as it is not defined), the value of the function becomes larger and larger without limit. > [@](#): > > The infinitesimal x can be also derived from the same formula: > Z = -log(x) , log is base 10 logarithm > you can easily make sure that the smaller the value of x is, the larger will be the > corresponding value of Z Yes, but as previously stated, the _only_ point at which the above calculation works is Z = infinity (or some other bizarre construct; Z is clearly not a real number). So what we’re dealing with is: Z = -log(x) \<=\> Infinity = -log(x) \<=\> log(x) = -infinity \<=\> 10^(log(x)) = 10^-Infinity \<=\> x = 10^-infinity \<=\> x = 0 And of course, that whole equation is entirely shorthand, because using infinity like that will make a mathematician cringe and want to start cutting, but it gets the point across. ---
**Author:** ![The\_Great\_Unwashed](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/the_great_unwashed/32/4791_2.png) [@The\_Great\_Unwashed](https://boards.straightdope.com/u/The_Great_Unwashed)\ **Post date:** [July 28, 2014, 10:35pm UTC](https://boards.straightdope.com/t/999-1/27517/1474 "2014-07-28T22:35:54Z")
> [@Budget\_Player\_Cadet](#): > > lim[x-\>0] (1/x) = ∞ \ lim[x-\>0[sup]+[/sup]] (1/x) = ∞ for lim[x-\>0[sup]-[/sup] ] (1/x) = -∞ because 1/x has a discontinuity when x = 0 ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 28, 2014, 11:12pm UTC](https://boards.straightdope.com/t/999-1/27517/1475 "2014-07-28T23:12:54Z")
One of many choices one may make in reifying infinity as an arithmetic quantity is whether to [distinguish](http://en.wikipedia.org/wiki/Extended_real_number_line) positive and negative infinity or [not](http://en.wikipedia.org/wiki/Real_projective_line) (each choice having circumstances in which it is the most convenient approach for the analysis one is interested in). ---
**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:** [July 28, 2014, 11:59pm UTC](https://boards.straightdope.com/t/999-1/27517/1476 "2014-07-28T23:59:54Z")
> [@](#): > > A wrong answer is that there is no Z. Why? Axiom. The Reals are closed under addition. If x and y are elements of **R** , then x + y is an element of **R**. Theorem, there exists Z, which is an element of **R** , ∞ which is not an element of **R** , such that Z + 1 = ∞ By closure of **R** , Since 1 is an element of **R** , ∞ must also be an element of **R**. This contradicts the theorem, so - either Z is not an element of **R** , or there is no Z such that Z + 1 = ∞. ---
**Author:** ![sich\_hinaufwinden](https://avatars.discourse-cdn.com/v4/letter/s/c5a1d2/32.png) [@sich\_hinaufwinden](https://boards.straightdope.com/u/sich_hinaufwinden)\ **Post date:** [July 29, 2014, 12:32am UTC](https://boards.straightdope.com/t/999-1/27517/1477 "2014-07-29T00:32:54Z")
> [@7777777](#): > > Regarding the largest number, if you want to know what it is, you should > solve the equation: > > Z + 1 = ∞ > > Z is the variable representing the largest number, if you add number 1 to it > you don’t arrive at still a larger number, instead you will arrive at infinity. See, I was always taught that to find the largest number you solve this equation: Z + 0.1 = ∞ So you and I are pretty much on the same page. Except, you know, that my Z is just a little bit bigger than your Z. Don’t get me wrong - I mean you got a nice Z and all. It’s just a little bit smaller than my Z. NTTAWWT. > [@7777777](#): > > A wrong answer is that there is no Z. ETA: Don’t forget the corollary - A right answer is that there is no spoon. ---
**Author:** ![dropzone](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dropzone/32/7515_2.png) [@dropzone](https://boards.straightdope.com/u/dropzone)\ **Post date:** [July 29, 2014, 3:42am UTC](https://boards.straightdope.com/t/999-1/27517/1478 "2014-07-29T03:42:16Z")
> [@zoid](#): > > You should reinstate this policy. I dunno. I kinda like it. My motto is _Bonum satis est bonum satis._ As long as I’m not doing CAD/CAM work “yeah, pretty much” gets the job done. However, a difference of 0.00000000000001–a rounding error deep in the computer–can crash the laser cutter. It took some effort convincing young engineers, all smarter than some of our guests, that it was true. > [@7777777](#): > > Should we be satisfied that 1=0.999… because the teacher tells it just is so shut up. Did you ever went to or heard about a shut up and calculate school? Why yes, I did. So did you, I assume. 'Shut up. It just is," was usually couched in the more gentle phrase of, “It’s axiomatic that,” but it’s much the same thing. Axioms are generally pretty obvious if you look at them with an open mind, but some perfectly intelligent people get a bee in their bonnet and think they found a loophole. My father went to his grave convinced that he was this\>\ **Author:** ![7777777](https://avatars.discourse-cdn.com/v4/letter/7/9de0a6/32.png) [@7777777](https://boards.straightdope.com/u/7777777)\ **Post date:** [July 29, 2014, 6:09am UTC](https://boards.straightdope.com/t/999-1/27517/1479 "2014-07-29T06:09:38Z") > [@sich\_hinaufwinden](#): > > See, I was always taught that to find the largest number you solve this equation: > > Z + 0.1 = ∞ > > So you and I are pretty much on the same page. > > Except, you know, that my Z is just a little bit bigger than your Z. > > Don’t get me wrong - I mean you got a nice Z and all. It’s just a little bit smaller than my Z. > > NTTAWWT. > > ETA: Don’t forget the corollary - A right answer is that there is no spoon. A good try! It is good to find at least one person pretty much on the same page. But your Z is not large enough. Don’t also get me wrong, you got a nice Z except that it is not yet large enough. Try increasing Z a little more, a good excercise is you first add 0.1 then 0.11 then 0.111 then 0.1111 then…and finally 0.1111111111… Then multiply it by 9 and add it to Z. What have you done? Did you got to infinity? Remember you also need a bit faith in order to calculate up to infinity in this way. Or perhaps faith is not the right word, I mean you need to trust in what you are doing. Otherwise there might be something wrong, there might be a spoon. I could notice an error in your Z.😕 Your Z is equal to infinity. Remember, infinity is not a number. ---
**Author:** ![7777777](https://avatars.discourse-cdn.com/v4/letter/7/9de0a6/32.png) [@7777777](https://boards.straightdope.com/u/7777777)\ **Post date:** [July 29, 2014, 6:24am UTC](https://boards.straightdope.com/t/999-1/27517/1480 "2014-07-29T06:24:56Z")
> [@dropzone](#): > > Why yes, I did. So did you, I assume. 'Shut up. It just is," was usually couched in the more gentle phrase of, “It’s axiomatic that,” but it’s much the same thing. Axioms are generally pretty obvious if you look at them with an open mind, but some perfectly intelligent people get a bee in their bonnet and think they found a loophole. My father went to his grave convinced that he was this\>\ **Author:** ![j\_sum1](https://avatars.discourse-cdn.com/v4/letter/j/8baadc/32.png) [@j\_sum1](https://boards.straightdope.com/u/j_sum1)\ **Post date:** [July 29, 2014, 7:12am UTC](https://boards.straightdope.com/t/999-1/27517/1481 "2014-07-29T07:12:10Z") Wait 77777777, let’s see if I have this right. In the context of some side discussion that you raised about subtracting 1 from infinity to get Z, you are now subtracting an infintesimal from infinity. (Ok, subtracting 1 and then adding nine times 0.111111…) I am really confused as to what kind of point you are trying to make. I have tried to give you the benefit of doubt but i must conclude that your language, your reasoning and the point that you are trying to make are extremely muddled. And you are rubbing shoulders with some of the best mathematical minds that you are likely to encounter anywhere. (I am not including myself in their number.) You then accuse people of just saying shut-up without pointing out errors in your proof. In fact they have been far more gracious than that. **The Hamster King** offered a detailed and yet simple proof to the original problem that reaches a different conclusion from you, and then invited you to isolate the particular point that you disagree with. You have not responded. Several posters have referred back to the system that we call the real numbers which is the group most logically relevant to the question as posed. They have pointed out politely that the conclusion that you arrive at is inconsistent with the real system and have invited you to clearly explain the system that you are working in. you have’t responded. **Indistinguishible** has talked about the difference between notation and concept. he has also introduced the possibilities of systems where there is a difference between 0.99999… and 1 but pointed out that there are different axioms underlying these systems and there are implications for notation. He has said he considers you a lost cause, but has invited you to engage more appropriately with the material by providing some kind of a framework to the system you envisage and/or tightening your notation by providing definitions. You have not responded. So, I am at a loss to understand what you are trying to achieve by this persistent and as yet unfounded assertion that 0.9999… ≠ 1. Worse than that, in your last couple of posts I am not even sure what kind of point you are trying to make. It certainly is not clear to me. You are not the first numbervangelist with unorthodox conclusions to fly by this thread and you probably won’t be the last. You will find within the 30 pages of this thread at least half a dozen different lucid mathematical proofs that what you are claiming is in fact incorrect. If you want a meaningful discussion I strongly recommend you locate these and ask questions about them. Or point out the precise line where you disagree from orthodox understandings of the situation. I am not holding my breath though. ---
**Author:** ![7777777](https://avatars.discourse-cdn.com/v4/letter/7/9de0a6/32.png) [@7777777](https://boards.straightdope.com/u/7777777)\ **Post date:** [July 29, 2014, 7:25am UTC](https://boards.straightdope.com/t/999-1/27517/1482 "2014-07-29T07:25:07Z")
> [@Francis\_Vaughan](#): > > If x and y are elements of **R** , then x + y is an element of **R**. You are using the word “if”. I told you already, if 1=0.99999…is true, it is true. _IF_ there is no infinitesimal difference between 1 and 0.99999…these numbers are the same. The word if is magical, you can prove anything with it. _IF_ 2=1 is true, it is true. [QUOTE=Francis Vaughan] Since 1 is an element of R [/QUOTE] Now you are telling that 1 is an element of R, although at first, when you write that if x+y are elements of R, you are implying that _IF_ 1 is an element of R. ---
**Author:** ![j\_sum1](https://avatars.discourse-cdn.com/v4/letter/j/8baadc/32.png) [@j\_sum1](https://boards.straightdope.com/u/j_sum1)\ **Post date:** [July 29, 2014, 8:41am UTC](https://boards.straightdope.com/t/999-1/27517/1483 "2014-07-29T08:41:38Z")
> [@](#): > > Originally Posted by Francis Vaughan > If x and y are elements of R, then x + y is an element of R. The “if” there is to identify that the statement that follows is applicable to the Real Number system. In other words, Francis Vaughan is clearly elucidating the axioms that he is using. For The Real Number system, 0.999… = 1. You are right. The “if” is important. Your conclusion is dependent on the axioms used. As **Indistinguishable** has pointed out, you are free to use other axioms and come to a different but logical conclusion. What you haven’t done is elucidate which axioms you are using, much less show that your conclusion logically follows from those axioms. ---
**Author:** ![j\_sum1](https://avatars.discourse-cdn.com/v4/letter/j/8baadc/32.png) [@j\_sum1](https://boards.straightdope.com/u/j_sum1)\ **Post date:** [July 29, 2014, 8:53am UTC](https://boards.straightdope.com/t/999-1/27517/1484 "2014-07-29T08:53:22Z")
I should add… Francis Vaughan’s axiom may be written just as clearly without the word “if”. x + y is a member of the set of R for all x and y that are themselves members of the set of R. Or more succinctly (x+y) ∈ R ∀ x,y ∈ R ---
**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:** [July 29, 2014, 9:23am UTC](https://boards.straightdope.com/t/999-1/27517/1485 "2014-07-29T09:23:21Z")
Indeed, I was trying to spell out the axiom. The work “if” isn’t needed for what I wrote, and can be ignored. Attacking the “if” is not helpful or pertinent. > [@](#): > > Now you are telling that 1 is an element of R, although at first, when you write that if x+y are elements of R, you are implying > that IF 1 is an element of R. I’m not even sure ho to parse what you wrote. So lets try again. This time more formally. Axiom 1 - The Reals are closed under addition: x+y is a member of the Reals for all x and y that are members of the Reals : (x+y) ∈ ℝ ∀ x,y ∈ ℝ Theorem 1 - There exists Z, Z is an element of R, there exists ∞, ∞ is not an element of R, such that Z + 1 = ∞ ∃ Z ∈ ℝ, ∃ ∞ ∌ ℝ : Z + 1 = ∞ 1 ∈ ℝ By Axiom 1 - 1 ∈ ℝ ⋀ Z ∈ ℝ -\> Z + 1 ∈ ℝ By Theorem 1 Z + 1 = ∞ -\> ∞ ∈ ℝ This contradicts Theorem 1. Either Z is not an element of R, or there is no Z such that Z + 1 = ∞. [Previous page](https://boards.straightdope.com/t/999-1/27517.md?page=73) [Next page](https://boards.straightdope.com/t/999-1/27517.md?page=75)