# .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:** 12

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**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\
**Post date:** [August 6, 2012, 9:33am UTC](https://boards.straightdope.com/t/999-1/27517/221 "2012-08-06T09:33:27Z")

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> [@erik150x](#):
>
> I want to repeat this becuase I think bears repeating:
> 
> \*\*In other words, people who ask this quesiton .999… = 1? Are not asking can you show with limits how we define this to be one in the same number? Most of them I can assure you are not asking this. They want to know form the purley intuitve knowledge that we all posess that .9 \< 1 and .99 \< 1 and .999 \< 1 and .9999 \< 1 and so on… how is it that at some point on further exapanison (even infinite) they become equal? \*\*

They don’t become equal at any finite point, as you correctly point out. And it is, in a sense, meaningless to say they become equal “at infinity”, because you can never get to that infinity. When people say they are “equal at infinity”, they are talking about limits.

Here’s a fairly rigorous definition of what that means:

“.999…” is the limit of a series of numbers, where the nth term of the series is the decimal point followed by n 9s, i.e., the series:  
.9, .99, .999, .9999, etc.

When we say that the limit of that series is 1 we mean that for every positive number ε, no matter how small ε is, there is a positive integer n such that every member of the series from the nth term on is less than ε different from 1. (The Greek letter epsilon (ε) is conventionally used for this arbitrarily small number.)

In other words, you can get as close as you like to the limit 1 by going as far as necessary in the series.

This is basically the usual definition of infinite decimal fractions, though most will have other limits – only .999… and 1.000… have 1 as a limit.

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**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\
**Post date:** [August 6, 2012, 9:34am UTC](https://boards.straightdope.com/t/999-1/27517/222 "2012-08-06T09:34:01Z")

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FV,

I was simply replying hopfully just infintesmialy less rediculously to Monimonika’s previous deomstration that he could add .999… + .999… + .999… + … ten times and show that it equaled 9.999…

Giles your point is well taken. Niether my nor Monimonika’s version makes perfect sense. It is not a simple issue to deal with. But if the proof was as easy as Monimonika makes it out to be, it would have been made long ago. Poeple accused me of thinking I am smarter than all the great mathematicians in history. I certainly do not even compair my self… There is no proof I know of in a very elemantry manner describing a proof for .999… = 1. If there was it would be all over the interent. Everyone i have seen uses Limit based calculus. If your going to use limit based calculus then you have already made the assumption they are equal, so what’s the point of even considering the question.

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**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\
**Post date:** [August 6, 2012, 9:38am UTC](https://boards.straightdope.com/t/999-1/27517/223 "2012-08-06T09:38:29Z")

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> [@Giles](#):
>
> They don’t become equal at any finite point, as you correctly point out. And it is, in a sense, meaningless to say they become equal “at infinity”, because you can never get to that infinity. When people say they are “equal at infinity”, they are talking about limits.
> 
> Here’s a fairly rigorous definition of what that means:
> 
> “.999…” is the limit of a series of numbers, where the nth term of the series is the decimal point followed by n 9s, i.e., the series:  
> .9, .99, .999, .9999, etc.
> 
> When we say that the limit of that series is 1 we mean that for every positive number ε, no matter how small ε is, there is a positive integer n such that every member of the series from the nth term on is less than ε different from 1. (The Greek letter epsilon (ε) is conventionally used for this arbitrarily small number.)
> 
> In other words, you can get as close as you like to the limit 1 by going as far as necessary in the series.
> 
> This is basically the usual definition of infinite decimal fractions, though most will have other limits – only .999… and 1.000… have 1 as a limit.

Sigh… 😉

Once again the limit definition here of the sum of the infinite series .999… offers us no how or why, it simply says since we can get infinitely close to the answer we supposedly want, think is correct, then we will allow are selves to just say it is so. It may be so, but it is never proven. To some people maybe this rings true. To me and MANY others it does not at least NECCESSARILY ring true. I am not totally closed to the idea that they are equal. But I do not take it for granted as the limit theory does.

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**Author:** ![allotrope](https://avatars.discourse-cdn.com/v4/letter/a/edb3f5/32.png) [@allotrope](https://boards.straightdope.com/u/allotrope)\
**Post date:** [August 6, 2012, 9:39am UTC](https://boards.straightdope.com/t/999-1/27517/224 "2012-08-06T09:39:01Z")

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> [@erik150x](#):
>
> FV,
> 
> I was simply replying hopfully just infintesmialy less rediculously to Monimonika’s previous deomstration that he could add .999… + .999… + .999… + … ten times and show that it equaled 9.999…
> 
> Giles your point is well taken. Niether my nor Monimonika’s version makes perfect sense. It is not a simple issue to deal with. But if the proof was as easy as Monimonika makes it out to be, it would have been made long ago. Poeple accused me of thinking I am smarter than all the great mathematicians in history. I certainly do not even compair my self.. There is no proof I know of in a very elemantry manner describing a proof for .999… = 1. If there was it would be all over the interent. Everyone i have seen uses Limit based calculus. If your going to use limit based calculus then you have already made the assumption they are equal, so what’s the point of even considering the question.

You’re avoiding my question. What about AT infinity

If you want us to assume an infinitely long decimal representation of a number. That’s fine. But then you want to turn around and say we can’t consider what happens at infinity.

I’m sorry, but that hardly seem fair, now does it?

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<div class="post-metadata">

**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\
**Post date:** [August 6, 2012, 9:42am UTC](https://boards.straightdope.com/t/999-1/27517/225 "2012-08-06T09:42:05Z")

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Even after infinity 9s I tend to think we are only infinitely close to 1.

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**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\
**Post date:** [August 6, 2012, 9:43am UTC](https://boards.straightdope.com/t/999-1/27517/226 "2012-08-06T09:43:39Z")

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AT infinity the difference is .000000…1 or if you prefer 1/infinity or if you prefer somethign we don’t understand, but I don’t see the difference being Zero.

And why should it be…

In my mind i see this:

1.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000…0  
0.99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999…9

How ever far you want to go, even infinity you still have a Zero above and a 9 below.

Hence the difference .000…1 or whatever.

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**Author:** ![allotrope](https://avatars.discourse-cdn.com/v4/letter/a/edb3f5/32.png) [@allotrope](https://boards.straightdope.com/u/allotrope)\
**Post date:** [August 6, 2012, 9:45am UTC](https://boards.straightdope.com/t/999-1/27517/227 "2012-08-06T09:45:38Z")

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> [@erik150x](#):
>
> AT infinity the difference is .000000…1 or if you prefer 1/infinity or if you prefer somethign we don’t understand, but I don’t see the difference being Zero.

Then I would submit that you don’t understand the concept of infinity.

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<div class="post-metadata">

**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\
**Post date:** [August 6, 2012, 9:52am UTC](https://boards.straightdope.com/t/999-1/27517/228 "2012-08-06T09:52:32Z")

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> [@allotrope](#):
>
> Then I would submit that you don’t understand the concept of infinity.

Not entirely… no. Do you? Could you explain please?

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**Author:** ![allotrope](https://avatars.discourse-cdn.com/v4/letter/a/edb3f5/32.png) [@allotrope](https://boards.straightdope.com/u/allotrope)\
**Post date:** [August 6, 2012, 9:56am UTC](https://boards.straightdope.com/t/999-1/27517/229 "2012-08-06T09:56:01Z")

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> [@erik150x](#):
>
> Not entirely… no. Do you? Could you explain please?

Certainly.

You want to go from point A to point B. I tell you that you can go, but only if you move half the distance at a time.

So we’ll call the full distance AB. First you go 1/2 of AB. Then you go 1/2 of 1/2 of AB and so on.

Will you ever reach point B?

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<div class="post-metadata">

**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\
**Post date:** [August 6, 2012, 9:59am UTC](https://boards.straightdope.com/t/999-1/27517/230 "2012-08-06T09:59:29Z")

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Let me be more precise because there are differnet levels of infinity…

in this instance how is it that the series .999… is less than one before we get to infinity, but at infinity it is = 1. Is is because the difference is infinitely small? And you think 1/infinity = 0?

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<div class="post-metadata">

**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\
**Post date:** [August 6, 2012, 10:03am UTC](https://boards.straightdope.com/t/999-1/27517/231 "2012-08-06T10:03:18Z")

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> [@allotrope](#):
>
> Certainly.
> 
> You want to go from point A to point B. I tell you that you can go, but only if you move half the distance at a time.
> 
> So we’ll call the full distance AB. First you go 1/2 of AB. Then you go 1/2 of 1/2 of AB and so on.
> 
> Will you ever reach point B?

Your talking about physical reality, yes. But I can’t explain how can you? Wihtout just making the assumption made in the limit defintion?

The pardox is essentially the same as Zeno’s. Can you explain how you got to point B?

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<div class="post-metadata">

**Author:** ![allotrope](https://avatars.discourse-cdn.com/v4/letter/a/edb3f5/32.png) [@allotrope](https://boards.straightdope.com/u/allotrope)\
**Post date:** [August 6, 2012, 10:03am UTC](https://boards.straightdope.com/t/999-1/27517/232 "2012-08-06T10:03:50Z")

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Infinity is just a construct. It doesn’t matter what you apply it to - infinitely large, small, long, tall, etc.

So no, there aren’t different levels.

It turns out that some infinities are larger than others, but that’s a topic for another day.

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<div class="post-metadata">

**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\
**Post date:** [August 6, 2012, 10:07am UTC](https://boards.straightdope.com/t/999-1/27517/233 "2012-08-06T10:07:03Z")

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> [@allotrope](#):
>
> Infinity is just a construct. It doesn’t matter what you apply it to - infinitely large, small, long, tall, etc.
> 
> So no, there aren’t different levels.
> 
> It turns out that some infinities are larger than others, but that’s a topic for another day.

So tell me how you got to point B. I have no doubt you can, but how?

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<div class="post-metadata">

**Author:** ![allotrope](https://avatars.discourse-cdn.com/v4/letter/a/edb3f5/32.png) [@allotrope](https://boards.straightdope.com/u/allotrope)\
**Post date:** [August 6, 2012, 10:07am UTC](https://boards.straightdope.com/t/999-1/27517/234 "2012-08-06T10:07:20Z")

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> [@erik150x](#):
>
> Your talking about physical reality, yes. But I can’t explain how can you? Wihtout just making the assumption made in the limit defintion?
> 
> The pardox is essentially the same as Zeno’s. Can you explain how you got to point B?

You asked me to explain infinity to you. That’s what I tried to do.

The answer was, no you would never reach B, except AT infinity - i.e., after you had divided the distance an infinite number of times.

Xeno’s paradox was a bit sophistic since it falsely assumed that you could only move half the distance at a time, which obviously isn’t how we tend to move about.

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<div class="post-metadata">

**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\
**Post date:** [August 6, 2012, 10:08am UTC](https://boards.straightdope.com/t/999-1/27517/235 "2012-08-06T10:08:26Z")

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> [@allotrope](#):
>
> Infinity is just a construct. It doesn’t matter what you apply it to - infinitely large, small, long, tall, etc.
> 
> So no, there aren’t different levels.
> 
> It turns out that some infinities are larger than others, but that’s a topic for another day.

Not different levels, just larger ones… ok. I guess you can define that distinction.

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<div class="post-metadata">

**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:** [August 6, 2012, 10:08am UTC](https://boards.straightdope.com/t/999-1/27517/236 "2012-08-06T10:08:50Z")

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This has probably been tried, but: between any two distinct reals, there lies another; in fact, there even always lies a rational number. So if you claim 0.999… does not equal 1, then you should be able to exhibit a number of the form m/n which is both larger than 0.999… and smaller than 1. But it’s easy to see that no such number exists; hence, the two can’t be distinct reals. So they are equal.

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<div class="post-metadata">

**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\
**Post date:** [August 6, 2012, 10:10am UTC](https://boards.straightdope.com/t/999-1/27517/237 "2012-08-06T10:10:41Z")

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> [@allotrope](#):
>
> You asked me to explain infinity to you. That’s what I tried to do.
> 
> The answer was, no you would never reach B, except AT infinity - i.e., after you had divided the distance an infinite number of times.
> 
> Xeno’s paradox was a bit sophistic since it falsely assumed that you could only move half the distance at a time, which obviously isn’t how we tend to move about.

You made no definition or explination of infinity, you simply said what happens in your example fo going from point A ot B. I never siad you don’t reach point B. You did. Nor did you explain how you got there. I guess your explaination is it just is, it just happens. If that is satifactory to you, we have nothign to discuss really.

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<div class="post-metadata">

**Author:** ![allotrope](https://avatars.discourse-cdn.com/v4/letter/a/edb3f5/32.png) [@allotrope](https://boards.straightdope.com/u/allotrope)\
**Post date:** [August 6, 2012, 10:11am UTC](https://boards.straightdope.com/t/999-1/27517/238 "2012-08-06T10:11:00Z")

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> [@erik150x](#):
>
> Not different levels, just larger ones… ok. I guess you can define that distinction.

It’s not in any way relevant to this discussion or the distinction you were trying to make.

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<div class="post-metadata">

**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\
**Post date:** [August 6, 2012, 10:13am UTC](https://boards.straightdope.com/t/999-1/27517/239 "2012-08-06T10:13:58Z")

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> [@Half\_Man\_Half\_Wit](#):
>
> This has probably been tried, but: between any two distinct reals, there lies another; in fact, there even always lies a rational number. So if you claim 0.999… does not equal 1, then you should be able to exhibit a number of the form m/n which is both larger than 0.999… and smaller than 1. But it’s easy to see that no such number exists; hence, the two can’t be distinct reals. So they are equal.

It has been brought up and a valid argument to make. But the diefinition is imposed artifially if you ask by the definition we place in reals using limits. It doesn;t have to be that way. Look at the natural numbers. You don’t say because there are no numbers between 2 and 3, that they are the same number do you? Why must this be for reals… except that we impose this via the limit definiton of real numbers.

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<div class="post-metadata">

**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\
**Post date:** [August 6, 2012, 10:15am UTC](https://boards.straightdope.com/t/999-1/27517/240 "2012-08-06T10:15:03Z")

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> [@allotrope](#):
>
> It’s not in any way relevant to this discussion or the distinction you were trying to make.

How do you know what distinciton I was trying to make?

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