# 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:** 26
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [November 10, 2016, 2:22pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/501 "2016-11-10T14:22:08Z")
> [@netzweltler](#): > > I did copy and paste it several times. We need to identify a **point** 0.999… on the number line to be able to define the distance |1 - 0.999…|. Ah … calculus uses limits, not number lines … please try again, and please use math instead of hokey irrelevant grammar school descriptions. ---
**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 10, 2016, 9:47pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/502 "2016-11-10T21:47:11Z")
> [@septimus](#): > > I haven’t read the thread. 😛 > > Did Mr. **Netz** ever indicate whether his argument against 0.99999 applies against 0.33333 ? > > If yes, does 1/3 have a decimal representation, or is it just another Not\_a\_number? > > If no, why is the pen moving from 0.9999 to 0.99999 not mimicked by the pen moving from 0.3333 to 0.33333? > > Did anyone ever explain to Mr. **Netz** that 1 = 0.99999 ? There’s something back there about all infinite strings being undefined as numbers, except of course things like pi, cause you can just stop at 3.14159 and that’s close enough for most purposes … ---
**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:** [November 11, 2016, 12:41am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/503 "2016-11-11T00:41:21Z")
> [@septimus](#): > > I haven’t read the thread. 😛 > > Did Mr. **Netz** ever indicate whether his argument against 0.99999 applies against 0.33333 ? > > If yes, does 1/3 have a decimal representation, or is it just another Not\_a\_number? > > If no, why is the pen moving from 0.9999 to 0.99999 not mimicked by the pen moving from 0.3333 to 0.33333? > > Did anyone ever explain to Mr. **Netz** that 1 = 0.99999 ? …and they have said that they don’t know what the decimal expansion of 1/3 is (I took that to mean they felt it wasn’t 0.333… – notwithstanding that when one moves one’s pen from 0 to 0.9 one passes right through it – because, for pen reasons, it isn’t a number). ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 11, 2016, 7:59am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/504 "2016-11-11T07:59:55Z")
> [@septimus](#): > > Did anyone ever explain to Mr. **Netz** that 1 = 0.99999 ? I have seen lots of claims that 1 = 0.999… but you can be the first one who solves this: If the claim is that we can reach a point 0.999… on the number line if we move 0.9 + 0.09 + 0.009 + … digit by digit, then the task is to define a procedure (a list) that shows how this point can be reached. ---
**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 11, 2016, 9:02am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/505 "2016-11-11T09:02:02Z")
> [@netzweltler](#): > > If the claim is that we can reach a point 0.999… on the number line if we move 0.9 + 0.09 + 0.009 + … digit by digit. It isn’t. Infinities can’t be dealt with this way. Are you ready to acknowledge we are indeed going in circles yet? ---
**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 11, 2016, 10:53am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/506 "2016-11-11T10:53:13Z")
> [@netzweltler](#): > > If the claim is that we can reach a point 0.999… on the number line if we move 0.9 + 0.09 + 0.009 + … digit by digit, then the task is to define a procedure (a list) that shows how this point can be reached. We know you’re confused but we’re not sure how. Are you claiming that 0.9999 is not a number? How about 0.3333 ? Does it avoid the problem you perceive with 0.9999 or does it suffer from the same problem? How about π = 3.141592653… which also has an infinite sequence in its decimal expansion? Is it OK, or does it also fail? Don’t just keep repeating your rubric; answer these questions so we can move forward. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 11, 2016, 11:46am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/507 "2016-11-11T11:46:37Z")
> [@septimus](#): > > We know you’re confused but we’re not sure how. > > Are you claiming that 0.9999 is not a number? > How about 0.3333 ? Does it avoid the problem you perceive with 0.9999 or does it suffer from the same problem? > > How about π = 3.141592653… which also has an infinite sequence in its decimal expansion? Is it OK, or does it also fail? > > Don’t just keep repeating your rubric; answer these questions so we can move forward. (0.3, 0.33, 0.333, …) does suffer from the same problem as (0.9, 0.99, 0.999, …). The limit that cannot be reached on the number line is 1/3 then instead of 1. The infinite sequence of 3.141592653… does suffer from the same problem. The limit that cannot be reached on the number line is π then. ---
**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 11, 2016, 12:04pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/508 "2016-11-11T12:04:13Z")
So … you can’t “move your pencil” to 1/3. You can’t “move your pencil” to π. Perhaps the problem isn’t the numbers, but rather with your pencil and how you are trying to move it. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 11, 2016, 12:16pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/509 "2016-11-11T12:16:11Z")
> [@septimus](#): > > So … you can’t “move your pencil” to 1/3. You can’t “move your pencil” to π. > > Perhaps the problem isn’t the numbers, but rather with your pencil and how you are trying to move it. Of course it is a problem how I am trying to move it. I can reach 1/3 if I am moving it in a single step. Please define the procedure how to move the pen to 1/3 in infinitely many steps! ---
**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 11, 2016, 12:36pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/510 "2016-11-11T12:36:39Z")
> [@netzweltler](#): > > Of course it is a problem how I am trying to move it. I can reach 1/3 if I am moving it in a single step. And you can reach 0.9999 in a single step. > [@netzweltler](#): > > Please define the procedure how to move the pen to 1/3 in infinitely many steps! 😕 (Unless you can produce some cogent response, this will be my last post.) ---
**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 11, 2016, 1:19pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/511 "2016-11-11T13:19:36Z")
> [@netzweltler](#): > > Please define the procedure how to move the pen to 1/3 in infinitely many steps! Please define a situation where this would be an interesting question. ---
**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:** [November 11, 2016, 1:59pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/512 "2016-11-11T13:59:35Z")
At best, our hero is simply repeating (ad infinitum) Zeno’s so-called paradox – despite cogent arguments that this approach is flawed, and has been known to be flawed for some centuries (I’d suppose that Zeno himself knew his argument was flawed). These discussions aren’t always the most amusing, but this one has become one of the most boring. ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [November 11, 2016, 2:00pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/513 "2016-11-11T14:00:05Z")
> [@netzweltler](#): > > Please define the procedure how to move the pen to 1/3 in infinitely many steps! You just did, with an infinite number of steps … if you’re not getting there, then your pen is what’s fucked up … This is like a riddle to you, you state something impossible and demand we make be possible … ridiculous … your logic is unsound, your premise is juvenile and if you think you’ll chase everyone away … [wolfish grin] … you’d be wrong … ---
**Author:** ![Ignotus](https://avatars.discourse-cdn.com/v4/letter/i/ecc23a/32.png) [@Ignotus](https://boards.straightdope.com/u/Ignotus)\ **Post date:** [November 11, 2016, 2:00pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/514 "2016-11-11T14:00:55Z")
> [@netzweltler](#): > > Of course it is a problem how I am trying to move it. **I can reach 1/3 if I am moving it in a single step.** Please define the procedure how to move the pen to 1/3 in infinitely many steps! You’d have to pass through 1/6 though. And before that, you’d have to pass through 1/12. And before _that_… ---
**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 11, 2016, 3:22pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/515 "2016-11-11T15:22:21Z")
How do you get from 0.3 to 0.33 anyway? You first have to move the pen from 0.3 to 0.315, then to 0.3225, then to 0.32525, then to … You may think that 0.33 is already a defined point on the number line, but so is 0.333… Your inability to treat infinities as anything other than a big finite doesn’t change this. ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [November 11, 2016, 3:44pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/516 "2016-11-11T15:44:00Z")
Well … we never stop appending 9’s to the decimal … but if we did stop, see see, then it’s not 1 … like duh … ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 12, 2016, 8:55am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/517 "2016-11-12T08:55:56Z")
> [@Telemark](#): > > > [@netzweltler](#): > > > > Please define the procedure how to move the pen to 1/3 in infinitely many steps! > > Please define a situation where this would be an interesting question. If the claim is that we can reach a point 0.333… on the number line if we move 0.3 + 0.03 + 0.003 + … digit by digit. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 12, 2016, 9:00am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/518 "2016-11-12T09:00:05Z")
> [@Ignotus](#): > > > [@netzweltler](#): > > > > Of course it is a problem how I am trying to move it.\*\* I can reach 1/3 if I am moving it in a single step.\*\* Please define the procedure how to move the pen to 1/3 in infinitely many steps! > > You’d have to pass through 1/6 though. And before that, you’d have to pass through 1/12. And before _that_… If this is a procedure of infinitely many steps, at what step number do you pass 1/6, or 1/12? ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 12, 2016, 9:06am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/519 "2016-11-12T09:06:58Z")
> [@naita](#): > > How do you get from 0.3 to 0.33 anyway? You first have to move the pen from 0.3 to 0.315, then to 0.3225, then to 0.32525, then to … > > You may think that 0.33 is already a defined point on the number line, but so is 0.333… You cannot get from a point of the number line to any other point of the number line in infinitely many steps. You can easily get there in a finite number of steps. So, to go from 0.3 to 0.33 in a single step is no problem. Show the procedure how to get there in an infinite number of steps! > [@naita](#): > > Your inability to treat infinities as anything other than a big finite doesn’t change this. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 12, 2016, 9:08am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/520 "2016-11-12T09:08:26Z")
> [@watchwolf49](#): > > Well … we never stop appending 9’s to the decimal … but if we did stop, see see, then it’s not 1 … like duh … And if we don’t stop we see we are nowhere on the number line. [Previous page](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711.md?page=25) [Next page](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711.md?page=27)