# 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:** 40
**Author:** ![Trinopus](https://avatars.discourse-cdn.com/v4/letter/t/2bfe46/32.png) [@Trinopus](https://boards.straightdope.com/u/Trinopus)\ **Post date:** [December 6, 2016, 6:40pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/781 "2016-12-06T18:40:31Z")
> [@leahcim](#): > > . . . [0, 1) doesn’t _contain_ (one of) its endpoints. It still _has_ both endpoints, but one of them is not an element of the set. Per my old topology course, it contains every neighborhood around 1. One could, if desired, define a new topology that has new rules. But, as **leachim** noted, it would force us to give up certain properties of numbers and sets, which we’re pretty well dependent on. As he notes, these properties are useful, where a new, proposed, hypothetical mathematics with different rules has not displayed any possible use. **netzweltler** is perfectly free to make up his own mathematics, just as Dr. Seuss was free to make up his own alphabet (in On Beyond Zebra, a darn fine exploration of crypto-orthography.) What he cannot do is insist that his definitions are correct, as accepted by mathematicians in the real world. ---
**Author:** ![leahcim](https://avatars.discourse-cdn.com/v4/letter/l/b4bc9f/32.png) [@leahcim](https://boards.straightdope.com/u/leahcim)\ **Post date:** [December 6, 2016, 6:54pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/782 "2016-12-06T18:54:08Z")
> [@Trinopus](#): > > Per my old topology course, it contains every neighborhood around 1. Actually, every neighbourhood of 1 contains some points in [0, 1). There is no neighbourhood of 1 that is contained entirely in [0, 1). #pedant See [Limit Point](https://en.wikipedia.org/wiki/Limit_point) and [Closure](https://en.wikipedia.org/wiki/Closure_(topology)). ---
**Author:** ![Saltire](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saltire/32/553_2.png) [@Saltire](https://boards.straightdope.com/u/Saltire)\ **Post date:** [December 6, 2016, 7:27pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/783 "2016-12-06T19:27:27Z")
I’m not a mathematician, but on reading **leahcim** ’s link about the greatest lower bound property and surfing to the [Completeness of the real numbers](https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers) article, I have realized that this line that **netzweltler** keeps talking about can’t be the real number line at all. ---
**Author:** ![Trinopus](https://avatars.discourse-cdn.com/v4/letter/t/2bfe46/32.png) [@Trinopus](https://boards.straightdope.com/u/Trinopus)\ **Post date:** [December 6, 2016, 8:32pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/784 "2016-12-06T20:32:02Z")
> [@leahcim](#): > > Actually, every neighbourhood of 1 contains some points in [0, 1). There is no neighbourhood of 1 that is contained entirely in [0, 1). #pedant > > See [Limit Point](https://en.wikipedia.org/wiki/Limit_point) and [Closure](https://en.wikipedia.org/wiki/Closure_(topology)). Oopsie, very right, I misphrased it. ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [December 6, 2016, 9:09pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/785 "2016-12-06T21:09:20Z")
> [@Saltire](#): > > I’m not a mathematician, but on reading **leahcim** ’s link about the greatest lower bound property and surfing to the [Completeness of the real numbers](https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers) article, I have realized that this line that **netzweltler** keeps talking about can’t be the real number line at all. It’s the **surreal number line** … which includes irrational numbers but not repeating decimals … nor (apparently) the number one … nor any number that can be a limit of any kind … but the most important feature of the surreal number line is that infinity is a very very large finite number … and it is useless … you’re welcome … ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [December 6, 2016, 9:48pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/786 "2016-12-06T21:48:25Z")
> [@watchwolf49](#): > > It’s the **surreal number line** … [No, I don’t think it is.](https://en.wikipedia.org/wiki/Surreal_number) ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [December 6, 2016, 11:32pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/787 "2016-12-06T23:32:58Z")
> [@watchwolf49](#): > > It’s the **surreal number line** … [snip] > [@Thudlow\_Boink](#): > > [No, I don’t think it is.](https://en.wikipedia.org/wiki/Surreal_number) [whimper] … just not fair … [wipes tear off check] … Fine fine … The **Not-quite-as-Surreal-as-the-Actual-Surreals-Which-are-Pretty-Damn-Surreal-if-You’re-Askin’-Me number line** … Actually, thanx for this info … now I can sling around _dx_ as a number and have my lame excuse in hand !!! ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [December 7, 2016, 5:02am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/788 "2016-12-07T05:02:10Z")
> [@leahcim](#): > > If you define the distance between a set and a point, as is conventionally done, as the infimum of the distances between the points in the set and the given point, then yes. So, without this definition (possibly axiomatically defined and therefore “useful”) it is still true to say “If none of the elements in [0, 1) is as close as zero distance to point 1 then the set cannot be as close as zero distance to point 1”, right? > [@leahcim](#): > > [0, 1) doesn’t _contain_ (one of) its endpoints. It still _has_ both endpoints, but one of them is not an element of the set. > [@watchwolf49](#): > > Are you seriously claiming the endpoint doesn’t exist? You wouldn’t say “it still has endpoint 1” for any of the finite sets { 0, 0.9, 1 } { 0, 0.9, 0.99, 1 } { 0, 0.9, 0.99, 0.999, 1 } … if we remove point 1 from the set, right? Why do you think it is true for the infinite set [0, 1]? ---
**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:** [December 7, 2016, 8:16am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/789 "2016-12-07T08:16:39Z")
> [@netzweltler](#): > > You wouldn’t say “it still has endpoint 1” for any of the finite sets > { 0, 0.9, 1 } > { 0, 0.9, 0.99, 1 } > { 0, 0.9, 0.99, 0.999, 1 } > … > if we remove point 1 from the set, right? > > Why do you think it is true for the infinite set [0, 1]? Neither would one say those sets have a length. Infinities need to be treated differently. ---
**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:** [December 7, 2016, 10:03am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/790 "2016-12-07T10:03:19Z")
> [@netzweltler](#): > > So, without this definition (possibly axiomatically defined and therefore “useful”) it is still true to say “If none of the elements in [0, 1) is as close as zero distance to point 1 then the set cannot be as close as zero distance to point 1”, right? > > You wouldn’t say “it still has endpoint 1” for any of the finite sets > { 0, 0.9, 1 } > { 0, 0.9, 0.99, 1 } > { 0, 0.9, 0.99, 0.999, 1 } > … > if we remove point 1 from the set, right? > > Why do you think it is true for the infinite set [0, 1]? Um… What? You’ve confused two totally different things - a _set_, which is a defined group of things, and a _range_, which is all the numbers between a high and a low value. What’s more, it doesn’t map to a set unless you do more to define it, such as telling us what set of numbers _within that range_ you want. And in this case, if the set you’re talking about is {x ∈ ℝ, 0 ≥ x ≥ 1} (all real numbers between 0 and 1), then we run into entirely different problems, because the real numbers are _uncountably_ infinite - meaning you literally cannot get from point A to point B by counting. If you remove 1 from that set, changing it to {x ∈ ℝ, 0 ≥ x \> 1}, then you literally cannot define a “greatest number” for the set, making the situation quite different. ---
**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:** [December 7, 2016, 2:08pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/791 "2016-12-07T14:08:31Z")
> [@Budget\_Player\_Cadet](#): > > And in this case, if the set you’re talking about is {x ∈ ℝ, 0 ≥ x ≥ 1} (all real numbers between 0 and 1), then we run into entirely different problems, because the real numbers are _uncountably_ infinite - meaning you literally cannot get from point A to point B by counting. If you remove 1 from that set, changing it to {x ∈ ℝ, 0 ≥ x \> 1}, then you literally cannot define a “greatest number” for the set, making the situation quite different. It’s early in the morning, but shouldn’t those “greater thans” be “less thans”? ---
**Author:** ![leahcim](https://avatars.discourse-cdn.com/v4/letter/l/b4bc9f/32.png) [@leahcim](https://boards.straightdope.com/u/leahcim)\ **Post date:** [December 7, 2016, 3:28pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/792 "2016-12-07T15:28:55Z")
> [@netzweltler](#): > > So, without this definition (possibly axiomatically defined and therefore “useful”) it is still true to say “If none of the elements in [0, 1) is as close as zero distance to point 1 then the set cannot be as close as zero distance to point 1”, right? Without _some_ definition, it is impossible to say _anything_ about distance between sets and points. That is how math works – you define things and then use those definitions to prove things. There is no, “I don’t want to define this thing, but I am sure I know something about it”. > [@](#): > > You wouldn’t say “it still has endpoint 1” for any of the finite sets > { 0, 0.9, 1 } > { 0, 0.9, 0.99, 1 } > { 0, 0.9, 0.99, 0.999, 1 } > … > if we remove point 1 from the set, right? > > Why do you think it is true for the infinite set [0, 1]? Again, if you posit a definition for “endpoint” as “the greatest lower bound or the least upper bound of the set” then it is obvious. { 0, 0.9, 0.99, 0.999 } has endpoints 0 and 0.999. [0, 1) has endpoints 0 and 1. Start with the definition, derive the theorem. If you don’t like that definition, propose another, but you can’t get away without one. This “work from analogy” thing is great for English common law, but it is not mathematics. ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [December 7, 2016, 4:07pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/793 "2016-12-07T16:07:00Z")
> [@netzweltler](#): > > [snip] … You wouldn’t say “it still has endpoint 1” for any of the finite sets > { 0, 0.9, 1 } … [snip] … > … > if we remove point 1 from the set, right? > > Why do you think it is true for the infinite set [0, 1]? The only comment I wish to add here is to remind **netzweltler** that the question at hand is the _length_ of a _ **line segment** _ that is _defined_ as having endpoints at 0 and 1 … these endpoint themselves are _points_ and are dimensionless … thus have no part in the evaluation of the _length_ … Welcome to the **Straight Dope Message Boards** … if you choose to log in with your pride … you can expect it to be bruised … sometimes it’s best to leave your pride back in Real Life so you can actually _learn_ something new … there’s _way_ too many people on these boards a hell of a lot smarter than you and I combined … ---
**Author:** ![Trinopus](https://avatars.discourse-cdn.com/v4/letter/t/2bfe46/32.png) [@Trinopus](https://boards.straightdope.com/u/Trinopus)\ **Post date:** [December 7, 2016, 6:20pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/794 "2016-12-07T18:20:47Z")
> [@watchwolf49](#): > > . . . there’s _way_ too many people on these boards a hell of a lot smarter than you and I combined … “me” not “I.” (And definitely not me!) 😉 ---
**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:** [December 7, 2016, 6:58pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/795 "2016-12-07T18:58:53Z")
I suppose it’s been pointed out that 0.999~=0.9 + 0.09 + 0.009 + … = 9\*Sum[sub]n=1[/sub][sup]infinity/sup[sup]n[/sup], where the sum is of the form Sum[sub]n=1[/sub][sup]infinity/sup[sup]n[/sup], which, for 1/x\<1, converges to 1/(x-1), i.e. 1/9 in our case? ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [December 7, 2016, 7:30pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/796 "2016-12-07T19:30:15Z")
> [@Trinopus](#): > > “me” not “I.” > > (And definitely not me!) 😉 Grammering is as overratted as spelting … phaw … I couldn’t be less proud of believing the subject of an subordinating conjunctivitis clauses takes the nominative case … it just sounds better … ---
**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:** [December 7, 2016, 7:34pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/797 "2016-12-07T19:34:43Z")
> [@Half\_Man\_Half\_Wit](#): > > I suppose it’s been pointed out that 0.999~=0.9 + 0.09 + 0.009 + … = 9\*Sum[sub]n=1[/sub][sup]infinity/sup[sup]n[/sup], where the sum is of the form Sum[sub]n=1[/sub][sup]infinity/sup[sup]n[/sup], which, for 1/x\<1, converges to 1/(x-1), i.e. 1/9 in our case? Yes. netzwelter’s whole shtick is that no such series sum to an actual number. Not that you can’t complete an infinite sum, mind you, but that such a sum is by necessity undefined and somehow not on the number line … ---
**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:** [December 7, 2016, 7:43pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/798 "2016-12-07T19:43:59Z")
> [@naita](#): > > Yes. netzwelter’s whole shtick is that no such series sum to an actual number. Not that you can’t complete an infinite sum, mind you, but that such a sum is by necessity undefined and somehow not on the number line … Ah, the ‘I don’t understand math so it’s all bullshit’-line, I see. Well, I’ll be on my way then… ---
**Author:** ![Marvin\_the\_Martian](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/marvin_the_martian/32/2898_2.png) [@Marvin\_the\_Martian](https://boards.straightdope.com/u/Marvin_the_Martian)\ **Post date:** [December 7, 2016, 7:59pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/799 "2016-12-07T19:59:33Z")
> [@Half\_Man\_Half\_Wit](#): > > Ah, the ‘I don’t understand math so it’s all bullshit’-line, I see. Well, I’ll be on my way then… You have chosen wisely. ---
**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:** [December 7, 2016, 10:13pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/800 "2016-12-07T22:13:04Z")
> [@watchwolf49](#): > > Grammering is as overratted as spelting … phaw … I couldn’t be less proud of believing the subject of an subordinating conjunctivitis clauses takes the nominative case … it just sounds better … I believe you’re correct when “than” is considered a conjunction, but the issue is that sometimes “than” is considered a preposition. FWIW, I agree with you in this case. However, shouldn’t it have been “…there are way too many people…” rather than “…there’s (there is) way too many people…”? I am fully prepared to be wrong because reasons. [Previous page](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711.md?page=39) [Next page](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711.md?page=41)