# Does .99(repeating) = 1 **URL:** \ **Category:** Factual Questions\ **Created:** [April 24, 2003, 11:03am UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821 "2003-04-24T11:03:48Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![GSV\_Consolation\_of\_Dreams](https://avatars.discourse-cdn.com/v4/letter/g/f1d935/32.png) [@GSV\_Consolation\_of\_Dreams](https://boards.straightdope.com/u/GSV_Consolation_of_Dreams)\ **Post date:** [April 24, 2003, 12:15pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/21 "2003-04-24T12:15:07Z")
**Mort Furd** : Huh? Let x=.333… 10x=3.333… 10x-x=3.333… - .333… 9x=3 x = 3/9 = 1/3 ---
**Author:** ![Mort\_Furd](https://avatars.discourse-cdn.com/v4/letter/m/8e8cbc/32.png) [@Mort\_Furd](https://boards.straightdope.com/u/Mort_Furd)\ **Post date:** [April 24, 2003, 12:18pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/22 "2003-04-24T12:18:34Z")
> [@](#): > > \*Originally posted by atarian \* > \*\ ***Mort Furd** : Huh? > > Let x=.333… > > 10x=3.333… > > 10x-x=3.333… - .333… > > 9x=3 > > x = 3/9 = 1/3 \*\* Erp. Shuts face and slinks off to hide in a cave. ---
**Author:** ![Derleth](https://avatars.discourse-cdn.com/v4/letter/d/b9e5f3/32.png) [@Derleth](https://boards.straightdope.com/u/Derleth)\ **Post date:** [April 24, 2003, 12:19pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/23 "2003-04-24T12:19:08Z")
**atarian** did with mathematical rigor what I alluded to in prose: There is a nonzero difference between 0.333… and 1.0, therefore the two numbers are not equal. No such difference exists between 0.999… and 1.0, as **Libertarian** proved. ---
**Author:** ![GSV\_Consolation\_of\_Dreams](https://avatars.discourse-cdn.com/v4/letter/g/f1d935/32.png) [@GSV\_Consolation\_of\_Dreams](https://boards.straightdope.com/u/GSV_Consolation_of_Dreams)\ **Post date:** [April 24, 2003, 12:19pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/24 "2003-04-24T12:19:09Z")
10x - x = 3x ??? ---
**Author:** ![GSV\_Consolation\_of\_Dreams](https://avatars.discourse-cdn.com/v4/letter/g/f1d935/32.png) [@GSV\_Consolation\_of\_Dreams](https://boards.straightdope.com/u/GSV_Consolation_of_Dreams)\ **Post date:** [April 24, 2003, 12:23pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/25 "2003-04-24T12:23:14Z")
Thanks **Derleth**. Don’t worry about it **Mort** , for some real mathematicl ignorance, do a google groups search for “Spaceman Driscoll” whom i referenced previously. He argues; apparently seriously, that -4 x -4 = -16. ---
**Author:** ![Mort\_Furd](https://avatars.discourse-cdn.com/v4/letter/m/8e8cbc/32.png) [@Mort\_Furd](https://boards.straightdope.com/u/Mort_Furd)\ **Post date:** [April 24, 2003, 12:31pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/26 "2003-04-24T12:31:54Z")
Not so much ignorance at work here as asleep at the switch. Stupidity is a horrible thing to see onself commit. ---
**Author:** ![lucwarm](https://avatars.discourse-cdn.com/v4/letter/l/e19adc/32.png) [@lucwarm](https://boards.straightdope.com/u/lucwarm)\ **Post date:** [April 24, 2003, 1:04pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/27 "2003-04-24T13:04:44Z")
Let x = 1 + 2 + 4 + 8 +16 . . . Then x-1 = 2 + 4 + 8 + 16 . . . and (x-1)/2 = 1 + 2 + 4 + 8 . . . so (x-1)/2 = x so x - 1 = 2x so -x = 1 and x = -1 so 1 + 2 + 4 + 8 + 16 . . . = -1 😃 ---
**Author:** ![GSV\_Consolation\_of\_Dreams](https://avatars.discourse-cdn.com/v4/letter/g/f1d935/32.png) [@GSV\_Consolation\_of\_Dreams](https://boards.straightdope.com/u/GSV_Consolation_of_Dreams)\ **Post date:** [April 24, 2003, 1:13pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/28 "2003-04-24T13:13:31Z")
Neat. Tricky but neat. I’m thinking the flaw lies in the difference between an infinite sequence tending to a finite vs. an infinity. I’ll leave it someone else to put it in purty math talk. ---
**Author:** ![Phage](https://avatars.discourse-cdn.com/v4/letter/p/a4c791/32.png) [@Phage](https://boards.straightdope.com/u/Phage)\ **Post date:** [April 24, 2003, 1:15pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/29 "2003-04-24T13:15:17Z")
The difference lies in theory, and how things get odd when you talk about infinity. The number .9999… approaches 1, but would never actually reach it. However, because it is infinitely close it is ok to call it 1 because you will probably never need to be infinitely accurate. It does not matter that there is no number between .9999… and 1, because .9999… is as close to 1 as a number less than 1 can be. ---
**Author:** ![CalMeacham](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/calmeacham/32/35_2.png) [@CalMeacham](https://boards.straightdope.com/u/CalMeacham)\ **Post date:** [April 24, 2003, 1:35pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/30 "2003-04-24T13:35:03Z")
> [@](#): > > The difference lies in theory, and how things get odd when you talk about infinity. The number .9999… approaches 1, but would never actually reach it. However, because it is infinitely close it is ok to call it 1 because you will probably never need to be infinitely accurate. It does not matter that there is no number between .9999… and 1, because .9999… is as close to 1 as a number less than 1 can be. I don’t think this is true. The proofs given above are correct, and the weird fact that 0.99999… = 1 is a quirk of the decimal system. Another way to look at it is that 1/3 = 0.333333… unambiguously – there’s no other way to represent 1/3 in our decimal system. And 3\* (1/3) = 1, so 3 \* (0.3333…) = 0.999…, so 0.99999… = 1.0 . ---
**Author:** ![nogginhead](https://avatars.discourse-cdn.com/v4/letter/n/5e9695/32.png) [@nogginhead](https://boards.straightdope.com/u/nogginhead)\ **Post date:** [April 24, 2003, 1:52pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/31 "2003-04-24T13:52:55Z")
> [@](#): > > \*Originally posted by CalMeacham \* > \*_I don’t think this is true. The proofs given above are correct, and the weird fact that 0.99999… = 1 is a quirk of the decimal system. Another way to look at it is that 1/3 = 0.333333… unambiguously – there’s no other way to represent 1/3 in our decimal system. And 3_ (1/3) = 1, so 3 \* (0.3333…) = 0.999…, so 0.99999… = 1.0 . \*\* I don’t think it’s a quirk of the decimal system. It has to do with the implied infinite series that we casually denote with the ‘…’ after the last 9. If we stopped at any number fo decimal places, we’d have a number less than 1, and we could find a number (actually an infinite number fo numbers) between that fixed-decimal-place doohickey and 1, just by adding decimal places. But the infinite series denoted by ‘…’ means that we never stop at a given number of decimal places. I would say that loosely this means that .999… is as close to 1 as it is possible to get, and that in any real application (if such a thing exists) it is close enought to 1 that you can use 1 in its place. ---
**Author:** ![sailor](https://avatars.discourse-cdn.com/v4/letter/s/a587f6/32.png) [@sailor](https://boards.straightdope.com/u/sailor)\ **Post date:** [April 24, 2003, 1:58pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/32 "2003-04-24T13:58:39Z")
> [@](#): > > \*Originally posted by Dolomite21 \* > Given that .111(repeating) = 1/9 > > .222(repeating) = 2/9 > .333(repeating) = 3/9 > .444(repeating) = 4/9 > > and on and on > > would 9/9 equal 1 and .99(repeating)? That is begging the question. IF (big operative term IF) for you .22222(repeating) =2/9 then yes, in the same measure, .9999(repeating) = 9/9 . But you have not proven that .22222(repeating) = 2/9 so it is not a given. And if it is a given then it is also a give that .9999 = 9/9 and it requires no proof. The whole premise and OP is flawed but if you want to play games you could say .99999~ = .44444~ + .55555~ and use that as proof. As I say, it does not stand scrutiny. ---
**Author:** ![Spiritus\_Mundi](https://avatars.discourse-cdn.com/v4/letter/s/c68b51/32.png) [@Spiritus\_Mundi](https://boards.straightdope.com/u/Spiritus_Mundi)\ **Post date:** [April 24, 2003, 1:58pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/33 "2003-04-24T13:58:51Z")
Nasty **lucwarm**. Pretending that the sum of one divergent series can be subtracted from the sum of another divergent series and leave a number behind. I bet you learned that in “Mean Math Tricks 101”. 😃 ---
**Author:** ![Spiritus\_Mundi](https://avatars.discourse-cdn.com/v4/letter/s/c68b51/32.png) [@Spiritus\_Mundi](https://boards.straightdope.com/u/Spiritus_Mundi)\ **Post date:** [April 24, 2003, 2:03pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/34 "2003-04-24T14:03:12Z")
Oh, and **Lib** , never let it be said that you do not learn from your mistakes, eh? 😉 ---
**Author:** ![Orbifold](https://avatars.discourse-cdn.com/v4/letter/o/779978/32.png) [@Orbifold](https://boards.straightdope.com/u/Orbifold)\ **Post date:** [April 24, 2003, 2:20pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/35 "2003-04-24T14:20:59Z")
> [@](#): > > \*Originally posted by lucwarm \* > Let x = 1 + 2 + 4 + 8 +16 . . . > > [stuff omitted] > > so > > 1 + 2 + 4 + 8 + 16 . . . = -1 > > 😃 I probably shouldn’t mention the 2-adic system of numbers where is this in fact a perfectly valid proof, and the limit of the infinite series 1+2+4+… is indeed -1, should I? (NOTE TO ANY LURKING WEB KOOKS: the 2-adic numbers are not the real numbers. They are an entirely separate number field used pretty much exclusively by number theorists. Don’t be a web kook! Thank you.) ---
**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:** [April 24, 2003, 2:44pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/36 "2003-04-24T14:44:46Z")
> [@](#): > > \*Originally posted by sailor \* > \*\*That is begging the question. IF (big operative term IF) for you .22222(repeating) =2/9 then yes, in the same measure, .9999(repeating) = 9/9 . But you have not proven that .22222(repeating) = 2/9 so it is not a given. \*\* It is not difficult to show by induction (IMHO, IANAM) that 2/9 = .22222 even though it was presented as a given in the referenced post. I think anyone that can do long division can be convinced of this. When I started reading this thread I was highly skeptical :dubious: . However, now I agree and I do think it’s a quirk of the decimal system. Suppose we do this in base 9, then the apparent paradox disappears. 1/9 = 0.1111…[sub]10[/sub] = 0.1[sub]9[/sub] 9/9 = 0.9999…[sub]10[/sub] = 1.0[sub]9[/sub] 1 = 1.0[sub]10[/sub] = 1.0[sub]9[/sub] Further, using base 9 you can create the same paradox for a different number: 1/8 = 0.1111…[sub]9[/sub] 3/8 = 0.3333…[sub]9[/sub] 4/8 = 0.4444…[sub]9[/sub] 8/8 = 0.8888…[sub]9[/sub] = 1.0[sub]9[/sub] In fact, this works in _any_ base where the divisor is one less than the radix. ---
**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:** [April 24, 2003, 2:45pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/37 "2003-04-24T14:45:56Z")
> [@](#): > > \*Originally posted by CookingWithGas \* > \*\*It is not difficult to show by induction (IMHO, IANAM) that 2/9 = .22222 \*\* Make that .2222… ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [April 24, 2003, 2:49pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/38 "2003-04-24T14:49:09Z")
For those who think that .9… is not equal to 1, please reconcile your thoughts with the axiom of completeness. It would be nice if you could use this version: “A monotonically increasing sequence converges to its least upper bound”. ---
**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:** [April 24, 2003, 2:51pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/39 "2003-04-24T14:51:36Z")
Here are two of the many threads where we discussed this subject to death: [http://boards.straightdope.com/sdmb/showthread.php?threadid=82064](http://boards.straightdope.com/sdmb/showthread.php?threadid=82064) [http://boards.straightdope.com/sdmb/showthread.php?threadid=15832](http://boards.straightdope.com/sdmb/showthread.php?threadid=15832) ---
**Author:** ![Dolomite21](https://avatars.discourse-cdn.com/v4/letter/d/e5b9ba/32.png) [@Dolomite21](https://boards.straightdope.com/u/Dolomite21)\ **Post date:** [April 24, 2003, 2:57pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/40 "2003-04-24T14:57:09Z")
Great…but _is_ it equal [Previous page](https://boards.straightdope.com/t/does-99-repeating-1/170821.md?page=1) [Next page](https://boards.straightdope.com/t/does-99-repeating-1/170821.md?page=3)