# 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:** 30
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [November 16, 2016, 9:51pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/581 "2016-11-16T21:51:18Z")
> [@gnoitall](#): > > You’re missing the point. [Literally](https://en.wikipedia.org/wiki/Point_(geometry)).0 dimensions means no length, no breadth, no height. No size whatsoever. > > A non-zero-sized point would be like saying “kinda approximately 1.” If you mean 1 precisely, than the point defining 1 is a true zero-dimensional point. > > Likewise, if you mean 3 \* 1/3, you mean the precise zero-dimensional point which precisely defines 1. Or .9 (repeating infinitely). It’s the same exact point, with exactly the same dimensionality (zero). > > This is not subject to debate. It’s fundamental mathematics. It’s not required to fit into your intuition in order to be correct. Wrong [context](https://en.wikipedia.org/wiki/Pen) … ---
**Author:** ![gnoitall](https://avatars.discourse-cdn.com/v4/letter/g/bb73d2/32.png) [@gnoitall](https://boards.straightdope.com/u/gnoitall)\ **Post date:** [November 16, 2016, 9:55pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/582 "2016-11-16T21:55:04Z")
> [@watchwolf49](#): > > Wrong [context](https://en.wikipedia.org/wiki/Pen) … Unless involving a pen is a wild hijack of the thread, a pen is a terrible analogue of a point, because it can’t be zero-dimensional. So maybe I was yelling at the wrong person? Wouldn’t be the first time, or the last. :dubious: The point, misdirected or not, stands. A precise number is a zero-dimensional point. ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [November 16, 2016, 10:01pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/583 "2016-11-16T22:01:28Z")
> [@netzweltler](#): > > [snip] …t = 0.99: I move my pen from point 0.81 to point 0.9 of the number line > t = 0.999: I move my pen from point 0.9 to point 0.891 of the number line … [snip] This is just a snippet of something that has been posted in this thread well over a hundred times … we’re taking issue with this _as a process_ … since as you correctly “point” out the number (0.999…) is in fact a point, and not a process. Now, if you’re supporting **Netzweltler’s** position that (0.999…) ≠ 1 … then that’s a different conversation … ---
**Author:** ![John\_W.Kennedy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/john_w.kennedy/32/1031_2.png) [@John\_W.Kennedy](https://boards.straightdope.com/u/John_W.Kennedy)\ **Post date:** [November 17, 2016, 2:37am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/584 "2016-11-17T02:37:22Z")
Can we bring in the infinite hotel at this point? ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 17, 2016, 6:52am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/585 "2016-11-17T06:52:45Z")
> [@jepflast](#): > > > [@netzweltler](#): > > > > t = 0: I move my pen from point 0 to point 0.3 of the number line > > t = 0.9: I move my pen from point 0.3 to point 0.33 of the number line > > t = 0.99: I move my pen from point 0.33 to point 0.333 of the number line > > … > > > > By t = 1 the pen has moved all the distances { 0.3, 0.03, 0.003, … }. If there is a point 0.333… the pen should have reached it - simply by executing the actions on the list. > > Well, there’s two different things going on here with what you’re saying. If you want to bring time into this and simply note where the pen is at any time, then yes the pen points to 0.333~ at t=1. > > The other thing is that, no, you can’t get to 0.333~ by executing the actions on the list that we’ve already determined is an infinite list of points that fall short of the target, being partial sums. That assertion is simply false. Isn’t saying that I can’t get to 0.333… by moving the distances { 0.3, 0.03, 0.003, … } the same as saying that I can’t get to 0.333… by adding 0.3 + 0.03 + 0.003 + …? > [@jepflast](#): > > There, that should clear things up, for sure this time. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 17, 2016, 6:59am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/586 "2016-11-17T06:59:34Z")
> [@Budget\_Player\_Cadet](#): > > t = 0: I move my pen from 1 to 2 > t = 0.5: I move my pen from 2 to 3 > t = 0.75: I move my pen from 3 to 4 > t = 0.875: I move my pen from 4 to 5 > > By t = 1, the pen has move all the distances in ℕ. If there is a point “infinity”… the pen should have reached it - simply by executing the actions on the list. Clearly, that means that when we finish the supertask of counting all the natural numbers, we won’t be at infinity, but rather at some natural number on the list. If we finished the task by t = 1 we are neither at a point infinity nor at some natural number on the list. If you are at some natural number on the list you haven’t finished the task. So, you are at t \< 1. The state of the pen is not defined for t ≥ 1. > [@Budget\_Player\_Cadet](#): > > Except that’s not how infinity works, and that’s not how limits work. The series {1, 2, 3…} clearly has a limit of infinity. If we “finish” the infinite series, we will, for lack of better terminology\*, be at infinity… Even though infinity is not present at any step on that list. Because if you look only to the steps taken for an infinite process, you will miss the big picture. > \*Okay technically it’s a divergent limit and there is better terminology here but I’m trying to explain the absolute basics to you here so we’ll be a little bit wrong to make a point. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 17, 2016, 7:02am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/587 "2016-11-17T07:02:10Z")
> [@watchwolf49](#): > > Then it does not exist? If width = 0, then how does it point to anything? If the center of the pen has no width the pen doesn’t exist? ---
**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 17, 2016, 7:59am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/588 "2016-11-17T07:59:58Z")
> [@netzweltler](#): > > Isn’t saying that I can’t get to 0.333… by moving the distances { 0.3, 0.03, 0.003, … } the same as saying that I can’t get to 0.333… by adding 0.3 + 0.03 + 0.003 + …? Yes. So by that set of rules we can’t define 0.333… that way, but by way of logic (there is no last member of the set {0.3, 0.33, 0.333, 0.333…}) the number 0.333… must exist, and it’s quite obvious that it is equal to 1/3. To some people this shows that infinities must be treated as actual infinities, and that one has to find sensible axioms that give answers that fit with reality, others choose to harp on and on about pens moving fractions of a distance while refusing to give credit to Zeno. ---
**Author:** ![Telemark](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/telemark/32/372_2.png) [@Telemark](https://boards.straightdope.com/u/Telemark)\ **Post date:** [November 17, 2016, 1:22pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/589 "2016-11-17T13:22:33Z")
> [@netzweltler](#): > > If you are at some natural number on the list you haven’t finished the task. Perhaps you’ve failed to notice that you’re talking about infinities and you concept of finishing a task doesn’t apply? As was said many pages ago, you’re still treating an infinite as a very large finite and as a result everything you propose is meaningless. ---
**Author:** ![Riemann](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/riemann/32/3133_2.png) [@Riemann](https://boards.straightdope.com/u/Riemann)\ **Post date:** [November 17, 2016, 1:38pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/590 "2016-11-17T13:38:08Z")
We established 300 posts ago that **netzweltler** is simply reprising Zeno’s paradox, he is unwilling to accept that there have been any valid developments in mathematics since Ancient Greece, and for 300 posts he has obstinately ignored any input, just reiterating variations on Zeno’s paradox over and over again. Einstein’s (apocryphal) definition of insanity: doing the same thing over and over again and expecting a different result. Does anyone seriously doubt that if this thread reaches 3,000 posts, **netzweltler** will still just be repeating Zeno’s paradox over and over again? Is this performance art, or what are you people trying to achieve here? You are starting to worry me. ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [November 17, 2016, 2:43pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/591 "2016-11-17T14:43:13Z")
You’re starting to worry _just now ???_ … We’re a group of peoples doing the same thing over and over again and expecting a different result. That makes all the difference in the world. > [@](#): > > If the center of the pen has no width the pen doesn’t exist? No pen made of molecules … we’d expect the tip to be at least as wide as an electron … very very small but still non-zero … ---
**Author:** ![Trinopus](https://avatars.discourse-cdn.com/v4/letter/t/2bfe46/32.png) [@Trinopus](https://boards.straightdope.com/u/Trinopus)\ **Post date:** [November 18, 2016, 2:21am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/592 "2016-11-18T02:21:47Z")
> [@John\_W.Kennedy](#): > > Can we bring in the infinite hotel at this point? It’s been mentioned already a couple of times. “We’ll leave infinity on for ya.” ---
**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, 3:34am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/593 "2016-11-18T03:34:13Z")
> [@naita](#): > > > [@netzweltler](#): > > > > Isn’t saying that I can’t get to 0.333… by moving the distances { 0.3, 0.03, 0.003, … } the same as saying that I can’t get to 0.333… by adding 0.3 + 0.03 + 0.003 + …? > > Yes. So, 0.3 + 0.03 + 0.003 + … ≠ 0.333… 😕 Do you mean both is true or both is wrong? > [@naita](#): > > So by that set of rules we can’t define 0.333… that way, but by way of logic (there is no last member of the set {0.3, 0.33, 0.333, 0.333…}) the number 0.333… must exist, and it’s quite obvious that it is equal to 1/3. To some people this shows that infinities must be treated as actual infinities, and that one has to find sensible axioms that give answers that fit with reality, others choose to harp on and on about pens moving fractions of a distance while refusing to give credit to Zeno. ---
**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, 3:36am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/594 "2016-11-18T03:36:43Z")
> [@Telemark](#): > > Perhaps you’ve failed to notice that you’re talking about infinities and you concept of finishing a task doesn’t apply? As was said many pages ago, you’re still treating an infinite as a very large finite and as a result everything you propose is meaningless. The task is finished by t = 1. And of course it applies **because** we are talking about infinity. ---
**Author:** ![TriPolar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tripolar/32/3008_2.png) [@TriPolar](https://boards.straightdope.com/u/TriPolar)\ **Post date:** [November 18, 2016, 3:39am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/595 "2016-11-18T03:39:24Z")
I keep trying to read all the way through this thread but it keeps getting longer. ---
**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, 3:40am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/596 "2016-11-18T03:40:05Z")
> [@watchwolf49](#): > > No pen made of molecules … we’d expect the tip to be at least as wide as an electron … very very small but still non-zero … Saying that the center of the pen has some size is like saying that the center of gravity of the pen has a minimum size. ---
**Author:** ![Riemann](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/riemann/32/3133_2.png) [@Riemann](https://boards.straightdope.com/u/Riemann)\ **Post date:** [November 18, 2016, 4:19am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/597 "2016-11-18T04:19:24Z")
> [@TriPolar](#): > > I keep trying to read all the way through this thread but it keeps getting longer. Yes, but posts #101-#596 only added 3 more bytes of complexity. ---
**Author:** ![Telemark](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/telemark/32/372_2.png) [@Telemark](https://boards.straightdope.com/u/Telemark)\ **Post date:** [November 18, 2016, 4:29am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/598 "2016-11-18T04:29:20Z")
> [@netzweltler](#): > > The task is finished by t = 1. And of course it applies **because** we are talking about infinity. Talking about infinity doesn’t mean you understand infinity. Which you clearly don’t. ---
**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, 5:37am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/599 "2016-11-18T05:37:41Z")
> [@TriPolar](#): > > I keep trying to read all the way through this thread but it keeps getting longer. > [@Riemann](#): > > Yes, but posts #101-#596 only added 3 more bytes of complexity. It will be a long thread, but finitely long. (Shhh, don’t tell that to netzweiler. And, we haven’t covered the concepts of countably infinite, and uncountably infinite, either.) ---
**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, 6:50am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/600 "2016-11-18T06:50:32Z")
> [@netzweltler](#): > > So, 0.3 + 0.03 + 0.003 + … ≠ 0.333… 😕 > Do you mean both is true or both is wrong? Neither is mathematically rigorous. “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. [Previous page](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711.md?page=29) [Next page](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711.md?page=31)