# An infinite question: Why doesn't .999~ = 1? **URL:** \ **Category:** Cecil's Columns/Staff Reports\ **Created:** [September 24, 2016, 7:29am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711 "2016-09-24T07:29:45Z")\ **Posts on this page:** 20\ **Page:** 31
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 18, 2016, 7:41am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/601 "2016-11-18T07:41:53Z")
> [@naita](#): > > > [@netzweltler](#): > > > > So, 0.3 + 0.03 + 0.003 + … ≠ 0.333… 😕 > > Do you mean both is true or both is wrong? > > Neither is mathematically rigorous. Well, at least Wikipedia thinks it’s rigorous. There you can read: 0.999… = 9(⅒) + 9(⅒)² + 9(⅒)³ + … [0.999... - Wikipedia](https://en.wikipedia.org/wiki/0.999)… > [@naita](#): > > “Get to” requires a different approach for infinities than finites. In your process you claim we can’t get to 0.999…, and occasionally you realise that then we can’t get to t=1 either. But then you turn around and talk about the state at t=1 like now. > > It _is_ a restating of zeno’s paradoxes. Your “process” can’t reach t=1, except by accepting one of the solutions to Zeno’s paradoxes and acknowledging that 0.999… = 1. A different approach is required. If any of us had an sense we’d just repeat that last sentence from now on. ---
**Author:** ![naita](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/naita/32/5862_2.png) [@naita](https://boards.straightdope.com/u/naita)\ **Post date:** [November 18, 2016, 9:16am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/602 "2016-11-18T09:16:06Z")
> [@netzweltler](#): > > Well, at least Wikipedia thinks it’s rigorous. There you can read: 0.999… = 9(⅒) + 9(⅒)² + 9(⅒)³ + … > [0.999... - Wikipedia](https://en.wikipedia.org/wiki/0.999)… Wikipedia doesn’t include the whole explanation and refers to a convergence theorem for convergent series. You can read about that [here](https://en.wikipedia.org/wiki/Convergent_series), but to summarize it equates convergent series with their limits. Since that’s the step you’ve consistently refused to accept it would be wrong to _ **simply** _ state that 0.999… = 9(⅒) + 9(⅒)² + 9(⅒)³ + … ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 18, 2016, 9:45am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/603 "2016-11-18T09:45:25Z")
> [@naita](#): > > Wikipedia doesn’t include the whole explanation and refers to a convergence theorem for convergent series. You can read about that [here](https://en.wikipedia.org/wiki/Convergent_series), but to summarize it equates convergent series with their limits. Since that’s the step you’ve consistently refused to accept it would be wrong to _ **simply** _ state that 0.999… = 9(⅒) + 9(⅒)² + 9(⅒)³ + … 0.999… is either equal to 9(⅒) + 9(⅒)² + 9(⅒)³ + … or it is not. Everything in this equation is clearly defined. “…” means “(countably) infinitely many”. We are still doing math here, no? If it is _ **simply** _ not equal, what is the difference between 0.999… and 9(⅒) + 9(⅒)² + 9(⅒)³ + … then? ---
**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:** [November 18, 2016, 10:43am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/604 "2016-11-18T10:43:36Z")
> [@netzweltler](#): > > 0.999… is either equal to 9(⅒) + 9(⅒)² + 9(⅒)³ + … or it is not. It is, but only if convergent series equal their limits. Which you have consistently rejected. ---
**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:** [November 18, 2016, 11:01am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/605 "2016-11-18T11:01:01Z")
Interestingly enough, from that wikipedia article… > [@](#): > > Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the limit concept and disagreements over the nature of infinitesimals. There are many common contributing factors to the confusion: > > Students are often “mentally committed to the notion that a number can be represented in one and only one way by a decimal.” Seeing two manifestly different decimals representing the same number appears to be a paradox, which is amplified by the appearance of the seemingly well-understood number 1.[35] > Some students interpret “0.999…” (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 “at infinity”.[36] > Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read “0.999…” as meaning the sequence rather than its limit.[37] > > These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive counterexamples to better understand 0.999… Sound familiar? ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 18, 2016, 11:06am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/606 "2016-11-18T11:06:24Z")
> [@Budget\_Player\_Cadet](#): > > It is, but only if convergent series equal their limits. Which you have consistently rejected. To me 9(⅒) + 9(⅒)² + 9(⅒)³ + … is simply another notation for 0.999… All I rejected is that we can find a point 0.999… on the number line. ---
**Author:** ![Bullitt](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bullitt/32/5725_2.png) [@Bullitt](https://boards.straightdope.com/u/Bullitt)\ **Post date:** [November 18, 2016, 11:12am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/607 "2016-11-18T11:12:50Z")
Guys, I have only been casually following this thread. I looked up netzweltler, who joined SDMB on 09 October. Since joining s/he has posted 154 times to date, and all 154 posts to date have been in this thread. S/he has no other posts in any other SDMB thread. First post, post #98, on 09 October, is here: [http://boards.straightdope.com/sdmb/showthread.php?p=19686578#post19686578](http://boards.straightdope.com/sdmb/showthread.php?p=19686578#post19686578) Don’t know if this tells us anything, but it’s a little interesting. Thought I’d share. **Bullitt** ---
**Author:** ![naita](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/naita/32/5862_2.png) [@naita](https://boards.straightdope.com/u/naita)\ **Post date:** [November 18, 2016, 12:18pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/608 "2016-11-18T12:18:20Z")
> [@netzweltler](#): > > To me 9(⅒) + 9(⅒)² + 9(⅒)³ + … is simply another notation for 0.999… > All I rejected is that we can find a point 0.999… on the number line. That is equivalent to rejecting that convergent series equal their limits. ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [November 18, 2016, 1:22pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/609 "2016-11-18T13:22:23Z")
> [@netzweltler](#): > > Well, at least Wikipedia thinks it’s rigorous. There you can read: 0.999… = 9(⅒) + 9(⅒)² + 9(⅒)³ + … > [0.999... - Wikipedia](https://en.wikipedia.org/wiki/0.999)… We have to be very very careful here quoting Wikipedia … we have a circular reference thing going on … if Wikipedia is citing SDMB for their information … then we can’t in turn use that information as a citation on SDMB … I can’t say “the sky was yellow and the sun was blue” … and use [this citation](http://boards.straightdope.com/sdmb/showpost.php?p=19790913&postcount=609) from my post #609: > [@](#): > > … the sky was yellow and the sun was blue … Anyway, I’d like **netzwletler’s** opinion of the information in the subsection “Infinite series and sequences”, fourth equation … ---
**Author:** ![Trinopus](https://avatars.discourse-cdn.com/v4/letter/t/2bfe46/32.png) [@Trinopus](https://boards.straightdope.com/u/Trinopus)\ **Post date:** [November 19, 2016, 1:53am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/610 "2016-11-19T01:53:01Z")
Just as a friendly digression, one of my favorite demonstrations of how infinity is counter-intuitive is to take the curve 1/x and rotate it about the y axis. You get a long funnel shape. (Infinitely long…) This shape has an infinite surface…and a finite volume. You can fill it with paint…but you cannot paint it. ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [November 19, 2016, 2:46am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/611 "2016-11-19T02:46:35Z")
> [@Trinopus](#): > > Just as a friendly digression, one of my favorite demonstrations of how infinity is counter-intuitive is to take the curve 1/x and rotate it about the y axis. You get a long funnel shape. (Infinitely long…) > > This shape has an infinite surface…and a finite volume. > > You can fill it with paint…but you cannot paint it. Historical note: This is [Gabriel’s Horn](https://en.wikipedia.org/wiki/Gabriel's_Horn) (sometimes called Torricelli’s trumpet since Evangelista Torricelli discovered it circa 1644.) > [@](#): > > The name refers to the tradition identifying the Archangel Gabriel as the angel who blows the horn to announce Judgment Day, associating the divine, or infinite, with the finite. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 19, 2016, 3:58am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/612 "2016-11-19T03:58:44Z")
> [@naita](#): > > > [@netzweltler](#): > > > > To me 9(⅒) + 9(⅒)² + 9(⅒)³ + … is simply another notation for 0.999… > > All I rejected is that we can find a point 0.999… on the number line. > > That is equivalent to rejecting that convergent series equal their limits. That’s true. The limit must be a point on the number line. This point is 1. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 19, 2016, 4:11am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/613 "2016-11-19T04:11:21Z")
> [@watchwolf49](#): > > Anyway, I’d like **netzwletler’s** opinion of the information in the subsection “Infinite series and sequences”, fourth equation … There’s nothing new. _“The last step, that  (⅒)ⁿ → 0 as n → ∞, is often justified by the Archimedean property of the real numbers.”_ What _last step_? We are dealing with infinity, no? ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 19, 2016, 4:45am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/614 "2016-11-19T04:45:52Z")
> [@netzweltler](#): > > There’s nothing new. _“The last step, that  (⅒)ⁿ → 0 as n → ∞, is often justified by the Archimedean property of the real numbers.”_ What _last step_? We are dealing with infinity, no? To make some sense we should be able to write \* (⅒)ⁿ = 0 as n = ∞\*. This just doesn’t happen, since n ∈ ℕ. There is no last number n = ∞. ---
**Author:** ![naita](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/naita/32/5862_2.png) [@naita](https://boards.straightdope.com/u/naita)\ **Post date:** [November 19, 2016, 9:45am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/615 "2016-11-19T09:45:10Z")
> [@netzweltler](#): > > That’s true. The limit must be a point on the number line. This point is 1. Yes, and according to math that point is also the value of the infinite series. Every single one of your posts equate to “But it isn’t.” ---
**Author:** ![naita](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/naita/32/5862_2.png) [@naita](https://boards.straightdope.com/u/naita)\ **Post date:** [November 19, 2016, 10:08am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/616 "2016-11-19T10:08:18Z")
> [@netzweltler](#): > > To make some sense we should be able to write \* (⅒)ⁿ = 0 as n = ∞\*. This just doesn’t happen, since n ∈ ℕ. There is no last number n = ∞. I’m going to try this approach once more, cause I’m a glutton for punishment. Try to look beyond your step-by-step-approach. You are describing a process that never ends. It’s obvious that **on it’s own** this approach doesn’t tell us all there is to know about infinities. For instance if we move from 0 to 1 on the number line by any continuous process that actually gets us to 1, we have to pass by any and all numbers between 0 and 1, including those with infinite digits, such as 0.333… Now please take the time to either accept this or refute it with actual mathematics and not the return to your monomania. Either we pass by 0.333… and pi to infinity digits, or we pass by some number with a finite number of 3s, or a finite expansion of pi, and the latter is quite obviously absurd. Now looking at your process we can’t **just** say “well at t=1 we’re at 1”, because by the rules of your process there’s no reaching t=1. But due to what I’ve described above, we also can’t just say “you can’t ‘reach’” infinity or “0.999… is not defined”. We can reach infinity if we look at the continuous process (just as Achilles can in fact overtake the tortoise and the arrow can reach the target) and 0.999… has to behave the same way whether it’s an instance in time in a continuous process or the, somewhat miraculous, “endpoint” of an infinite process. So try to grasp this. By the rule set you are operating on, there’s no reaching infinity. But the rule set is obviously limited, as it can’t reach 1, and we know 1 exists. So we **have** to look beyond your limited set of rules. How do **you** suggest we expand the rules? Or alternatively, refute the convergence theorem using something **other** than your tiresome repetition of zeno’s paradoxes. ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [November 19, 2016, 3:22pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/617 "2016-11-19T15:22:53Z")
> [@watchwolf49](#): > > [snip] … Anyway, I’d like **netzwletler’s** opinion of the information in the subsection “Infinite series and sequences”, fourth equation … > [@netzweltler](#): > > There’s nothing new. _“The last step, that  (⅒)ⁿ → 0 as n → ∞, is often justified by the Archimedean property of the real numbers.”_ What _last step_? We are dealing with infinity, no? … [glue] … To make some sense we should be able to write \* (⅒)ⁿ = 0 as n = ∞\*. This just doesn’t happen, since n ∈ ℕ. There is no last number n = ∞. I asked for **YOU** to comment on the equation, not repeat “heuristics” you don’t understand. This equation lays out the derivation of (0.999…) = 1 clearly, accurately and unambiguously … please state the exact step you think is in error. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 19, 2016, 4:51pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/618 "2016-11-19T16:51:12Z")
> [@naita](#): > > I’m going to try this approach once more, cause I’m a glutton for punishment. Try to look beyond your step-by-step-approach. > > You are describing a process that never ends. It’s obvious that **on it’s own** this approach doesn’t tell us all there is to know about infinities. For instance if we move from 0 to 1 on the number line by any continuous process that actually gets us to 1, we have to pass by any and all numbers between 0 and 1, including those with infinite digits, such as 0.333… > > Now please take the time to either accept this or refute it with actual mathematics and not the return to your monomania. Either we pass by 0.333… and pi to infinity digits, or we pass by some number with a finite number of 3s, or a finite expansion of pi, and the latter is quite obviously absurd. > > Now looking at your process we can’t **just** say “well at t=1 we’re at 1”, because by the rules of your process there’s no reaching t=1. But due to what I’ve described above, we also can’t just say “you can’t ‘reach’” infinity or “0.999… is not defined”. We can reach infinity if we look at the continuous process (just as Achilles can in fact overtake the tortoise and the arrow can reach the target) and 0.999… has to behave the same way whether it’s an instance in time in a continuous process or the, somewhat miraculous, “endpoint” of an infinite process. > > So try to grasp this. By the rule set you are operating on, there’s no reaching infinity. But the rule set is obviously limited, as it can’t reach 1, and we know 1 exists. So we **have** to look beyond your limited set of rules. > > How do **you** suggest we expand the rules? > > Or alternatively, refute the convergence theorem using something **other** than your tiresome repetition of zeno’s paradoxes. You don’t give up your notion that 9(⅒) + 9(⅒)² + 9(⅒)³ + … is a continuous process, and I don’t give up the notion that 9(⅒) + 9(⅒)² + 9(⅒)³ + … is a step-by-step-approach. I think we stuck in a loop. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 19, 2016, 4:52pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/619 "2016-11-19T16:52:45Z")
> [@watchwolf49](#): > > I asked for **YOU** to comment on the equation, not repeat “heuristics” you don’t understand. This equation lays out the derivation of (0.999…) = 1 clearly, accurately and unambiguously … please state the exact step you think is in error. I think I did. Otherwise be more specific, please. ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [November 19, 2016, 5:41pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/620 "2016-11-19T17:41:01Z")
> [@netzweltler](#): > > You don’t give up your notion that 9(⅒) + 9(⅒)² + 9(⅒)³ + … is a continuous process, and I don’t give up the notion that 9(⅒) + 9(⅒)² + 9(⅒)³ + … is a step-by-step-approach. I think we stuck in a loop. It’s addition … not a recipe … > [@netzweltler](#): > > I think I did. Otherwise be more specific, please. You copy/pasted what’s posted at Wikipedia … I’m asking about the equation and nothing more … see there, nowhere in the equation is it specified that n equal infinity, only that n approaches infinity … there’s a big difference. I understand you answered a question that wasn’t asked, so to be more specific would be to ask you to answer the questions that _are_ being asked … which step in the equation’s derivation do you think is in error? That question is open to anybody, I make no claim that this is a perfect proof … … [Previous page](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711.md?page=30) [Next page](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711.md?page=32)