# 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:** 34
**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 24, 2016, 12:20pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/661 "2016-11-24T12:20:27Z")
> [@netzweltler](#): > > So, what’s the difference? One is a number. The other is a segment. ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [November 24, 2016, 1:40pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/662 "2016-11-24T13:40:46Z")
> [@netzweltler](#): > > Please answer the 3 questions and you will see what I mean. I’m not familiar with your notation … maybe I was presented with this material way back when I was a lad … but that was half a century ago … I do know that whatever your trying to say can be said using functions … … but **The Great Unwashed** has answered your questions … if you have a point, then make it … frankly, I’m not sure you understand your notation … if I understand this correctly, then I am particularly fond of **TGU** answer to #3 … [0,1] … that one makes me smile … Why do you think the value of our previous function cannot equal it’s limit? ETA: If I may modify **BCP** answer a bit … 0.9 is a point on the number line … ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 25, 2016, 8:06am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/663 "2016-11-25T08:06:51Z")
> [@watchwolf49](#): > > I’m not familiar with your notation … maybe I was presented with this material way back when I was a lad … but that was half a century ago … I do know that whatever your trying to say can be said using functions … It’s basic set theoretical notation. 0.999… can be written as [0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… and there is no segment that includes point 1 in [0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… You cannot find a segment which is not already present in the list [0, 0.9] [0, 0.9]∪[0.9, 0.99] [0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999] … (4) Usually we are saying that a number is greater than a set of smaller numbers, if it contains at least one segment which is not a segment of one of the smaller numbers. Do you agree to that? > [@watchwolf49](#): > > … but **The Great Unwashed** has answered your questions … if you have a point, then make it … frankly, I’m not sure you understand your notation … if I understand this correctly, then I am particularly fond of **TGU** answer to #3 … [0,1] … that one makes me smile … > > Why do you think the value of our previous function cannot equal it’s limit? > > ETA: If I may modify **BCP** answer a bit … 0.9 is a point on the number line … ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 25, 2016, 10:26am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/664 "2016-11-25T10:26:14Z")
A little exercise for those who are not familiar with set theoretical notation: We agree that 1 is greater than the numbers of the set { 0.5, 0.8, 0.95 }. In set theoretical notation 1 can be written as [0, 0.5]∪[0.5, 0.8]∪[0.8, 0.95]∪[0.95, 1], the union of these segments. [0.95, 1] is a segment which is not present in one of the numbers { 0.5, 0.8, 0.95 }. The segment [0.95, 1] shows that 1 is greater than the numbers 0.5, 0.8, and 0.95. Now try that with the set { 0.9, 0.99, 0.999, … } an show that 0.999… is greater than all numbers of this set. What is the segment in 0.999… which is not present in one of the numbers 0.9, 0.99, 0.999, …? ---
**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 25, 2016, 10:54am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/665 "2016-11-25T10:54:04Z")
> [@netzweltler](#): > > It’s basic set theoretical notation. > > 0.999… can be written as [0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… > and there is no segment that includes point 1 in [0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… > > You cannot find a segment which is not already present in the list > > [0, 0.9] > [0, 0.9]∪[0.9, 0.99] > [0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999] > … This is only the case if you reject that a convergent infinite series is equal to its limit. In which case you will find a segment which is not necessarily present in the list: _the limit_. ---
**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 25, 2016, 11:40am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/666 "2016-11-25T11:40:57Z")
> [@netzweltler](#): > > It’s basic set theoretical notation. So you’ve finally stopped repeating your nursery rhyme, and are speaking mathematics. My impression is you are unfamiliar with [open sets](https://en.wikipedia.org/wiki/Open_set) and [closed sets](https://en.wikipedia.org/wiki/Closed_set). And you are correct: 1 (a synonym of 0.9999) is not present in the half-open interval [0,1). Ho hum. This is all in chapter 1 of any Freshman Topology text. We could have saved a lot of time if you didn’t feel the need to first cut-paste your nursery rhyme 27 times. :smack: ---
**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 25, 2016, 7:02pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/667 "2016-11-25T19:02:51Z")
> [@netzweltler](#): > > It’s basic set theoretical notation. > > 0.999… can be written as [0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… > > … In which case you’re just finding another way in which your mysteriously-moving-pen-non-argument can be written. Were you hoping no-one was going to notice? Anyway, speaking of what 0.999… can be written as, it can be written as 1 (didn’t we already mention this?). I think I’ll go with that – it uses less screenspace. ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [November 25, 2016, 7:08pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/668 "2016-11-25T19:08:34Z")
**Why do you think the value of our previous function cannot equal it’s limit?** Near as I can figure, you’re just saying the exact same thing here … if you could please explicitly state what the new information is that you are introducing … > [@netzweltler](#): > > It’s basic set theoretical notation. > > 0.999… can be written as [0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… > and there is no segment that includes point 1 in [0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… > > You cannot find a segment which is not already present in the list > > [0, 0.9] > [0, 0.9]∪[0.9, 0.99] > [0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999] > … > > (4) Usually we are saying that a number is greater than a set of smaller numbers, if it contains at least one segment which is not a segment of one of the smaller numbers. > Do you agree to that? Hold on here … excuse my ignorance, but since when can the union of two finite line segments ever be exactly equivalent to a point? Do you mean the _length_ of the line segment is equal to (0.999…) … then as the number of segments approaches infinity, then the length of the union approaches 1 … which leads us back to my question at the top of this post … the one in bold typeface … in case you didn’t see it there … (4) - If the set contains points exclusively, then it contains no line segments, so I don’t know what you’re asking me to agree with. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 26, 2016, 8:02am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/669 "2016-11-26T08:02:26Z")
> [@Budget\_Player\_Cadet](#): > > This is only the case if you reject that a convergent infinite series is equal to its limit. In which case you will find a segment which is not necessarily present in the list: _the limit_. There is no “not necessarily present” in mathematics. Mathematics clearly states that there is no segment that includes point 1 in [0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 26, 2016, 8:20am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/670 "2016-11-26T08:20:26Z")
> [@watchwolf49](#): > > **Why do you think the value of our previous function cannot equal it’s limit?** > > Near as I can figure, you’re just saying the exact same thing here … if you could please explicitly state what the new information is that you are introducing … > > Hold on here … excuse my ignorance, but since when can the union of two finite line segments ever be exactly equivalent to a point? > > Do you mean the _length_ of the line segment is equal to (0.999…) … then as the number of segments approaches infinity, then the length of the union approaches 1 … which leads us back to my question at the top of this post … the one in bold typeface … in case you didn’t see it there … > > (4) - If the set contains points exclusively, then it contains no line segments, so I don’t know what you’re asking me to agree with. Of course we are talking about the length of segments. Since you know that now, please try to answer: In set theoretical notation 1 can be written as [0, 0.5]∪[0.5, 0.8]∪[0.8, 0.95]∪[0.95, 1], the union of these segments. [0.95, 1] is a segment which is not present in one of the numbers { 0.5, 0.8, 0.95 }. The segment [0.95, 1] shows that 1 is greater than the numbers 0.5, 0.8, and 0.95. Now try that with the set { 0.9, 0.99, 0.999, … } an show that 0.999… is greater than all numbers of this set. What is the segment in 0.999… which is not present in one of the numbers 0.9, 0.99, 0.999, …? ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [November 26, 2016, 10:30am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/671 "2016-11-26T10:30:36Z")
**Why do you think the value of our previous function cannot equal it’s limit?** Why do you think the limit point of the set doesn’t exist in the set? ---
**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 27, 2016, 3:41am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/672 "2016-11-27T03:41:35Z")
Aren’t we a little too deep into Monty Python’s “Argument Sketch” by now? ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 27, 2016, 9:17am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/673 "2016-11-27T09:17:04Z")
> [@watchwolf49](#): > > **Why do you think the value of our previous function cannot equal it’s limit?** > > Why do you think the limit point of the set doesn’t exist in the set? The set { 0.9, 0.99, 0.999, … } does neither contain 0.999… nor 1 as one of its elements. You would call me stupid, if I would claim that there are more segments in 1 than the segments [0, 0.25], [0.25, 0.5], [0.5, 0.75], [0.75, 1], wouldn’t you? And if you ask me what the additional segment is and I don’t answer - wouldn’t it be suspicious? So, what is the segment in 0.999… which is not present in one of the numbers 0.9, 0.99, 0.999, …? ---
**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 27, 2016, 11:34am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/674 "2016-11-27T11:34:12Z")
So I was right! Mr. **Netz** had never heard of closed sets and open sets. What do I win? ---
**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:** [November 27, 2016, 4:04pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/675 "2016-11-27T16:04:02Z")
> [@septimus](#): > > So I was right! Mr. **Netz** had never heard of closed sets and open sets. > > What do I win? You win? I think I pointed out that Netz didn’t know what set theory was sometime during the Bush administration. The first Bush administration. ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [November 27, 2016, 7:47pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/676 "2016-11-27T19:47:21Z")
> [@netzweltler](#): > > The set { 0.9, 0.99, 0.999, … } does neither contain 0.999… nor 1 as one of its elements … [snip] I understand what you’re saying … my question is _ **why** _ you are saying this … what property of the set { 0.9, 0.99, 0.999, … } precludes 1 for being one of it’s elements … … or are you asking me to assume first what it is you are concluding? > [@John\_W.Kennedy](#): > > Aren’t we a little too deep into Monty Python’s “Argument Sketch” by now? Oh, right, two doors down and on the left … this is the “Pointless Word Banter” sketch … so sorry about that … ---
**Author:** ![Trinopus](https://avatars.discourse-cdn.com/v4/letter/t/2bfe46/32.png) [@Trinopus](https://boards.straightdope.com/u/Trinopus)\ **Post date:** [November 28, 2016, 1:25am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/677 "2016-11-28T01:25:56Z")
> [@netzweltler](#): > > . . . So, what is the segment in 0.999… which is not present in one of the numbers 0.9, 0.99, 0.999, …? That’s what we’re asking you. If .999~ \<\> 1.0, then you need to construct the non-zero difference. You need to develop some kind of notation to indicate what the difference consists of. The fact is that you neither can, nor apparently will. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 29, 2016, 5:58am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/678 "2016-11-29T05:58:10Z")
> [@septimus](#): > > So I was right! Mr. **Netz** had never heard of closed sets and open sets. > > What do I win? [0, 1] is a closed set, [0, 1) is a half-open set (point 1 excluded). So what? ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 29, 2016, 6:00am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/679 "2016-11-29T06:00:15Z")
> [@watchwolf49](#): > > I understand what you’re saying … my question is _ **why** _ you are saying this … what property of the set { 0.9, 0.99, 0.999, … } precludes 1 for being one of it’s elements … […] Because the sequence 0.9, 0.99, 0.999, … doesn’t include its limit? ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [November 29, 2016, 6:01am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/680 "2016-11-29T06:01:29Z")
> [@Trinopus](#): > > That’s what we’re asking you. If .999~ \<\> 1.0, then you need to construct the non-zero difference. You need to develop some kind of notation to indicate what the difference consists of. > > The fact is that you neither can, nor apparently will. The fact is that I can’t, because 0.999… is not a defined point on the number line. [Previous page](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711.md?page=33) [Next page](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711.md?page=35)