# Why doesn't .9999~ = 1?

**URL:** <https://boards.straightdope.com/t/why-doesnt-9999-1/179019>\
**Category:** Factual Questions\
**Created:** [June 1, 2003, 11:12pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019 "2003-06-01T23:12:58Z")\
**Posts on this page:** 20\
**Page:** 3

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**Author:** ![Orbifold](https://avatars.discourse-cdn.com/v4/letter/o/779978/32.png) [@Orbifold](https://boards.straightdope.com/u/Orbifold)\
**Post date:** [June 2, 2003, 1:56am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/41 "2003-06-02T01:56:28Z")

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> [@](#):
>
> \*Originally posted by x-ray vision \*  
> \*\*That’s logical? \*\*

**Q.E.D.** has already expounded on this a bit, but let me elaborate as well.

Given any two distinct real numbers, there is always a third real number lying strictly between the first two. Between 0 and 1, there is 1/2. Between 1 and 1/2, there is 3/4. Between x and y, there is (x+y)/2.

By the same token, if there is no such third number then the first two numbers can’t be distinct. There’s no number x such that 1 \< x \< 1, which is why 1 and 1 are the same number.

Now, what about 0.999… and 1? I claim there is no number between them. Indeed, suppose 0.999… \< x \< 1; I claim this leads to a contradiction.

For starters, it must be true that 0 \< x-0.999… \< 1-0.999… . But 1-0.999… is clearly less than 1-0.9=0.1. It’s also less than 1-0.99=0.01, 1-0.999=0.001, 1-0.9999=0.0001, and so on. In fact, given any positive number y I can show that 1-0.999… is less than y by writing out enough 9’s. But somehow 1-0.999… _isn’t_ less than x-0.999…, even though x-0.999… is a positive number. This is the contradiction.

So there does _not_ exist any such number x.

The same thing happens with the sequence 0.1, 0.01, 0.001, … . No term in the sequence is 0. But the limit _must_ be zero. After all, the limit certainly isn’t negative. And the limit can’t be postive either, for if it were then it would be smaller than 0.000…00001 for some number of zeroes, and how can the limit of a decreasing sequence be smaller than any of the terms in the sequence? Zero is the only possibility remaining.

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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:** [June 2, 2003, 1:57am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/42 "2003-06-02T01:57:37Z")

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**X-Ray Vision** , the whole point behind infinite arguments is that is you go on **forever** you **do** get 0.

These threads always come down to this one basic factor. Either you get the idea of infinity and understand this equality or you don’t. What’s ironic is that your argument - that we keep getting smaller and smaller but never reach zero - is arguing for an infinitesimal. But infinitesimals and limits are incompatible ideas in our arithmetic.

The limit as .999999… goes to infinity is 1.

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**Author:** ![3\_14159265358979323846](https://avatars.discourse-cdn.com/v4/letter/3/eada6e/32.png) [@3\_14159265358979323846](https://boards.straightdope.com/u/3_14159265358979323846)\
**Post date:** [June 2, 2003, 2:03am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/43 "2003-06-02T02:03:26Z")

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> [@](#):
>
> \*Originally posted by x-ray vision \*  
> \*\*1 - .9 = 0.1
> 
> 1- .99 = 0.01
> 
> 1 - .999 = 0.001
> 
> \*\*I can go on forever \*\*and never get 0, however, 1 -1 =0.
> 
> My conclusion is 1 cannot be the same as .999… \*\*

No, you _can’t_. That’s just it. You’ll die, eventually.  
0.999… has an _infinite_ number of 9s.  
Using .9, or .99, or .999 is an _approximation_, which is why you don’t get exactly zero. But the approximation gets closer by a factor of 10 for each digit you add. So, what happens when you use an _infinite_ number of digits?

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**Author:** ![Donut](https://avatars.discourse-cdn.com/v4/letter/d/e19adc/32.png) [@Donut](https://boards.straightdope.com/u/Donut)\
**Post date:** [June 2, 2003, 2:05am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/44 "2003-06-02T02:05:54Z")

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> [@](#):
>
> My conclusion is 1 cannot be the same as .999…

Your conclusion is based on an intuitive understanding of what 0.999… means, an intuitive understanding which is simply _wrong_. 0.999… is generally understood to mean the limit of the sequence 0.9, 0.99, 0.999, 0.9999, … and so on. That limit is 1, as inarguably as any mathematical fact, no matter how strange it seems to you.

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**Author:** ![3\_14159265358979323846](https://avatars.discourse-cdn.com/v4/letter/3/eada6e/32.png) [@3\_14159265358979323846](https://boards.straightdope.com/u/3_14159265358979323846)\
**Post date:** [June 2, 2003, 2:31am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/45 "2003-06-02T02:31:42Z")

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Let me expand on my previous response.

.9 differs from 1 by (1-.9)/1, which is .1 or 1/10.  
.99 differs from 1 by (1-.99)/1, which is .01 or 1/100.  
.999 differs from 1 by (1-.999)/1, which is .001 or 1/1000.

Let n be the number of 9s you are using in the approximation. Thus, the difference between 1 and .9999… is 1/(10[sup]n[/sup]).

For n=infinity, the difference is 1/(10[sup]infinity[/sup]) or 1/infinity.

Since 1/infinity equals zero\*, the difference between .999… and 1 is zero. Saying that the difference between two numbers is zero is equivalent to saying that they are the same number. QED.

\*Yes, I know that it isn’t technically correct to say that 1/inf _equals_ 0, but rather lim[sub]x–\>inf[/sub] 1/x = 0. Small technicality.

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**Author:** ![x-ray\_vision](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/x-ray_vision/32/351_2.png) [@x-ray\_vision](https://boards.straightdope.com/u/x-ray_vision)\
**Post date:** [June 2, 2003, 2:50am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/46 "2003-06-02T02:50:30Z")

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> [@](#):
>
> _Originally posted by_ **Orbifold**
> 
> In fact, given any positive number y I can show that 1-0.999… is less than y by writing out enough 9’s. But somehow 1-0.999… isn’t less than x-0.999…, even though x-0.999… is a positive number. This is the contradiction.

That’s a contradiction I just can’t grasp.

> [@](#):
>
> _Originally posted by_ **3\_14159265358979323846**
> 
> No, you can’t. That’s just it. You’ll die, eventually.

C’mon now.

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**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\
**Post date:** [June 2, 2003, 2:55am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/47 "2003-06-02T02:55:33Z")

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Let’s leave infinite arithmetic out of this, and not cloud the issue with technicalities.

Others have already said most of what I would, and I’ve been there and done that on this topic, so let me just add that the fact that .9… = 1 is a direct consequence of the most important property of the real numbers, and if they’re not equal, **everything** falls apart.

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**Author:** ![InLikeFlynn](https://avatars.discourse-cdn.com/v4/letter/i/f19dbf/32.png) [@InLikeFlynn](https://boards.straightdope.com/u/InLikeFlynn)\
**Post date:** [June 2, 2003, 3:07am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/48 "2003-06-02T03:07:58Z")

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Q.E.D. your wrong again!!  
If you went .999 of the distance to 1.0, then guess what?  
You never got there!!  
Terribly obviously wrong again!!  
If you went .99999999999~~ way around the world you never reached your starting point so .99999~ will never equal 1!!  
I don’t care who tells you it does!!  
WRONG!!

got your back again x-ray!!

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<div class="post-metadata">

**Author:** ![Q.E.D](https://avatars.discourse-cdn.com/v4/letter/q/51bf81/32.png) [@Q.E.D](https://boards.straightdope.com/u/Q.E.D)\
**Post date:** [June 2, 2003, 3:18am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/49 "2003-06-02T03:18:30Z")

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> [@](#):
>
> \*Originally posted by InLikeFlynn \*  
> \*\*Q.E.D. your wrong again!!  
> If you went .999 of the distance to 1.0, then guess what?  
> You never got there!!  
> Terribly obviously wrong again!!  
> If you went .99999999999~~ way around the world you never reached your starting point so .99999~ will never equal 1!!  
> I don’t care who tells you it does!!  
> WRONG!!
> 
> got your back again x-ray!! \*\*

Well, I tell you what. If you can find ONE source with a correct proof that .9999… != 1, i’ll eat my computer.

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**Author:** ![dejahma](https://avatars.discourse-cdn.com/v4/letter/d/839c29/32.png) [@dejahma](https://boards.straightdope.com/u/dejahma)\
**Post date:** [June 2, 2003, 3:28am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/50 "2003-06-02T03:28:13Z")

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To all the mathematicians out there give it up; those who don’t understand limits and calculus and choose not to study will never understand. To all of the math challenged out there let’s discuss something a bit more on your level: How many angels can dance on the head of a pin?

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**Author:** ![5-HT](https://avatars.discourse-cdn.com/v4/letter/5/f19dbf/32.png) [@5-HT](https://boards.straightdope.com/u/5-HT)\
**Post date:** [June 2, 2003, 3:39am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/51 "2003-06-02T03:39:12Z")

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> [@](#):
>
> \*Originally posted by RealityChuck \*  
> \*\*We can always bring out the formula for those not bothering with the other threads. We use the basic method for turning a repeating decimal into a fraction.
> 
> let x = 0.9999…  
> then 10x = 9.99999…
> 
> Now we subtract  
> 10x - x = 9.99999… - 0.99999…
> 
> Solving both sides  
> 9x = 9.00000
> 
> Reducing  
> x = 9/9 = 1
> 
> Therefore 0.9999… = 1
> 
> No calculus involved. \*\*

my head just exploded

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**Author:** ![x-ray\_vision](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/x-ray_vision/32/351_2.png) [@x-ray\_vision](https://boards.straightdope.com/u/x-ray_vision)\
**Post date:** [June 2, 2003, 3:40am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/52 "2003-06-02T03:40:10Z")

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I’m tryin’ real hard to be open minded about this because I respect the intelligence of you guys, but some of the rationale you guys are giving I just can’t jive with.  
Allow me to repeat the quote from the second web-page I quoted from earlier.

> [@](#):
>
> The numbers in the “Answer” column do indeed get closer and closer to some number, namely 1. **The fact that they never get there is irrelevant**. 1 is by definition the exact answer to the infinite addition problem.
> 
> So in conclusion, Point-Nine-Repeating is indeed equal to one. In order to see it, we needed to know how to add up infinitely many small numbers. By the way, this knowledge itself is probably more important that the little fact that .999…=1.

It seems the writer is contradicting himself. First, saying that they never get there, then saying 1 is the exact answer to the infinite addition problem.

I know **Q.E.D.** tried to explain this by writing that the writer is talking about a finite number of 9s, but how is that so? We’re talking about an infinite number of 9s.

I’m not looking for any more explanations, but I am going to delve into this further on my own. I’ll get back to you all when it sinks in, or I have proof I was right all along. 😉

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**Author:** ![x-ray\_vision](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/x-ray_vision/32/351_2.png) [@x-ray\_vision](https://boards.straightdope.com/u/x-ray_vision)\
**Post date:** [June 2, 2003, 3:45am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/53 "2003-06-02T03:45:32Z")

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Ok **dejahma** , I’m no math whiz, but if you tell me how big each angel’s feet are, how close each angel can stand next to another, and how big the head of the pin is, I bet I can figure it out.

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**Author:** ![Q.E.D](https://avatars.discourse-cdn.com/v4/letter/q/51bf81/32.png) [@Q.E.D](https://boards.straightdope.com/u/Q.E.D)\
**Post date:** [June 2, 2003, 3:47am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/54 "2003-06-02T03:47:54Z")

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It’s making that transition from understanding what a _finite_ number of 9s after the decimal point means (which clearly you do, **x-ray vision** , to understanding what an _infinite_ series means. **RealityChuck** ’s algebraic proof is pretty good towards making this step. The hard part to accept about it is that even though the decimal point has been shifted, the _number_\* of 9s to the right of the decimal point _hasn’t changed_. It’s still infinite, since the expression _infinity - 1_ is meaningless.

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**Author:** ![Speaker\_for\_the\_Dead](https://avatars.discourse-cdn.com/v4/letter/s/c77e96/32.png) [@Speaker\_for\_the\_Dead](https://boards.straightdope.com/u/Speaker_for_the_Dead)\
**Post date:** [June 2, 2003, 3:48am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/55 "2003-06-02T03:48:22Z")

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Why, .999… can!

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**Author:** ![desdinova](https://avatars.discourse-cdn.com/v4/letter/d/8baadc/32.png) [@desdinova](https://boards.straightdope.com/u/desdinova)\
**Post date:** [June 2, 2003, 3:50am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/56 "2003-06-02T03:50:00Z")

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Tell ya what, **x-ray** , let me quote you back the same cite you posted, except with my own bolding added:

> [@](#):
>
> The numbers in the “Answer” column do indeed get closer and closer to some number, namely 1. The fact that they never get there is irrelevant. 1 is by definition the exact answer to the infinite addition problem.
> 
> **So in conclusion, Point-Nine-Repeating is indeed equal to one.** In order to see it, we needed to know how to add up infinitely many small numbers. By the way, this knowledge itself is probably more important that the little fact that .999…=1.

Like **Q.E.D.** , I’m willing to do any number of disgusting things if anyone can provide an actual _mathematical_ proof that 0.999… != 1.

But, instead of digging around for, well, forever, you might find it faster to just read the previous threads and find out why you’re wrong.

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<div class="post-metadata">

**Author:** ![x-ray\_vision](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/x-ray_vision/32/351_2.png) [@x-ray\_vision](https://boards.straightdope.com/u/x-ray_vision)\
**Post date:** [June 2, 2003, 4:01am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/57 "2003-06-02T04:01:28Z")

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I didn’t need different bolding **desdinova** , my whole point was the writer is saying things that contradict one another.

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**Author:** ![desdinova](https://avatars.discourse-cdn.com/v4/letter/d/8baadc/32.png) [@desdinova](https://boards.straightdope.com/u/desdinova)\
**Post date:** [June 2, 2003, 4:11am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/58 "2003-06-02T04:11:17Z")

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> [@](#):
>
> \*Originally posted by x-ray vision \*  
> \*\*I didn’t need different bolding **desdinova** , my whole point was the writer is saying things that contradict one another. \*\*

So what? My point is:

A.) You’re wrong.  
B.) We’ve explained why this is over and over again, as the previous threads cited in the second and third posts demonstrate.  
C.) The writer doesn’t say anything that contradicts anything. **By definition, the fact that “they never get there” is irrelevant.** If you’d like to come up with some new definitions to use in mathematics, feel free to.

Let me put it to you this way: if 0.999…\<1, then what number comes between them?

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<div class="post-metadata">

**Author:** ![Q.E.D](https://avatars.discourse-cdn.com/v4/letter/q/51bf81/32.png) [@Q.E.D](https://boards.straightdope.com/u/Q.E.D)\
**Post date:** [June 2, 2003, 4:17am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/59 "2003-06-02T04:17:45Z")

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> [@](#):
>
> \*Originally posted by x-ray vision \*  
> \*\*I didn’t need different bolding **desdinova** , my whole point was the writer is saying things that contradict one another. \*\*

No, he’s not. He’s saying that successive iterations of the sum .9 + .09 +.009 +.0009… will never equal 1 with a _finite_ number of iterations. And it won’t. But with an _infinite_ number of iterations, it will _exactly_ equal 1. This is the difficult part of this concept, mainly because there is no smooth, simple transition from finite to infinite. There is no end to the series I just outlined, but it’s limit is clearly 1. It cannot be less than 1, because any finite number x of iterations will always be exceeded by x + 1 iterations. So that .99999999999999999999 can be exeeded by .999999999999999999999 and so on. It cannot be greater than 1 because that would be silly. So, if it’s not less than and not greater than, it _must_ be equal to. There is no other choice.

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**Author:** ![Flymaster](https://avatars.discourse-cdn.com/v4/letter/f/c4cdca/32.png) [@Flymaster](https://boards.straightdope.com/u/Flymaster)\
**Post date:** [June 2, 2003, 4:38am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/60 "2003-06-02T04:38:22Z")

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I think the biggest thing to realize here is this: Infinite does NOT mean “a lot”. It does not mean “more than you can imagine.” It does not mean “Three lifetimes worth.” It means that it simply DOESN’T end.

You know how you start subtraction on right hand side of a number, and then carry to the left? With .999… there IS no right side. None. Because no matter where you decide the right side is, there’s infinitely more 9s to the right. Not a lot more 9s. Not more 9s than you can imagine. An infinite number. The same number as are to the right of the decimal point.

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