# .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:** 18
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [August 6, 2012, 4:29pm UTC](https://boards.straightdope.com/t/999-1/27517/341 "2012-08-06T16:29:47Z")
> [@President\_Johnny\_Gentle](#): > > Not to pick on you, since I’ve seen other people make similar comments elsewhere, but it’s important to note that we typically add from right to left because of a specific algorithm that we learn in elementary school. We use it because it works, but there’s nothing exceptional about this algorithm (or the typical algorithms for subtracting, multiplying, dividing…) What I’m lecturing, for example, I typically add from left to right, since that’s the way we read the numbers. I just make sure to scan one place ahead to see if it may become necessary to carry. There are many other algorithms that can be used as well, although some aren’t as effective. Yeah, actually, that brings to mind the fact that you can subtract from left to right, it’s just a little more complicate. If I’m subtracting 0.9 from 1.0, I can start by subtracting the leftmost digits (1-0) to get 1. I write that down followed by a decimal point. Next I try to subtract the tenths digits (0-9). I can’t because 9 is greater than 0. So I take one away from the 1 I already wrote down, reducing it to zero, and I now subtract 10-9 for the tenths digits, leaving me with one. Repeat down the line, and each digit of the answer turns out to be 0. 0.000… ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 6, 2012, 4:31pm UTC](https://boards.straightdope.com/t/999-1/27517/342 "2012-08-06T16:31:43Z")
ok guys, I do appreciate that add info. But I don’t want to lose sight of finding the error in my proof. Informal as it may be. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [August 6, 2012, 4:32pm UTC](https://boards.straightdope.com/t/999-1/27517/343 "2012-08-06T16:32:32Z")
Oops, I switched from subtractin 1.0-0.9 to subtracting 1.000…-0.999… in the middle of the above post, but hopefully my point was clear anyway. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 6, 2012, 4:33pm UTC](https://boards.straightdope.com/t/999-1/27517/344 "2012-08-06T16:33:14Z")
So we are talking abou the same infinity 0.999… or 1.000… or the halves of Zeno’s Paradox, no arguments? ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [August 6, 2012, 4:33pm UTC](https://boards.straightdope.com/t/999-1/27517/345 "2012-08-06T16:33:37Z")
> [@erik150x](#): > > ok guys, I do appreciate that add info. But I don’t want to lose sight of finding the error in my proof. Informal as it may be. Well, that’s an error. You’re using an algorithm to produce a number in your proof, when that algorithm can not actually be used. ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [August 6, 2012, 4:34pm UTC](https://boards.straightdope.com/t/999-1/27517/346 "2012-08-06T16:34:22Z")
> [@Frylock](#): > > No, it’s the same infinity. Both are aleph null–simple, countable infinity. > > That’s in terms of cardinality. I now forget how to compare infinities for ordinality. Aleph-null plus one is the same cardinality as aleph null (namely, aleph null) but has a higher ordinality (it is one greater). I forget how to determine the ordinality of an infinite number in general. So I can not say for certain whether the two sequences have the same ordinality. > > But they have the same cardinality, and that is almost certainly what’s relevant to this discussion. It should be noted, though, that infinity MINUS infinity doesn’t work because of the ability to add any non-infinite (and some infinite) value to infinity and get infinity. It’s a indeterminate form. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 6, 2012, 4:35pm UTC](https://boards.straightdope.com/t/999-1/27517/347 "2012-08-06T16:35:28Z")
> [@Frylock](#): > > Well, that’s an error. You’re using an algorithm to produce a number in your proof, when that algorithm can not actually be used. Can you explain that further? ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 6, 2012, 4:37pm UTC](https://boards.straightdope.com/t/999-1/27517/348 "2012-08-06T16:37:07Z")
> [@Jragon](#): > > It should be noted, though, that infinity MINUS infinity doesn’t work because of the ability to add any non-infinite (and some infinite) value to infinity and get infinity. It’s a indeterminate form. We’re not talking about adding or subtracting any infinities here, just 1 and .999… ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [August 6, 2012, 4:38pm UTC](https://boards.straightdope.com/t/999-1/27517/349 "2012-08-06T16:38:59Z")
> [@erik150x](#): > > Can you explain that further? I’m not sure how to. I believe I explained it as well as I can in post 328, quoted below: > [@](#): > > Under the rules of “normal math,” the numerical representation 0.999… does not have a final digit, and neither does the numerical representation 1.000… This is just a matter of definition–it is how the ellipses are defined in this context. > > But under the rules of “normal math,” the subtraction algorithm you were using requires that you start with the final digit of the numerical representations you are working on. > > It requires a final digit–but by definition (see above) there is no final digit. Hence, the algorithm can not be applied. The above makes mention of “normal math.” If it’s your intention not to use “normal math,” then the flaw consists in your not having laid out what mathematical system you _are_ using. But if you mean for the proof to work for the kind of math the educated non-specialist understands, then the abovequoted lays out a specific flaw in your proof. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 6, 2012, 4:42pm UTC](https://boards.straightdope.com/t/999-1/27517/350 "2012-08-06T16:42:52Z")
> [@Frylock](#): > > I’m not sure how to. I believe I explained it as well as I can in post 328, quoted below: > > The above makes mention of “normal math.” If it’s your intention not to use “normal math,” then the flaw consists in your not having laid out what mathematical system you _are_ using. > > But if you mean for the proof to work for the kind of math the educated non-specialist understands, then the abovequoted lays out a specific flaw in your proof. Well, I don’t know about the normal rules of math and no final digit. To me this is an artifact of the idea of infinity being endless… a reasonable assumption about an infinite sequence of numbers to make, but perhaps not entirely true. For if we can move from point A to point B and there are an infinite number of points in that movement, why can I not move to the end of an infinite string of 9s? or 0s? ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [August 6, 2012, 4:43pm UTC](https://boards.straightdope.com/t/999-1/27517/351 "2012-08-06T16:43:50Z")
> [@erik150x](#): > > Hmmm, I don’t know. I would potenitally argue there is a last digit at the infinite’th place. By the agreed upon definitions that everybody else is using there is not and cannot be any such thing as the infinite’th place. Infinite is defined as endless. It does not matter what size infinity you are talking about. They are all endless. They cannot be put into 1 to 1 correspondence but that is a property that is different than endlessness. There’s the flaw in your argument. No matter how often you repeat variations of it, they all come down to the same error. You are looking at our definition, throwing it out, and then saying that our answers are wrong. As many people have patiently explained, it is possible to construct arithmetics that start with a different definition of infinity. You are not doing so. You are simply stating that you don’t like the standard definition and attempting to apply an alternate one without any rigor. When we say that .999~ represents an unending series, you can never subtract .999~ from .999~ and find a place where the difference is .000~1. (By the definition we are using .999~ is an infinite sequence. I have no idea what you mean when you say it isn’t. The tilda at the end is equivalent to using … at the end. Both are notations that mean infinite, again by definition.) You can’t just want the one to be there. It’s not there. It can’t be there, because by definition the infinite’th place doesn’t exist. Whenever you try to stick an infinite’th place in, the rest of us using the standard definition will point out your error. The reason we like the standard definition is that it solves the problems that once were intractable, like Zero’s Paradox. (Also, please note that the paradox can be represented in a literally infinite number of ways. 1/2 + 1/4 + 1/8 + … = 1 is just another way of expressing 0.9 + 0.09 + 0.009 + … =1. You can find infinite expansions that grow much more slowly yet equal 1. And expansions can converse on any number. Pi must be calculated by an infinite sum. There are an infinite number of these expansions, I believe, yet they all exactly equal pi. Not approximately equal pi; not come close to pi in the terms we can calculate; exactly equal pi. All this is incredibly useful and all this is derivable only because of the definition of endless.) If you can’t accept that math is entirely axioms, definitions, and proofs derived from them in a rigorous and non-contradictory way, then you can’t accept math at all. You can change the axioms and definitions and get whole and consistent sets of proofs. Everybody agrees that’s true. But once you choose your axioms and definitions, the rest follows and you don’t get to pick which you like and which you don’t. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 6, 2012, 4:45pm UTC](https://boards.straightdope.com/t/999-1/27517/352 "2012-08-06T16:45:42Z")
By the way in Computer Science we talk about computing infinite strings all the time, despite the fact that a computer never could. I would refer you to Turing Machines. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 6, 2012, 4:47pm UTC](https://boards.straightdope.com/t/999-1/27517/353 "2012-08-06T16:47:50Z")
> [@Exapno\_Mapcase](#): > > By the agreed upon definitions that everybody else is using there is not and cannot be any such thing as the infinite’th place. Infinite is defined as endless. It does not matter what size infinity you are talking about. They are all endless. They cannot be put into 1 to 1 correspondence but that is a property that is different than endlessness. > > There’s the flaw in your argument. No matter how often you repeat variations of it, they all come down to the same error. You are looking at our definition, throwing it out, and then saying that our answers are wrong. > > As many people have patiently explained, it is possible to construct arithmetics that start with a different definition of infinity. You are not doing so. You are simply stating that you don’t like the standard definition and attempting to apply an alternate one without any rigor. > > When we say that .999~ represents an unending series, you can never subtract .999~ from .999~ and find a place where the difference is .000~1. (By the definition we are using .999~ is an infinite sequence. I have no idea what you mean when you say it isn’t. The tilda at the end is equivalent to using … at the end. Both are notations that mean infinite, again by definition.) You can’t just want the one to be there. It’s not there. It can’t be there, because by definition the infinite’th place doesn’t exist. Whenever you try to stick an infinite’th place in, the rest of us using the standard definition will point out your error. > > The reason we like the standard definition is that it solves the problems that once were intractable, like Zero’s Paradox. (Also, please note that the paradox can be represented in a literally infinite number of ways. 1/2 + 1/4 + 1/8 + … = 1 is just another way of expressing 0.9 + 0.09 + 0.009 + … =1. You can find infinite expansions that grow much more slowly yet equal 1. And expansions can converse on any number. Pi must be calculated by an infinite sum. There are an infinite number of these expansions, I believe, yet they all exactly equal pi. Not approximately equal pi; not come close to pi in the terms we can calculate; exactly equal pi. All this is incredibly useful and all this is derivable only because of the definition of endless.) > > If you can’t accept that math is entirely axioms, definitions, and proofs derived from them in a rigorous and non-contradictory way, then you can’t accept math at all. You can change the axioms and definitions and get whole and consistent sets of proofs. Everybody agrees that’s true. But once you choose your axioms and definitions, the rest follows and you don’t get to pick which you like and which you don’t. How is the infinity in Zeno’s Paradox different than the infinity of a repeating decimal expansion? ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 6, 2012, 4:50pm UTC](https://boards.straightdope.com/t/999-1/27517/354 "2012-08-06T16:50:00Z")
People seem to want to say we cannot ever reach the end of an infinite sequence, but yet do so to solve Zeno’s Paradox. So I would say that is a contradiction there. You must throw out one or the other, from my perspective? which is incorrect or how are they different? ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [August 6, 2012, 4:52pm UTC](https://boards.straightdope.com/t/999-1/27517/355 "2012-08-06T16:52:16Z")
> [@septimus](#): > > I mentioned in three (3) separate posts that 0.999…=1 was a direct consequence of the Axiom of Archimedes – a simple common-sense proposition (which Archimedes attributes to his predecessor Eudoxus) which requires no mention whatsoever of words like “definition”, “limit”, “infinity”, or “infinitesimal.” That it uses the word “Axiom” rather than “Definition” is cold comfort… Furthermore, I see from your first post that the version of the Axiom of Archimedes you use is “For any positive ɛ there is a finite integer N such that Nɛ \> 1”. This is literally the statement “Every positive ɛ is non-infinitesimal”, or, in other words, “There are no (nonzero) infinitesimals”. It doesn’t use the word “infinitesimal”, but only because it hasn’t been translated in that fashion; the _definition_ of “infinitesimal” is “ɛ is infinitesimal if there is not a finite integer N such that Nɛ \> 1”. So the Axiom of Archimedes is not at all intuitive (or true) if you have a numeric system in mind that makes infinitesimal distinctions. It doesn’t bring anything to the table other than “Let’s agree not to draw infinitesimal distinctions”. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 6, 2012, 4:52pm UTC](https://boards.straightdope.com/t/999-1/27517/356 "2012-08-06T16:52:57Z")
> [@erik150x](#): > > By the way in Computer Science we talk about computing infinite strings all the time, despite the fact that a computer never could. I would refer you to Turing Machines. well, I shouldn’t say all the time… but we do. 😉 ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [August 6, 2012, 4:53pm UTC](https://boards.straightdope.com/t/999-1/27517/357 "2012-08-06T16:53:40Z")
> [@erik150x](#): > > We’re not talking about adding or subtracting any infinities here, just 1 and .999… I’m aware, I was pointing out how infinity (aleph null) is different from regular numbers. Since it’s not a single value, “infinitieth” place has no meaning. Which place is that? Whichever place you choose, **including** the ill-defined “infinitieth” will have another place after it. > [@erik150x](#): > > By the way in Computer Science we talk about computing infinite strings all the time, despite the fact that a computer never could. I would refer you to Turing Machines. Well, yeah, you can trivially make a TM or FSM (or whatever else) that can accept an infinite string, but if you **give** it a string that it can accept, I can guarantee you it won’t halt (for instance, an FSM with two states, a starting state acceptor with a self-loop on a, and a transition to a rejecting state on any other input will never halt given an infinite string of a’s). I’m not sure what your point is. In comp sci we talk about infinite strings, in math we do too. I’m not sure what you’re trying to say. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 6, 2012, 5:01pm UTC](https://boards.straightdope.com/t/999-1/27517/358 "2012-08-06T17:01:25Z")
> [@Jragon](#): > > I’m aware, I was pointing out how infinity (aleph null) is different from regular numbers. Since it’s not a single value, “infinitieth” place has no meaning. Which place is that? Whichever place you choose, **including** the ill-defined “infinitieth” will have another place after it. > > Well, yeah, you can trivially make a TM or FSM (or whatever else) that can accept an infinite string, but if you **give** it a string that it can accept, I can guarantee you it won’t halt (for instance, an FSM with two states, a starting state acceptor with a self-loop on a, and a transition to a rejecting state on any other input will never halt given an infinite string of a’s). I’m not sure what your point is. In comp sci we talk about infinite strings, in math we do too. I’m not sure what you’re trying to say. My comment about computer science was in hindsight irrevlivent. okay fair enough you point about another place after it as well. So how do reconcile the anology i am making to Zeno’S Parodox then? We progress through an infinite number of halves: 1/2 + 1/4 + 1/8 + … + 1/infinity to get from point A to point B I am relating each of these points to numeral in the number .999… or 1.000… how is this a different infinity? ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [August 6, 2012, 5:07pm UTC](https://boards.straightdope.com/t/999-1/27517/359 "2012-08-06T17:07:16Z")
You’re making the mistake of writing 1/infinity in the first place. In the Real number system, 1/infinity is undefined. 1/2 + 1/4 + 1/8 … There is no 1/infinity, only a series of smaller and smaller points, forever and ever, in perpetuity. Whichever numbered term you pick, there’s another one after it. That’s literally the definition of infinite. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 6, 2012, 5:09pm UTC](https://boards.straightdope.com/t/999-1/27517/360 "2012-08-06T17:09:55Z")
> [@Jragon](#): > > You’re making the mistake of writing 1/infinity in the first place. In the Real number system, 1/infinity is undefined. > > 1/2 + 1/4 + 1/8 … > > There is no 1/infinity, only a series of smaller and smaller points, forever and ever, in perpetuity. Whichever numbered term you pick, there’s another one after it. That’s literally the definition of infinite. Okay… is it fair to say there is no final point in this 1 mile walk anology of Zeno’s Paradox? We never reach a final point here? How do we get to B? [Previous page](https://boards.straightdope.com/t/999-1/27517.md?page=17) [Next page](https://boards.straightdope.com/t/999-1/27517.md?page=19)