# An infinite question: Why doesn't  .999~ = 1?

**URL:** <https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711>\
**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:** 24

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**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\
**Post date:** [November 8, 2016, 7:08am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/461 "2016-11-08T07:08:36Z")

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> [@The\_Great\_Unwashed](#):
>
> **netzweltler** , I am sure I am not the only one who isn’t convinced that your arguments are made in good faith.
> 
> You have side-stepped issue after issue and have showed no preparedness to play the game and argue directly to the points made. I feel obliged to do some mathematics for you, this isn’t how it should work, you should be doing this yourself.
> 
> We’ll suppose that you are right, that 0.999… \< 1

Ok. I see. It is still not clear what I said. When I say that the position of 0.999… is not defined on the number line, then this means you cannot do any defined comparison or calculation. So, I never said that 0.999… \< 1.

> [@The\_Great\_Unwashed](#):
>
> If p ∈ (0, 1) then p \< √p \< 1
> 
> So 0.9 \< √0.9 \< 1 so the first digit after the decimal point of √0.9 must be 0.9  
> and 0.99 \< √0.99 \< 1 so the first digits after the decimal point of √0.99 must be 0.99  
> and 0.999 \< √0.999 \< 1 so the first digits after the decimal point of √0.999 must be 0.999
> 
> It should be obvious that as 0.999… is an infinite string of 9s, that its square root must also be an infinite string of 9s.
> 
> i.e. 0.999… = √0.999…
> 
> What number(s) are there for which x = √x ?  
> There, no deltas, no epsilons, no limits as n approaches infinity. High School maths at most.

To take the square root of an undefined point on the number line will result in an undefined point again.

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<div class="post-metadata">

**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\
**Post date:** [November 8, 2016, 7:16am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/462 "2016-11-08T07:16:04Z")

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> [@watchwolf49](#):
>
> The nice thing about plunging down rabbit holes is there’s a hookah smoking caterpillar to be found.
> 
> 103rd …
> 
> > [@netzweltler](#):
> >
> > “…” is clearly defined:
> 
> Then please state the definition you are using in the following, because (…) can be defined as a long comma … requiring the reader to pause a bit longer than a comma but not so long as would be the case with a period.

Didn’t I define that when I said there is a line for each n ∈ ℕ?

> [@watchwolf49](#):
>
> Again, infinity is not a natural number … so asking us to pick a line n (∈ ℕ) is nonsense …

We don’t pick a particular line, we take all of them and see that none of them does get us to point 1.

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<div class="post-metadata">

**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\
**Post date:** [November 8, 2016, 7:40am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/463 "2016-11-08T07:40:03Z")

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> [@74westy](#):
>
> **netzweltler** , I understand that you don’t consider 0.999… to be a number. I’m curious what you do consider to be a number.
> 
> Is -1 a number?  
> Is 1/2 a number?  
> Is the square root of 2 a number?  
> Is the square root of -1 a number?
> 
> Is something like 0.999… objectively not a number or could we create a self-consistent definition that maps strings like 0.999… to numbers?

(Copy and paste again)

n = 1, t = 0: I move my pen from point 0 to point 0.9 of the number line  
n = 2, t = 0.9: I move my pen from point 0.9 to point 0.99 of the number line  
n = 3, t = 0.99: I move my pen from point 0.99 to point 0.999 of the number line  
…

For every n ∈ ℕ there is a line on the list.  
For every n ∈ ℕ is valid that we don’t reach 1.

If 0.999… would be the same point as point 1 on the number line we would have reached point 1 by t = 1.

Try that with -1:

n = 1, t = 0: I move my pen from point 0 to point -0.5 of the number line  
n = 2, t = 0.9: I move my pen from point -0.5 to point -0.75 of the number line  
n = 3, t = 0.99: I move my pen from point -0.75 to point -1 of the number line  
n = 4, t = 0.999: I move my pen from point -1 to point -1 of the number line  
n = 5, t = 0.9999: I move my pen from point -1 to point -1 of the number line  
…

For n ≥ 3 is valid that we reached -1.

At t = 1 we can state that the pen is pointing to -1. So, -1 is a well-defined point on the number line.

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<div class="post-metadata">

**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\
**Post date:** [November 8, 2016, 7:42am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/464 "2016-11-08T07:42:42Z")

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> [@Trinopus](#):
>
> Actually, here’s where you’re wrong. In your model, at t=1, then pen has “jumped” an infinite number of times, and has come to rest on 1.0

At t = 1 the pen has jumped an infinite number of times. Nowhere the pen jumped to point 1.0. Otherwise, show where.

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**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\
**Post date:** [November 8, 2016, 9:41am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/465 "2016-11-08T09:41:46Z")

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> [@netzweltler](#):
>
> At t = 1 the pen has jumped an infinite number of times. Nowhere the pen jumped to point 1.0. Otherwise, show where.

You haven’t shown it doesn’t … this is your case to make.

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<div class="post-metadata">

**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\
**Post date:** [November 8, 2016, 11:18am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/466 "2016-11-08T11:18:10Z")

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> [@watchwolf49](#):
>
> You haven’t shown it doesn’t … this is your case to make.

I did. Do you want me to copy and paste again?

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**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\
**Post date:** [November 8, 2016, 11:19am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/467 "2016-11-08T11:19:11Z")

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> [@74westy](#):
>
> Is the square root of -1 a number?

Not on the number line under discussion.

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**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 8, 2016, 1:24pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/468 "2016-11-08T13:24:12Z")

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> [@netzweltler](#):
>
> I did. Do you want me to copy and paste again?

God no! You’ve posted the same thing several dozen times in this thread and it hasn’t convinced anyone, and you’ve handwaved all the refutations. You’re out on an island here and that’s just fine. If you think you have something useful please demonstrate something useful you can do with your assertion. We’ll wait here.

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<div class="post-metadata">

**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\
**Post date:** [November 8, 2016, 1:41pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/469 "2016-11-08T13:41:48Z")

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> [@netzweltler](#):
>
> I did. Do you want me to copy and paste again?

Sure, go ahead, make sure you include the line where t = 1. Also, how wide is your pen?

This is a very cute story you tell … but now you have to convert your words into mathematical expressions …

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<div class="post-metadata">

**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 8, 2016, 2:06pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/470 "2016-11-08T14:06:17Z")

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> [@netzweltler](#):
>
> Didn’t I define that when I said there is a line for each n ∈ ℕ?  
> We don’t pick a particular line, we take all of them and see that none of them does get us to point 1.

So once we have done all of them, where is our pen? And don’t tell me “not defined”, because we are _clearly_ still on the number line - after all, no step took us off of it. :rolleyes:

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**Author:** ![74westy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/74westy/32/3950_2.png) [@74westy](https://boards.straightdope.com/u/74westy)\
**Post date:** [November 8, 2016, 2:54pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/471 "2016-11-08T14:54:49Z")

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> [@netzweltler](#):
>
> (Copy and paste again)
> 
> n = 1, t = 0: I move my pen from point 0 to point 0.9 of the number line  
> n = 2, t = 0.9: I move my pen from point 0.9 to point 0.99 of the number line  
> n = 3, t = 0.99: I move my pen from point 0.99 to point 0.999 of the number line  
> …
> 
> For every n ∈ ℕ there is a line on the list.  
> For every n ∈ ℕ is valid that we don’t reach 1.
> 
> If 0.999… would be the same point as point 1 on the number line we would have reached point 1 by t = 1.
> 
> Try that with -1:
> 
> n = 1, t = 0: I move my pen from point 0 to point -0.5 of the number line  
> n = 2, t = 0.9: I move my pen from point -0.5 to point -0.75 of the number line  
> n = 3, t = 0.99: I move my pen from point -0.75 to point -1 of the number line  
> n = 4, t = 0.999: I move my pen from point -1 to point -1 of the number line  
> n = 5, t = 0.9999: I move my pen from point -1 to point -1 of the number line  
> …
> 
> For n ≥ 3 is valid that we reached -1.
> 
> At t = 1 we can state that the pen is pointing to -1. So, -1 is a well-defined point on the number line.

> [@netzweltler](#):
>
> Not on the number line under discussion.

Am I understanding you correctly? You have some kind of rubric that separates real numbers from non-numbers and it involves moving a pen in discrete steps along a number line.

Can you define this procedure rigorously enough that others can use it and come to the same conclusions you do? Can I do this?  
n = 1, t = 0: I move my pen from point 0 to point 0.999… of the number line

Why not?

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<div class="post-metadata">

**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 8, 2016, 3:17pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/472 "2016-11-08T15:17:41Z")

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> [@74westy](#):
>
> Am I understanding you correctly? You have some kind of rubric that separates real numbers from non-numbers and it involves moving a pen in discrete steps along a number line.
> 
> Can you define this procedure rigorously enough that others can use it and come to the same conclusions you do? Can I do this?  
> n = 1, t = 0: I move my pen from point 0 to point 0.999… of the number line
> 
> Why not?

You’ve hit the nail on the head. We’re all trying to demonstrate the absurdity. but we can’t tell which of us is getting through: **Mr. Netz** just keeps copy-pasting his rubric.

Let me try. I’ll demonstrate that _ **zero is not a number** _.

At t = 0, my pencil is at 1.0  
At t = 1 I move my pencil to 0.9  
At t = 2 I move my pencil to 0.8  
At t = 3 I move my pencil to 0.7  
At t = 4 I move my pencil to 0.6 // think I’m going to make it? 🙂  
At t = 5 I move my pencil to Mars  
At t = 6 I move my pencil to Jupiter  
At t = 7 I move my pencil to Saturn  
At t = 8 I move my pencil to …

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<div class="post-metadata">

**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 8, 2016, 3:50pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/473 "2016-11-08T15:50:16Z")

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> [@netzweltler](#):
>
> (Copy and paste again)
> 
> n = 1, t = 0: I move my pen from point 0 to point 0.9 of the number line  
> n = 2, t = 0.9: I move my pen from point 0.9 to point 0.99 of the number line  
> n = 3, t = 0.99: I move my pen from point 0.99 to point 0.999 of the number line  
> …
> 
> For every n ∈ ℕ there is a line on the list.  
> For every n ∈ ℕ is valid that we don’t reach 1.

Yeah but, the astute reader will notice that as ∞ ∉ ℕ then 0.999… doesn’t appear on the right hand side. That’s your problem, right there. That’s your example that is insufficient to model the case of an infinitely recurring decimal, that’s not a problem with maths, that’s a problem with your argument.

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<div class="post-metadata">

**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 8, 2016, 3:54pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/474 "2016-11-08T15:54:42Z")

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> [@Budget\_Player\_Cadet](#):
>
> So once we have done all of them, where is our pen? And don’t tell me “not defined”, because we are _clearly_ still on the number line - after all, no step took us off of it. :rolleyes:

Just want to keep this one up there. I expect the answer is “An undefinable point undefinably close to 1”, but hope springs eternal.

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<div class="post-metadata">

**Author:** ![Chingon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chingon/32/17645_2.png) [@Chingon](https://boards.straightdope.com/u/Chingon)\
**Post date:** [November 8, 2016, 4:01pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/475 "2016-11-08T16:01:10Z")

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You need him to define a number. He is apparently considering any symbol not defined/denoted by a fraction of integers to be a non number.

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<div class="post-metadata">

**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 8, 2016, 4:01pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/476 "2016-11-08T16:01:21Z")

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> [@netzweltler](#):
>
> Ok. I see. It is still not clear what I said. When I say that the position of 0.999… is not defined on the number line, then this means you cannot do any defined comparison or calculation. So, I never said that 0.999… \< 1.

I’m pretty sure that you have accepted that 0.999… is greater than every element of the set {0.9,0.99,0.999, …} and whilst I cannot be arsed searching, I’m pretty sure you’ve held that 0.999… is indeed less than 1.

Still, if the whole of your argument is that your super-fine pen never reaches 0.999… therefore 0.999… is not well-defined, then you really have to explain why we should want to accept your bizarro definition of “well definedness” particularly because it excludes some very regular numbers, for example 0.333…, but yet somehow(?) allows for √2 and _π_.

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<div class="post-metadata">

**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\
**Post date:** [November 8, 2016, 4:29pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/477 "2016-11-08T16:29:33Z")

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His strawman keeps moving the goal posts … and Pluto is non-defined as an undefinable definition … wait for it … now we have Pluto is under-defined over the not-quite-defined non-definitions …

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<div class="post-metadata">

**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 8, 2016, 7:39pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/478 "2016-11-08T19:39:49Z")

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nm

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<div class="post-metadata">

**Author:** ![Trinopus](https://avatars.discourse-cdn.com/v4/letter/t/2bfe46/32.png) [@Trinopus](https://boards.straightdope.com/u/Trinopus)\
**Post date:** [November 8, 2016, 11:27pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/479 "2016-11-08T23:27:03Z")

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> [@septimus](#):
>
> nm

True dat.

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<div class="post-metadata">

**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 9, 2016, 12:10am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/480 "2016-11-09T00:10:10Z")

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> [@Trinopus](#):
>
> True dat.

> [@septimus](#):
>
> …  
> At t = 7 I move my pencil to Saturn  
> At t = 8 I move my pencil to …

Mine, before I nm’ed it, was going to be  
Why did nobody give me a _“I see what you did there!”_

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