# 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:** 9\ **Page:** 41
**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, 10:19pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/801 "2016-12-07T22:19:58Z")
I don’t understand grammering so it’s all bullshit to me … However, I do understand your username … shame on you … [giggle] But, yes, I’m not using `people` as a singular entity … so it should have the plural form of the verb `are` ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [December 8, 2016, 7:19am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/802 "2016-12-08T07:19:07Z")
> [@Budget\_Player\_Cadet](#): > > 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. I agree with that. We cannot define a “greatest number”. That’s why we cannot define an endpoint. ---
**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\ **Post date:** [December 8, 2016, 7:25am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/803 "2016-12-08T07:25:48Z")
> [@leahcim](#): > > 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”. Is there definition missing? Wouldn’t we rather have to define elements as close as zero distance to point 1, if we want the set [0, 1) to be as close as zero distance to point 1? I mean, shouldn’t we be able to name such elements? > [@leahcim](#): > > 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 8, 2016, 2:15pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/804 "2016-12-08T14:15:51Z")
> [@netzweltler](#): > > I agree with that. We cannot define a “greatest number”. That’s why we cannot define an endpoint. So, the line segment [0, 1) with the defined endpoints at 0 and 1 has no defined endpoints … interesting … that’s running loose with the definition of definition … that’s for sure … ---
**Author:** ![RealityChuck](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/realitychuck/32/195_2.png) [@RealityChuck](https://boards.straightdope.com/u/RealityChuck)\ **Post date:** [December 8, 2016, 7:21pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/805 "2016-12-08T19:21:12Z")
We’re still arguing this? It’s easily proven with basic math. for .999999… x = .99999… Multiply by 10 10x = 9.9999999… Subtract .999999… from both sides 10x - .9999999 = 9 From the first equation, it’s given that: x = .999999… So, substituting that in, you get 10x - x = 9 Simply 9x = 9 Divide both sides by 9 x = 1 QED This works for any repeating decimal as a way to find a whole number value for it. Try .333333… x = .333333… 10x = 3.33333… 10x - .333333… = 3 10x - x = 3 9x - 3 x = 3/9 = 1/3 Thus .333333… = 1/3 If the proof is wrong for .99999…, it’s also wrong for .3333333… But you can easily prove that 1/3 = .333333… by simple division. Therefore, the proof is correct. Any other argument ignores math. ---
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [December 8, 2016, 7:59pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/806 "2016-12-08T19:59:48Z")
Yes, this is taking longer than we thought. ---
**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:** [December 8, 2016, 9:34pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/807 "2016-12-08T21:34:13Z")
> [@RealityChuck](#): > > If the proof is wrong for .99999…, it’s also wrong for .3333333… But you can easily prove that 1/3 = .333333… by simple division. Therefore, the proof is correct. Honestly, that’s a horrible argument, a “proof” might well be wrong for both _Proposition A_ and _Proposition B_ but even if we could prove _Proposition B_ by independent other means that would have no bearing on the soundness of said previous argument for the truth of _Proposition A_. ---
**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:** [December 8, 2016, 9:36pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/808 "2016-12-08T21:36:54Z")
> [@netzweltler](#): > > Wouldn’t we rather have to define elements as close as zero distance to point 1, if we want the set [0, 1) to be as close as zero distance to point 1? I mean, shouldn’t we be able to name such elements? I said it sometime before, easily missed: 0.999… is not an element of [0,1) so I don’t know why you think it of the least importance that we cannot name its “closest to 1” element. ---
**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:** [December 10, 2016, 2:11am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/809 "2016-12-10T02:11:54Z")
> [@The\_Great\_Unwashed](#): > > I said it sometime before, easily missed: 0.999… is not an element of [0,1) so I don’t know why you think it of the least importance that we cannot name its “closest to 1” element. I may be wrong, but I think that, from his viewpoint, you’re begging the question. [Previous page](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711.md?page=40)