# 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:** 4
**Author:** ![FranticMad](https://avatars.discourse-cdn.com/v4/letter/f/c68b51/32.png) [@FranticMad](https://boards.straightdope.com/u/FranticMad)\ **Post date:** [April 25, 2003, 2:32am UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/61 "2003-04-25T02:32:46Z")
0.9999… is not a number in the same way that infinity is not a number. For any physical manifestation of it, 0.99999… will behave exactly the same as 1.0, yet, their mathematics can be different. _Frantic’s Law of infinitesimals:_ There is no difference between 0.9999… and 1.0, but they are not the same. (pun intended) ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [April 25, 2003, 2:44am UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/62 "2003-04-25T02:44:22Z")
**FranticMad** , could you give us an example of how the mathematics of .9999… and 1 are different? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [April 25, 2003, 2:53am UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/63 "2003-04-25T02:53:59Z")
Ooh boy, I can’t wait to hear this one. Didja know that there are no infinitesimals in the real number system? ---
**Author:** ![FranticMad](https://avatars.discourse-cdn.com/v4/letter/f/c68b51/32.png) [@FranticMad](https://boards.straightdope.com/u/FranticMad)\ **Post date:** [April 25, 2003, 3:12am UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/64 "2003-04-25T03:12:40Z")
> [@](#): > > \*Originally posted by Cabbage \* > \*\ ***FranticMad** , could you give us an example of how the mathematics of .9999… and 1 are different? \*\* No. Except that 1 is a real number, no question about it. But, .9999… is, um, not quite the same kind of number. Since there are no infinitesimals in the real number system, then 0.999… is not a real number since it must include the idea of an infinitesimal. Hence, they are not the same. I have a feeling that I’ve not answered your question, though I don’t know why. Could you be more specific? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [April 25, 2003, 3:29am UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/65 "2003-04-25T03:29:15Z")
> [@](#): > > \*Originally posted by FranticMad \* > \*\*No. > > Except that 1 is a real number, no question about it. > But, .9999… is, um, not quite the same kind of number. Since there are no infinitesimals in the real number system, then 0.999… is not a real number since it must include the idea of an infinitesimal. Hence, they are not the same. > > I have a feeling that I’ve not answered your question, though I don’t know why. Could you be more specific? \*\* .9… is a real number, in exactly the same way that 1.0… is. They’re both the limits of sequences of real numbers, and by the completeness of the real number system, they are real numbers. ---
**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:** [April 25, 2003, 3:35am UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/66 "2003-04-25T03:35:15Z")
So far, every person who understands math has said flatly, without equivocation, that 1 and .999999… are just two different representations of exactly the same number, just as 2/2 and 1/3 + 1/3+ 1/3 are. This happens to be completely correct. “But, .9999… is, um, not quite the same kind of number.” is not exactly a mathematical expression. Nor is it in any way correct. What it is about the concept of infinitely repeating numbers that you don’t understand? ---
**Author:** ![FranticMad](https://avatars.discourse-cdn.com/v4/letter/f/c68b51/32.png) [@FranticMad](https://boards.straightdope.com/u/FranticMad)\ **Post date:** [April 25, 2003, 3:40am UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/67 "2003-04-25T03:40:28Z")
> [@](#): > > \*Originally posted by ultrafilter \* > \*\*.9… is a real number, in exactly the same way that 1.0… is. They’re both the limits of sequences of real numbers, and by the completeness of the real number system, they are real numbers. \*\* Ah ha!! (says FranticMad, knowing that the ultrashark smells blood and is swimming fast to make the kill), but does the completeness of the real number system also specify that real numbers A, and B can be subtracted such that A - B = C where C is not a real number (i.e. an infinitesimal)?? (fish darts away from the shark, looking in vain for the rest of his school of fish which have disappeared for some reason) ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [April 25, 2003, 3:44am UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/68 "2003-04-25T03:44:15Z")
> [@](#): > > But, .9999… is, um, not quite the same kind of number. Since there are no infinitesimals in the real number system, then 0.999… is not a real number since it must include the idea of an infinitesimal. I don’t follow you here. Would you also claim, for example, that pi is not a real number? After all, it also has a never ending decimal expansion. Or perhaps you are claiming that the infinitesimal is introduced by the idea that, if .999… is a real number, we can subtract 1 - .999…, and in that way produce a real infinitesimal. Yes, that would be a problem if we could get a real infinitesimal that way. However, only you (and perhaps a few others) are claiming such an infinitesimal; the rest of us claim that 1 - .999… is exactly zero, not an infinitesimal. Also: > [@](#): > > Could you be more specific? Ahem. You said: > [@](#): > > For any physical manifestation of it, 0.99999… will behave exactly the same as 1.0, yet, their mathematics can be different. I was merely asking for an(y) example of how their mathematics can be different. Apparently it’s you who need to be more specific. ---
**Author:** ![FranticMad](https://avatars.discourse-cdn.com/v4/letter/f/c68b51/32.png) [@FranticMad](https://boards.straightdope.com/u/FranticMad)\ **Post date:** [April 25, 2003, 3:57am UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/69 "2003-04-25T03:57:21Z")
> [@](#): > > \*Originally posted by Exapno Mapcase \* > So far, every person who understands math has said flatly… Argumentum ad verecundium. A logical fallacy, which negates the validity of your argument. > [@](#): > > What it is about the concept of infinitely repeating numbers that you don’t understand? Implied argumentum ad hominem. Invalid. ---
**Author:** ![FranticMad](https://avatars.discourse-cdn.com/v4/letter/f/c68b51/32.png) [@FranticMad](https://boards.straightdope.com/u/FranticMad)\ **Post date:** [April 25, 2003, 4:08am UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/70 "2003-04-25T04:08:15Z")
> [@](#): > > \*Originally posted by Cabbage \* > \*\*I don’t follow you here. Would you also claim, for example, that pi is not a real number? After all, it also has a never ending decimal expansion. > \*\* I made no claims about pi. It was **ultrafilter** who said infinitesimals (not pi, BTW) are not part of the real number system. > [@](#): > > \*\* > Or perhaps you are claiming that the infinitesimal is introduced by the idea that, if .999… is a real number, we can subtract 1 - .999…, and in that way produce a real infinitesimal.\*\* No, in fact you support my point. My question was whether such a subtraction _can_ create a real number. If not, then how can 1.0 - 0/999 be a real number, i.e. 1? You are shadow boxing with your own shadow, not with me. > [@](#): > > \*\*I was merely asking for an(y) example of how their mathematics can be different. Apparently it’s you who need to be more specific. \*\* I posed a specific example and question to **ultrafilter** , who is more capable than you of arguing me into oblivion. Await his answer and learn how one truly destroys an imbecile, oh student. ---
**Author:** ![The\_Ryan](https://avatars.discourse-cdn.com/v4/letter/t/7feea3/32.png) [@The\_Ryan](https://boards.straightdope.com/u/The_Ryan)\ **Post date:** [April 25, 2003, 4:25am UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/71 "2003-04-25T04:25:15Z")
> [@](#): > > \*Originally posted by FranticMad \* > \*\*Argumentum ad verecundium. A logical fallacy, which negates the validity of your argument. > Implied argumentum ad hominem. Invalid. \*\* And that would be [Argumentum ad logicam](http://www.infidels.org/news/atheism/logic.html#logicam). Bippy the Beardless > [@](#): > > it can be seen that this set could have a countable infinite number of entities (0.9…7… would be a valid entity) each precisely equal to 1.0… 😕 If you have a set in which all of the elements are equal, how can there be an infinite number of elements? Isn’t there only one element? Aren’t you just creating an infinite number of names for the same thing? ultrafiller: > [@](#): > > “A [bounded] monotonically increasing sequence [of real numbers] converges to its least upper bound”. I’m a nitpickaholic. You can’t blame me; it’s a disease. Derleth: > [@](#): > > Sure. The number between 0.333… and 1.0 is 0.666… Therefore, 0.333… does not equal 1.0. As there is no number between 0.999… and 1.0, 1.0 is equal to 0.999… This proof is technically incomplete, as it relies on aspects of the real number ordering system which are implied but not stated. There are number systems in which there is no number between X and Y, but X !=Y.                                                                    \_ PS re putting line over numbers: how about this? .9 ---
**Author:** ![John\_Kentzel-Griffin](https://avatars.discourse-cdn.com/v4/letter/j/ccd318/32.png) [@John\_Kentzel-Griffin](https://boards.straightdope.com/u/John_Kentzel-Griffin)\ **Post date:** [April 25, 2003, 4:34am UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/72 "2003-04-25T04:34:56Z")
> [@](#): > > \*Originally posted by \*\*\*FranticMad \*\* > Ah ha!! (says FranticMad, knowing that the ultrashark smells blood and is swimming fast to make the kill), but does the completeness of the real number system also specify that real numbers A, and B can be subtracted such that > A - B = C > where C is not a real number (i.e. an infinitesimal)?? No. The real numbers are closed under subtraction. If A and B are real, then A-B is real also. For an explanation of Completeness see [Real Number Wikipedia](http://www.wikipedia.org/wiki/Real_number) (From That page under the topic Construction from the Rationals): “Notice that the sequence (0,0.9,0.99,0.999,0.9999,…) is equivalent to the sequence (1,1.0,1.00,1.000,1.0000,…); this shows that 0.9999… = 1.” Oh, and **FranticMad** , **Cabbage** can hold his own in a math discussion. You could possibly learn from him. This isn’t a debate. It’s about math facts. 0.999… is exactly equal to 1.000. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [April 25, 2003, 4:36am UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/73 "2003-04-25T04:36:56Z")
> [@](#): > > \*Originally posted by FranticMad \* > \*\*Ah ha!! (says FranticMad, knowing that the ultrashark smells blood and is swimming fast to make the kill), but does the completeness of the real number system also specify that real numbers A, and B can be subtracted such that > A - B = C > where C is not a real number (i.e. an infinitesimal)?? > > (fish darts away from the shark, looking in vain for the rest of his school of fish which have disappeared for some reason) \*\* No. That’s closure, which is something different. The real numbers are closed under addition, subtraction, multiplication, and division. Completeness means that every convergent sequence of real numbers converges to a real number. I trust it’s obvious that .9, .99, .999, … converges to .9… And because I know you’re going to ask, division by zero is not a counterexample. In order that the real numbers not be closed under division, you would have to find a pair r, s such that r and s are real numbers, r/s is defined, and r/s is not a real number. **The Ryan** : 😛 ---
**Author:** ![John\_Kentzel-Griffin](https://avatars.discourse-cdn.com/v4/letter/j/ccd318/32.png) [@John\_Kentzel-Griffin](https://boards.straightdope.com/u/John_Kentzel-Griffin)\ **Post date:** [April 25, 2003, 4:53am UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/74 "2003-04-25T04:53:16Z")
**ultrafilter** I disagree with your definition of closure. The reals are not closed under division. S is closed under an operation we’ll call @, if for every r, s in S, r@s is in S. If we haven’t yet defined negative integers, would you say the non-negative integers are closed under subtraction? ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [April 25, 2003, 4:55am UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/75 "2003-04-25T04:55:20Z")
**FranticMad** : > [@](#): > > I made no claims about pi. Of course not, I never claimed you did. I brought up pi in the hopes that you might use that example to clarify your own point. I was wrong. > [@](#): > > No, in fact you support my point. My question was whether such a subtraction can create a real number. If not, then how can 1.0 - 0/999 be a real number, i.e. 1? Perhaps you have not read my posts closely if you believe I support your point. To answer your question, yes, the subtraction of two real numbers is _always_ a real number. I’m claiming that .999… is a real number (in fact, it is equal to 1). You, as far as I can tell, are claiming that .999… is not a real number. Furthermore, you seem to claim that since .999… is not a real number, 1 - .999… is also not a real number (and possibly an infinitesimal). Note, however, that this is all based on your original _unfounded_ claim that .999… is not a real number. Nothing you have said supports this claim; all you have presented is a string of reasoning which _follows_ from your claim. Would you care to begin _supporting_ your claim now? ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [April 25, 2003, 5:01am UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/76 "2003-04-25T05:01:47Z")
**ultrafilter** : > [@](#): > > Completeness means that every convergent sequence of real numbers converges to a real number. I believe you actually meant to say, “Completeness means that every _Cauchy_ sequence of real numbers converges to a real number.” Oh, and thanks, **DrMatrix**! 😃 ---
**Author:** ![The\_Ryan](https://avatars.discourse-cdn.com/v4/letter/t/7feea3/32.png) [@The\_Ryan](https://boards.straightdope.com/u/The_Ryan)\ **Post date:** [April 25, 2003, 7:01am UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/77 "2003-04-25T07:01:14Z")
> [@](#): > > \*Originally posted by Cabbage \* > \*\ ***ultrafilter** : > I believe you actually meant to say, “Completeness means that every _Cauchy_ sequence of real numbers converges to a real number.” > > Oh, and thanks, **DrMatrix**! 😃 \*\* I see my nitpicks have been preempted. “Every convergent sequence converges” certainly is true, but it’s rather trivial. And one can simply define alpha to be 1/0 (and vice versa) and you have a number r/s which is defined but not part of R. No less valid than defining alpha to be a root of an irreducible polynomial. ---
**Author:** ![Liberal](https://avatars.discourse-cdn.com/v4/letter/l/848f3c/32.png) [@Liberal](https://boards.straightdope.com/u/Liberal)\ **Post date:** [April 25, 2003, 11:17am UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/78 "2003-04-25T11:17:35Z")
For the record, argument from authority is a fallacy only when the authority is illegitimate. ---
**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 25, 2003, 12:52pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/79 "2003-04-25T12:52:24Z")
Just for kicks, following up about repeating units in different base systems, you have in binary: 0.111111… = 1 ---
**Author:** ![RM\_Mentock](https://avatars.discourse-cdn.com/v4/letter/r/e274bd/32.png) [@RM\_Mentock](https://boards.straightdope.com/u/RM_Mentock)\ **Post date:** [April 25, 2003, 1:03pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/80 "2003-04-25T13:03:20Z")
> [@](#): > > \*Originally posted by Libertarian \* > \*\*Yep. This exact topic is one of the things I often cite when asked whether I’ve changed my mind about anything due to my activities here at Straight Dope. The proof is both valid and sound. After seeing it, there was nothing else to consider. I _had_ to change my mind — much like some atheists who have seen the modal ontological proof of God’s existence. \*\* [I always knew that there was some sort of ulterior motive](http://boards.straightdope.com/sdmb/showthread.php?s=&threadid=22371&perpage=40&pagenumber=1)! 😉 Wow, those old threads get all balled up don’t they? [Previous page](https://boards.straightdope.com/t/does-99-repeating-1/170821.md?page=3) [Next page](https://boards.straightdope.com/t/does-99-repeating-1/170821.md?page=5)