# .999 = 1?

**URL:** <https://boards.straightdope.com/t/999-1/27517>\
**Category:** Factual Questions\
**Created:** [July 31, 2000, 6:28am UTC](https://boards.straightdope.com/t/999-1/27517 "2000-07-31T06:28:03Z")\
**Posts on this page:** 20\
**Page:** 54

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**Author:** ![Gary\_Robson](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/gary_robson/32/3448_2.png) [@Gary\_Robson](https://boards.straightdope.com/u/Gary_Robson)\
**Post date:** [January 18, 2013, 3:58pm UTC](https://boards.straightdope.com/t/999-1/27517/1066 "2013-01-18T15:58:06Z")

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[mod note]  
**Leo Bloom** , we don’t allow post count parties in GQ. It’s fine to discuss this in other threads, but there was no need at all to bump this thread to do it.  
No warning issued, but please don’t post to dormant threads just to comment on their length.  
[/mod note]

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**Author:** ![Francis\_Vaughan](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/francis_vaughan/32/3093_2.png) [@Francis\_Vaughan](https://boards.straightdope.com/u/Francis_Vaughan)\
**Post date:** [January 20, 2013, 4:22am UTC](https://boards.straightdope.com/t/999-1/27517/1067 "2013-01-20T04:22:53Z")

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At the risk of stirring things up again, I recently came across a pertinent paper by Reuben Hersh, that is worth a read. [Here](http://www.math.umt.edu/tmme/vol8no1and2/2_Hersh_TMME2011_article2_pp.35_50.pdf) (PDF).

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**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:** [January 20, 2013, 6:01am UTC](https://boards.straightdope.com/t/999-1/27517/1068 "2013-01-20T06:01:57Z")

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> [@Francis\_Vaughan](#):
>
> At the risk of stirring things up again, I recently came across a pertinent paper by Reuben Hersh, that is worth a read. [Here](http://www.math.umt.edu/tmme/vol8no1and2/2_Hersh_TMME2011_article2_pp.35_50.pdf) (PDF).

That _is_ an interesting paper – I’ve downloaded it to reread at my leisure – worth the price of admission to this thread! 🆒 (Don’t be discouraged by the tedious dialog in first 3 pages.)

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**Author:** ![moriah](https://avatars.discourse-cdn.com/v4/letter/m/b2d939/32.png) [@moriah](https://boards.straightdope.com/u/moriah)\
**Post date:** [January 20, 2013, 6:13am UTC](https://boards.straightdope.com/t/999-1/27517/1069 "2013-01-20T06:13:33Z")

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And this is particularly germane:

[9.999… reasons that .999… = 1](http://www.youtube.com/watch?v=TINfzxSnnIE)

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**Author:** ![Freakenstein](https://avatars.discourse-cdn.com/v4/letter/f/b782af/32.png) [@Freakenstein](https://boards.straightdope.com/u/Freakenstein)\
**Post date:** [January 20, 2013, 11:16am UTC](https://boards.straightdope.com/t/999-1/27517/1070 "2013-01-20T11:16:39Z")

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OK, an honest question:  
There’s another thread where You dopers prove that “2 does not equal 3, even for very large values of 2.”  
Then how zero for very large values can equal one?  
As I see, You can’t have it both ways or am I missing something.

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**Author:** ![Stealth\_Potato](https://avatars.discourse-cdn.com/v4/letter/s/d78d45/32.png) [@Stealth\_Potato](https://boards.straightdope.com/u/Stealth_Potato)\
**Post date:** [January 20, 2013, 11:44am UTC](https://boards.straightdope.com/t/999-1/27517/1071 "2013-01-20T11:44:30Z")

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> [@Freakenstein](#):
>
> OK, an honest question:  
> There’s another thread where You dopers prove that “2 does not equal 3, even for very large values of 2.”  
> Then how zero for very large values can equal one?  
> As I see, You can’t have it both ways or am I missing something.

Can you explain what you’re asking? I haven’t seen that other thread so I don’t know how it’s relevant. Nobody’s talking about “large values of zero,” whatever that’s supposed to mean; 0.999… isn’t zero, it’s one.

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**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:** [January 20, 2013, 12:37pm UTC](https://boards.straightdope.com/t/999-1/27517/1072 "2013-01-20T12:37:21Z")

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I think it’s related to Colbert’s Antinomy:

> [@](#):
>
> After _One Direction_ tells the girl “You don’t know you’re beautiful; That’s what makes you beautiful!” the girl knows she’s beautiful and so … is beautiful no more.

ETA: Or maybe not. Perhaps I’ve conflated Antinomy with Antimony.

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**Author:** ![Freakenstein](https://avatars.discourse-cdn.com/v4/letter/f/b782af/32.png) [@Freakenstein](https://boards.straightdope.com/u/Freakenstein)\
**Post date:** [January 20, 2013, 1:06pm UTC](https://boards.straightdope.com/t/999-1/27517/1073 "2013-01-20T13:06:47Z")

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> [@Stealth\_Potato](#):
>
> Can you explain what you’re asking? I haven’t seen that other thread so I don’t know how it’s relevant. Nobody’s talking about “large values of zero,” whatever that’s supposed to mean; 0.999… isn’t zero, it’s one.

[The other thread.](http://boards.straightdope.com/sdmb/showthread.php?t=419965) 2+2=5, for large values of 2. I copied the wording “large values of …” from that thread, so I supposed it would be correct, though it seems odd to Me ( not a native English speaker ).

I’m not a math-hero, so maybe that’s why I don’t get it. And maybe it really isn’t relevant. But if people can’t prove 0.9999…=1 the way that I understand it, then they haven’t proven it to Me at all. The way I see is, that any given number is only itself and not another number ( even if it could be written in a different way ).  
And, yes, it’s been proven that the difference is infinitely small, but to Me that doesn’t make in non-existent.  
But I’m not good with infinities, big or small.

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**Author:** ![Leo\_Bloom](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/leo_bloom/32/10377_2.png) [@Leo\_Bloom](https://boards.straightdope.com/u/Leo_Bloom)\
**Post date:** [January 20, 2013, 3:23pm UTC](https://boards.straightdope.com/t/999-1/27517/1074 "2013-01-20T15:23:08Z")

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> [@moriah](#):
>
> And this is particularly germane:
> 
> [9.999… reasons that .999… = 1](http://www.youtube.com/watch?v=TINfzxSnnIE)

This video is terrific (“didn’t read the thread, saw the movie”).

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**Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\
**Post date:** [January 20, 2013, 3:43pm UTC](https://boards.straightdope.com/t/999-1/27517/1075 "2013-01-20T15:43:19Z")

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> [@Freakenstein](#):
>
> [The other thread.](http://boards.straightdope.com/sdmb/showthread.php?t=419965) 2+2=5, for large values of 2. I copied the wording “large values of …” from that thread, so I supposed it would be correct, though it seems odd to Me ( not a native English speaker ).
> 
> I’m not a math-hero, so maybe that’s why I don’t get it. And maybe it really isn’t relevant. But if people can’t prove 0.9999…=1 the way that I understand it, then they haven’t proven it to Me at all. The way I see is, that any given number is only itself and not another number ( even if it could be written in a different way ).  
> And, yes, it’s been proven that the difference is infinitely small, but to Me that doesn’t make in non-existent.  
> But I’m not good with infinities, big or small.

I’m guessing that you think that 0.999… is not equal to 1 because you suppose that every every real number has a unique decimal expansion? If that is the case, consider the following:

For a positive real number x, the following properties about it’s square root come from basic inequalities such as x\*y \< x if y\<1:

sqrt(x)\>x if x\<1

sqrt(x)=x if x=1

sqrt(x)\<x if x\>1

Also:

sqrt(x)\<1 if x\<1

sqrt(x)=1 if x=1

sqrt(x)\>1 if x\>1

Where sqrt(x) is the positive square root of x.

Now suppose 0.999… is less than 1, therefore:

0.999… \< sqrt(0.999…) \< 1

What would be the unique decimal expansion for sqrt(0.999…)?

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**Author:** ![Great\_Antibob](https://avatars.discourse-cdn.com/v4/letter/g/e47c2d/32.png) [@Great\_Antibob](https://boards.straightdope.com/u/Great_Antibob)\
**Post date:** [January 20, 2013, 4:27pm UTC](https://boards.straightdope.com/t/999-1/27517/1076 "2013-01-20T16:27:18Z")

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> [@Freakenstein](#):
>
> But if people can’t prove 0.9999…=1 the way that I understand it, then they haven’t proven it to Me at all.

That doesn’t mean it’s not true. It’s rather arrogant to assume your lack of understanding is tantamount to proof it’s not true. Basically your argument matches Asimov’s quip that democratic anti-intellectualism devolves to the idea that my ignorance is as good as your knowledge.

2 is not 3. That’s clear. 0.99… IS 1. There’s no “large values” approximation going on. As noted in that thread, 2+2 = 5 for large values of 2 is a JOKE. It’s not a serious statement.

And none of that has any bearing here unless you already begin with the assumption that 0.999… is not 1, which is rather tautological. You don’t conclude by re-stating your assumptions. That makes no sense.

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**Author:** ![Wendell\_Wagner](https://avatars.discourse-cdn.com/v4/letter/w/8491ac/32.png) [@Wendell\_Wagner](https://boards.straightdope.com/u/Wendell_Wagner)\
**Post date:** [January 20, 2013, 4:35pm UTC](https://boards.straightdope.com/t/999-1/27517/1077 "2013-01-20T16:35:05Z")

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Freakenstein, have you read all of this thread? If not, please do so. We’d rather not have to repeat all the answers to your questions when we’ve already answered them.

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**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:** [January 20, 2013, 5:13pm UTC](https://boards.straightdope.com/t/999-1/27517/1078 "2013-01-20T17:13:26Z")

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> [@Freakenstein](#):
>
> [The other thread.](http://boards.straightdope.com/sdmb/showthread.php?t=419965) 2+2=5, for large values of 2. I copied the wording “large values of …” from that thread, so I supposed it would be correct, though it seems odd to Me ( not a native English speaker ).
> 
> I’m not a math-hero, so maybe that’s why I don’t get it. And maybe it really isn’t relevant. But if people can’t prove 0.9999…=1 the way that I understand it, then they haven’t proven it to Me at all. The way I see is, that any given number is only itself and not another number ( even if it could be written in a different way ).  
> And, yes, it’s been proven that the difference is infinitely small, but to Me that doesn’t make in non-existent.  
> But I’m not good with infinities, big or small.

If you’re not a native English speaker then reading through this thread might be a real problem. The [video](http://www.youtube.com/watch?v=TINfzxSnnIE) covers all the same territory (except for some of the higher set theory) in a simple straightforward manner. The problem there is that the narrator talks very fast and includes many puns. But if you can follow her, she does a great job of hitting all the places that normally cause confusion for non-math people.

And you show one of them. The difference between 9.999… and 1 is not infinitely small. It is 0. Exactly 0. They are indeed the same number expressed in two different ways.

Try watching the video and see if that helps your understanding.

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**Author:** ![virtually\_yours](https://avatars.discourse-cdn.com/v4/letter/v/a587f6/32.png) [@virtually\_yours](https://boards.straightdope.com/u/virtually_yours)\
**Post date:** [January 20, 2013, 5:52pm UTC](https://boards.straightdope.com/t/999-1/27517/1079 "2013-01-20T17:52:52Z")

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> [@Exapno\_Mapcase](#):
>
> If you’re not a native English speaker then reading through this thread might be a real problem. The [video](http://www.youtube.com/watch?v=TINfzxSnnIE) covers all the same territory (except for some of the higher set theory) in a simple straightforward manner. The problem there is that the narrator talks very fast and includes many puns. But if you can follow her, she does a great job of hitting all the places that normally cause confusion for non-math people.
> 
> And you show one of them. The difference between 9.999… and 1 is not infinitely small. It is 0. Exactly 0. They are indeed the same number expressed in two different ways.
> 
> Try watching the video and see if that helps your understanding.

Jumping in to the fray. hope I don’t get friction burns. too much posts to read so if pointed out already please disregard..  
My contention is that a third of 1 is not _point_ anything.. The 1/3 of of one is 1/3A (ONE third Absolute), or ONE of the other two parts which are themselves one,used to make up one. If you took that third of one as it self and said you were going to divide it into thirds you wouldn’t say you were going to divide a third of one by a third (or .33333…), you would say you are dividing ONE into a third..(apples and oranges aside) I am using a piece of wood for example so to speak. My OBSERVATION of the two ends of a third of the wood shows it does not go on point threeing or nineing forever. It has an absolute ending.  
Invent a new terminology to explain it..  
I suggest: 1/3A, one third absolutUM not 1/3 infinitum!  
maybe I’m just a layman, what could I know?  
vyVY

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<div class="post-metadata">

**Author:** ![Wendell\_Wagner](https://avatars.discourse-cdn.com/v4/letter/w/8491ac/32.png) [@Wendell\_Wagner](https://boards.straightdope.com/u/Wendell_Wagner)\
**Post date:** [January 20, 2013, 6:04pm UTC](https://boards.straightdope.com/t/999-1/27517/1080 "2013-01-20T18:04:59Z")

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Again, please read the entire thread.

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<div class="post-metadata">

**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\
**Post date:** [January 20, 2013, 6:08pm UTC](https://boards.straightdope.com/t/999-1/27517/1081 "2013-01-20T18:08:50Z")

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> [@Wendell\_Wagner](#):
>
> Again, please read the entire thread.

Seconded.

It’s actually pretty interesting throughout.

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<div class="post-metadata">

**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\
**Post date:** [January 20, 2013, 6:10pm UTC](https://boards.straightdope.com/t/999-1/27517/1082 "2013-01-20T18:10:35Z")

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Look especially for posts by **Indistinguishable**.

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<div class="post-metadata">

**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:** [January 20, 2013, 6:14pm UTC](https://boards.straightdope.com/t/999-1/27517/1083 "2013-01-20T18:14:04Z")

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> [@virtually\_yours](#):
>
> Jumping in to the fray. hope I don’t get friction burns. too much posts to read so if pointed out already please disregard..  
> My contention is that a third of 1 is not _point_ anything.. The 1/3 of of one is 1/3A (ONE third Absolute), or ONE of the other two parts which are themselves one,used to make up one. If you took that third of one as it self and said you were going to divide it into thirds you wouldn’t say you were going to divide a third of one by a third (or .33333…), you would say you are dividing ONE into a third..(apples and oranges aside) I am using a piece of wood for example so to speak. My OBSERVATION of the two ends of a third of the wood shows it does not go on point threeing or nineing forever. It has an absolute ending.  
> Invent a new terminology to explain it..  
> I suggest: 1/3A, one third absolutUM not 1/3 infinitum!  
> maybe I’m just a layman, what could I know?  
> vyVY

If you had read the thread you’d have noticed that people have repeatedly posted that math is not a word problem. What you are saying is not math. It’s not anything at all, really, but it’s definitely not math.

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<div class="post-metadata">

**Author:** ![CookingWithGas](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cookingwithgas/32/485_2.png) [@CookingWithGas](https://boards.straightdope.com/u/CookingWithGas)\
**Post date:** [January 20, 2013, 6:22pm UTC](https://boards.straightdope.com/t/999-1/27517/1084 "2013-01-20T18:22:56Z")

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> [@moriah](#):
>
> And this is particularly germane:
> 
> [9.999… reasons that .999… = 1](http://www.youtube.com/watch?v=TINfzxSnnIE)

Miss .999…, please meet the [Vocal Fry thread](http://boards.straightdope.com/sdmb/showthread.php?t=644035).

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<div class="post-metadata">

**Author:** ![Freakenstein](https://avatars.discourse-cdn.com/v4/letter/f/b782af/32.png) [@Freakenstein](https://boards.straightdope.com/u/Freakenstein)\
**Post date:** [January 20, 2013, 6:41pm UTC](https://boards.straightdope.com/t/999-1/27517/1085 "2013-01-20T18:41:46Z")

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> [@Asympotically\_fat](#):
>
> I’m guessing that you think that 0.999… is not equal to 1 because you suppose that every every real number has a\*\* unique decimal expansion\*\*?

I know what all these words mean separately, I even checked them out to be sure, but I have absolutely no idea what they mean together here.

> [@](#):
>
> such as x\*y \< x if y\<1:

This is what’s My problem with higher math. To Me math is numbers, not letters. I’m not bad with numbers itself, but I hate this kind of stuff, which obviously contributes to My problem.  
Anyway I took a calculator and all was fine and well until I got this far:

> [@](#):
>
> Where sqrt(x) is the positive square root of x.

I know the meaning of every single word of this, but together - no idea.

However

> [@](#):
>
> Now suppose 0.999… is less than 1, therefore:  
> 0.999… \< sqrt(0.999…) \< 1

All I can say is that was a neat job. Thank You ( although I’m just 99.999… % convinced )

**Great Antibob** :  
I didn’t say it’s not true, I just didn’t get it. There are lots of scientific stuff that I don’t get, but as scientists work based on them, I just believe it’s true.

**Wendell Wagner**  
Sorry, two pages and and I didn’t understand a thing, especially as there was disagreement. If this would include only that posting by **Asympotically fat** all would be fine and dandy, be sure to bookmark that for future.

**Exapno Mapcase** :  
Video - thanks, but way too fast.

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