# Stupid math question

**URL:** <https://boards.straightdope.com/t/stupid-math-question/66283>\
**Category:** Factual Questions\
**Created:** [May 23, 2001, 12:50pm UTC](https://boards.straightdope.com/t/stupid-math-question/66283 "2001-05-23T12:50:42Z")\
**Posts on this page:** 12\
**Page:** 2

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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:** [May 24, 2001, 7:25pm UTC](https://boards.straightdope.com/t/stupid-math-question/66283/21 "2001-05-24T19:25:18Z")

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Thanks for the clarifications, **DrMatrix** and **curiousgeorgeordeadcat**. I e-mailed the professor in question, and am awaiting his response. I’ll let you know what he has to say.

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**Author:** ![C\_K\_Dexter\_Haven](https://avatars.discourse-cdn.com/v4/letter/c/b2d939/32.png) [@C\_K\_Dexter\_Haven](https://boards.straightdope.com/u/C_K_Dexter_Haven)\
**Post date:** [May 24, 2001, 7:50pm UTC](https://boards.straightdope.com/t/stupid-math-question/66283/22 "2001-05-24T19:50:10Z")

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Worth the reminder that in the real world, there is no such thing as an “infinite” decimal. After a few hundred thousand decimal places or so, you have quantities way far less than the width of an electron, way too small to be measured. From a real world perspective, if we think of .3333…[100,000 decimal places]…33 , it’s not different from .3333…[100,000 decimal places]…34  
since the difference cannot be measured. Thus, in the real world, there will always be a slight difference between .33333 and 1/3.

When we’re dealing with math, however, we have an abstract system, where we can indeed have infinite non-repeating decimals. In the abstract world of math, .3333… is identical to 1/3.

And, as noted, there is the simple but elegant proof already cited that .9999… = 1, namely:

X = 0.99999…  
10X = 9.99999… and subtract the equations, obtaining:

* * *

9X = 9  
X = 1 QED. But a six year old ain’t gonna get that, neither.

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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:** [May 25, 2001, 1:27am UTC](https://boards.straightdope.com/t/stupid-math-question/66283/23 "2001-05-25T01:27:04Z")

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OK, so apparently I was completely wrong on this one. My professor simply intended his comment to mean that I hadn’t **proved** that every real number has a decimal representation.

But part of being wrong is learning things, so I think I’m going to start another thread with a question I’ve got out of all this. Looking forward to the replies…and thanks for your patience with someone who thinks he knows more than he does.

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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:** [May 25, 2001, 1:45am UTC](https://boards.straightdope.com/t/stupid-math-question/66283/24 "2001-05-25T01:45:33Z")

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> [@](#):
>
> \*Originally posted by ultrafilter \*  
> **…so I think I’m going to start another thread with a question I’ve got…**

Nevermind, figured out the answer myself.

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**Author:** ![Cardinal](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cardinal/32/4000_2.png) [@Cardinal](https://boards.straightdope.com/u/Cardinal)\
**Post date:** [May 25, 2001, 4:12am UTC](https://boards.straightdope.com/t/stupid-math-question/66283/25 "2001-05-25T04:12:40Z")

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I _think_ the answer to ?Authority’s point is that 3/5 = .6 only in that everyone agrees to drop the zeros, as they don’t affect calculations in any way.

If you start dropping 3s from the end of 1/3 = .33333… **ever** , you’re only approximating the actual value of 1/3. It does matter that you stopped writing the additional threes after the decimal. You’re never, ever truly done.

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**Author:** ![Kalt](https://avatars.discourse-cdn.com/v4/letter/k/b9e5f3/32.png) [@Kalt](https://boards.straightdope.com/u/Kalt)\
**Post date:** [May 25, 2001, 5:12am UTC](https://boards.straightdope.com/t/stupid-math-question/66283/26 "2001-05-25T05:12:22Z")

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why did everyone ignore DynoSaur’s post… he proved the answer. Or… should i say his proof shows the answer. Or… his proof is the answer. Blah… his proof proves that 0.999… = 1

yeah, that’s what i meant.

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**Author:** ![sethdallob](https://avatars.discourse-cdn.com/v4/letter/s/e36b37/32.png) [@sethdallob](https://boards.straightdope.com/u/sethdallob)\
**Post date:** [May 25, 2001, 8:12am UTC](https://boards.straightdope.com/t/stupid-math-question/66283/27 "2001-05-25T08:12:03Z")

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[partial hijack]  
I don’t have a PhD in math, but I’m still a little troubled…

If we assume that .9999… = 1, is 1/.99999 = 1 or \> 1?

According to the prior logic, it should be = 1. But simply saying that because there is no number between them doesn’t make them equal (does it?). I can’t seem to mesh the two ideas.

[/partial hijack]

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**Author:** ![Ms2001](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ms2001/32/18463_2.png) [@Ms2001](https://boards.straightdope.com/u/Ms2001)\
**Post date:** [May 25, 2001, 10:00am UTC](https://boards.straightdope.com/t/stupid-math-question/66283/28 "2001-05-25T10:00:27Z")

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Between any two different numbers, there are always more numbers to be found. Between 1.0 and 2.0 is 1.5, between 0.0001 and 0.0002 is 0.00015, between 0.0000…\<insert one million zeros\>…1 and 0.0000…\<insert one million zeros\>…2, well, you get the point.

So if there are no numbers in between A and B, it follows that A and B are the same number. 1/0.9999… is 1 because 1 and 0.9999… are the same number.

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**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\
**Post date:** [May 25, 2001, 7:51pm UTC](https://boards.straightdope.com/t/stupid-math-question/66283/29 "2001-05-25T19:51:39Z")

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Quoth **?authoriy** :

> [@](#):
>
> Chronos, I still don’t get it. What do you mean 1/3 is not equal to 0.333…?

I don’t mean that. I was referring to your earlier post, where you said

> [@](#):
>
> I dont undrestand why 1/3 is different than 0.333…

You don’t understand why they’re different, because they’re _not_ different. That’s what I said.

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**Author:** ![Race\_Bannon](https://avatars.discourse-cdn.com/v4/letter/r/5f8ce5/32.png) [@Race\_Bannon](https://boards.straightdope.com/u/Race_Bannon)\
**Post date:** [May 26, 2001, 1:19pm UTC](https://boards.straightdope.com/t/stupid-math-question/66283/30 "2001-05-26T13:19:25Z")

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> [@](#):
>
> \*Originally posted by C K Dexter Haven \*  
> \*\*Thus, in the real world, there will always be a slight difference between .33333 and 1/3.
> 
> When we’re dealing with math, however, we have an abstract system, where we can indeed have infinite non-repeating decimals. In the abstract world of math, .3333… is identical to 1/3.  
> \*\*

So, please tell us, where is the boundary between the “real world” numbers and the “abstract” numbers. Are real numbers “real world” numbers, or only the rationals? How about negative numbers?

I always think of all numbers as being abstract, in the sense that you can show me 2 apples or 3 cars or 1/2 acre of land, but you “really” can’t show me just a 2, 3, or 1/2.

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**Author:** ![sethdallob](https://avatars.discourse-cdn.com/v4/letter/s/e36b37/32.png) [@sethdallob](https://boards.straightdope.com/u/sethdallob)\
**Post date:** [May 26, 2001, 4:43pm UTC](https://boards.straightdope.com/t/stupid-math-question/66283/31 "2001-05-26T16:43:09Z")

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> [@](#):
>
> \*Originally posted by Mr2001 \*  
> \*\*Between any two different numbers, there are always more numbers to be found. Between 1.0 and 2.0 is 1.5, between 0.0001 and 0.0002 is 0.00015, between 0.0000…\<insert one million zeros\>…1 and 0.0000…\<insert one million zeros\>…2, well, you get the point.
> 
> So if there are no numbers in between A and B, it follows that A and B are the same number. 1/0.9999… is 1 because 1 and 0.9999… are the same number. \*\*

I’m sorry, but I’m still not quite there yet. For the sake of simplicity, let’s assume a world where only whole numbers exist. Using the above logic, 1=2 2=3 3=4 4=5 etc. because there are no numbers between them. We clearly know that 2 \< 3, so why can it not be said that .9999… \< 1? And if .9999… \< 1, how can they be equal to each other?

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**Author:** ![curiousgeorgeordeadcat](https://avatars.discourse-cdn.com/v4/letter/c/4af34b/32.png) [@curiousgeorgeordeadcat](https://boards.straightdope.com/u/curiousgeorgeordeadcat)\
**Post date:** [May 26, 2001, 5:25pm UTC](https://boards.straightdope.com/t/stupid-math-question/66283/32 "2001-05-26T17:25:32Z")

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sethdallob:  
Using the standard definition of real numbers, you can prove  
that any two distinct real numbers have another real number,  
distinct from them, between them. That is:  
If x and y are real numbers, and x\<y, then there is a real  
number z such that x\<z\<y.  
And thus if x and y are real numbers, and there is no real  
number z such that either x\<z\<y or y\<z\<x, then x=y.  
No one has claimed a similar theorem holds for the integers.  
The set of integers is not the same as the set of real  
numbers, and so the fact that a similar theorem does not  
hold for the integers yields no contradiction.

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