# 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:** 3
**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, 3:01pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/41 "2003-04-24T15:01:47Z")
> [@](#): > > \*Originally posted by Dolomite21 \* > \*\*Great…but _is_ it equal \*\* Yes. ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [April 24, 2003, 3:52pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/42 "2003-04-24T15:52:51Z")
I just want to say that I do not like the proof that **Libertarian** gave. It’s the classic proof, but it’s informal, and you can use a similar-looking “proof” to “prove” things that are wrong. You shouldn’t be subtracting two series when you haven’t even proved they converge. But I won’t argue with the result. That’s all. 🙂 ---
**Author:** ![Bippy\_the\_Beardless](https://avatars.discourse-cdn.com/v4/letter/b/ac8455/32.png) [@Bippy\_the\_Beardless](https://boards.straightdope.com/u/Bippy_the_Beardless)\ **Post date:** [April 24, 2003, 4:15pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/43 "2003-04-24T16:15:00Z")
Since the OP has been pretty well answered, can I expand the question a little… 0.9… = 1.0… where … indicates the infinite repetition of the digit to its left. Has been shown above. But in considering the real number line, you can consider that 0.9… aproaches 1.0… as the repitition of the disgits tends to infinity, but can not become 1.0… Is there any value in equating 0.9… and 1.0… to two adjacent real numbers? That is two real numbers that have no real numbers between them. I find this a way to consider the real number continuum, and I wonder if this is terribly flawed. 1.0… = 0.9…9 = 0.9…8 = 0.9…7 where 0.9…8 would be the imagined number that begins 0.9… but whose ‘final’ ot infinitieth digit is an 8. 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… Which at least to me gives me a sense for the difference between a continuum (the reals) and a countable infinity. Please comment or point me to a web site that exposes the problems with this way of thinking. Cheers, Bippy ---
**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, 4:21pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/44 "2003-04-24T16:21:03Z")
There is no final digit in the standard theory of decimal representations. ---
**Author:** ![panamajack](https://avatars.discourse-cdn.com/v4/letter/p/47e85d/32.png) [@panamajack](https://boards.straightdope.com/u/panamajack)\ **Post date:** [April 24, 2003, 4:23pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/45 "2003-04-24T16:23:23Z")
Another one of the things for the web kooks to stay away from : surreal numbers. These are discussed in the first thread **Wendall Wagner** gave the link for; on the second page **MrDeath** gives a good explanation of what they are. Though 1.0 and 0.999… would be unique surreal numbers, again these are NOT real numbers (hence the name). As has been shown, these forms do represent the same number in the reals. A brief quote may help explain a bit without having to pull up that thread : _Originally posted by MrDeath in another thread_ > [@](#): > > But clearly we’re missing something if we try this out with .999999… = 1 - 1/omega. The problem is, we can get at 9.99999… two different ways - we can multiply .999999… by 10, or we can add 9 to it instead. (This is the trick that makes the proof work.) In the world of surreal numbers, the two methods are not compatible and give you different answers: 9 + (1 - 1/omega) = 10 - 1/omega, but 10(1 - 1/omega) = 10 - 10/omega, which is a distinct surreal number, although the two are only infinitesimally different. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [April 24, 2003, 5:20pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/46 "2003-04-24T17:20:00Z")
> [@](#): > > \*Originally posted by Bippy the Beardless \* > \*\*But in considering the real number line, you can consider that 0.9… aproaches 1.0… as the repitition of the disgits tends to infinity, but can not become 1.0… > > Is there any value in equating 0.9… and 1.0… to two adjacent real numbers? That is two real numbers that have no real numbers between them.\*\* Well, strictly speaking, a single number can’t “approach” anything; it just sits there having whatever value it has. A series or sequence of numbers (like .9, .99, .999, …) can approach a limit, and a repeating decimal like .9999… can be defined/understood to mean the limit of this sequence, which is 1. Under the standard conception of the real numbers, there is no such thing as “two real numbers that have no real numbers between them.” If two numbers are not the same, then there are always infinitely many other numbers between them (like (a+b)/2, for instance). You may be wanting to use “infinitessimals”: “infinitely small” numbers that are greater than 0 yet smaller than any other number. In the very early days of Calculus, mathematicians used this idea to explain and develop what they were doing, but it became apparent that this didn’t really make a whole lot of sense and didn’t hold water logically, so eventually the ideas of Calculus were reformulated in terms of the modern delta-epsilon definition of a limit, which put things on firmer ground. Fairly recently, as I understand it, something called “nonstandard analysis” has been developed, which is an attempt to actually define and use infinitessimals in a logically consistent way. ---
**Author:** ![Bippy\_the\_Beardless](https://avatars.discourse-cdn.com/v4/letter/b/ac8455/32.png) [@Bippy\_the\_Beardless](https://boards.straightdope.com/u/Bippy_the_Beardless)\ **Post date:** [April 24, 2003, 5:44pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/47 "2003-04-24T17:44:02Z")
Sorry about being unclear I meant by you can consider that 0.9… aproaches 1.0… as the repitition of the disgits tends to infinity The series 0.90…, 0.990…, 0.9990… continuing as the number of 9s tends to infinity. Is there a form of logic interpretation where we can say that 1.0… = 0.9… AND 1.0… \> 0.9… AND NOT 1.0… \< 0.9… in a consistant fashion? Clearly this doesn’t map all real numbers (as pi for minstance cannot be expressed in this way) but is there a sence in considering a last digit, or last set of digits after an infinite repitition of digits, even though they must have a value equal to zero or infinity. ( 1 integer as a real in decimal notation could be considered 0… 1. 0…, any number not starting 0… is numerically infinite, so we drop the 0… whn writing a finite number ) Cheers, Bippy p.s. does anyone have a method to overline a number on this board to allow for the more usual symbol for repeating digits. ---
**Author:** ![andymurph64](https://avatars.discourse-cdn.com/v4/letter/a/a88e57/32.png) [@andymurph64](https://boards.straightdope.com/u/andymurph64)\ **Post date:** [April 24, 2003, 5:49pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/48 "2003-04-24T17:49:06Z")
> [@](#): > > by ultrafilter > 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”. GAH!!! Must flush memories of Real Analysis from brain. GAH! I hate you ultrfilter! 😃 ---
**Author:** ![Mangetout](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mangetout/32/19_2.png) [@Mangetout](https://boards.straightdope.com/u/Mangetout)\ **Post date:** [April 24, 2003, 6:28pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/49 "2003-04-24T18:28:46Z")
10 / 3 = 3.333… 3.333… + 3.333… + 3.333… = 9.999… ---
**Author:** ![Liberal](https://avatars.discourse-cdn.com/v4/letter/l/848f3c/32.png) [@Liberal](https://boards.straightdope.com/u/Liberal)\ **Post date:** [April 24, 2003, 7:19pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/50 "2003-04-24T19:19:15Z")
> [@](#): > > \*Originally posted by Spiritus Mundi \* > \*\*Oh, and **Lib** , never let it be said that you do not learn from your mistakes, eh? 😉 \*\* 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. ---
**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, 7:24pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/51 "2003-04-24T19:24:36Z")
> [@](#): > > 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. I must again disagree. 0.999… is _not_ arbitrarily close to 1, or “as close to 1 as it is possible to get” or “close enough to 1 that you can use 1 in its place”. These imply that it is some sort of approximation. The math clearly shows that it is not, and that 0.9999… is _identical_ to 1. They are the same number. It certainly is a quirk of the number system that this is so. In base 9, for instance, you can use the same reasoning to show that 0.88888888 … = 1 ---
**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, 8:27pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/52 "2003-04-24T20:27:29Z")
> [@](#): > > \*Originally posted by CalMeacham \* > \*\*I must again disagree. 0.999… is _not_ arbitrarily close to 1, or “as close to 1 as it is possible to get” or “close enough to 1 that you can use 1 in its place”. These imply that it is some sort of approximation. The math clearly shows that it is not, and that 0.9999… is _identical_ to 1. They are the same number. It certainly is a quirk of the number system that this is so. In base 9, for instance, you can use the same reasoning to show that 0.88888888 … = 1 \*\* That’s why I said “loosely … in application” In application, you do **NOT** have an infinite series. Look, .999… means the sum, as i goes from 1 to infinity, of 9/(10^i). If you want to call that a number, it’s fine with me. But it’s an infinite series. Loosely, in practice, if you want to think of it as a number, and not an infinite series, you can think of it as a number as close to 1 as you need it to be. BTW, this is not a quirk of the decimal system, as the more formal representation of the infinite series above shows. You can replace 9/(10^i) with k/[(k+1)^i for any k and still get 1 as the value of the infinite sum. ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [April 24, 2003, 8:32pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/53 "2003-04-24T20:32:39Z")
> [@](#): > > \*Originally posted by nogginhead \* > Look, .999… means the sum, as i goes from 1 to infinity, of 9/(10^i). No, not exactly. _Exactly_, it’s the limit (as n goes to infinity) of the partial sums for i=1 to n. And a limit _is_ a number. It’s defined to be a number, with a definite value. When we say “the sum from 1 to infinity” it’s shorthand. We really mean a limit of partial sums. ---
**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, 9:09pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/54 "2003-04-24T21:09:02Z")
> [@](#): > > \*Originally posted by Achernar \* > \*\*No, not exactly. _Exactly_, it’s the limit (as n goes to infinity) of the partial sums for i=1 to n. And a limit _is_ a number. It’s defined to be a number, with a definite value. > > When we say “the sum from 1 to infinity” it’s shorthand. We really mean a limit of partial sums. \*\* You’re right. My bad. ---
**Author:** ![kanicbird](https://avatars.discourse-cdn.com/v4/letter/k/5f8ce5/32.png) [@kanicbird](https://boards.straightdope.com/u/kanicbird)\ **Post date:** [April 24, 2003, 11:05pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/55 "2003-04-24T23:05:32Z")
> [@](#): > > 2/9 = .222(repeating) = 2/9 this is not true 2/9 = 0.22222(repeating) and 2/9th at the end as a fraction of the last digit in the repeating series as the number of 2’s goes to infinity ---
**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, 11:13pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/56 "2003-04-24T23:13:17Z")
> [@](#): > > \*Originally posted by kanicbird \* > \*\*this is not true > > 2/9 = 0.22222(repeating) and 2/9th at the end as a fraction of the last digit in the repeating series as the number of 2’s goes to infinity \*\* I’m not at all sure what you’re trying to say here. ---
**Author:** ![MilTan](https://avatars.discourse-cdn.com/v4/letter/m/71e660/32.png) [@MilTan](https://boards.straightdope.com/u/MilTan)\ **Post date:** [April 24, 2003, 11:56pm UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/57 "2003-04-24T23:56:46Z")
> [@](#): > > \*Originally posted by Orbifold \* > \*\*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.) \*\* So I know you warned the Web Kooks, but I felt that I had to comment here. By briefly reading about p-adic numbers, it seems that they are, in some sense, akin to writing numbers in a given base. Thus, if we consider 2-adic numbers, 1 + 2 + 4 + 8 + … becomes 11111… Which set my little computer-science brain ticking: If this were a register, and we were dealing with twos-complement numbers, once we filled the register with 1s, we’d have the representation of a decimal -1. So extending this to the magic “infinite register”, it seems obvious that the only way to represent -1 is by having an infinite series of 1s, or, in standard numbers, an infinite summation of powers of 2. But then I figured that p-adic numbers probably came first, and so twos-complement notation uses the properties of 2-adic numbers to do its magic, instead of vice-versa. I always wondered how twos-complement worked (it always seemed sort of like magic), and this gives me at least some basis for understanding. Is this important? Is this even correct? I dunno, but everything’s spinning around in my head, and it’s beautiful! Woo! (oh, and **Lib**? Thanks for mentioning that argument. Now I’m gonna have trouble falling asleep tonight) ---
**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, 12:08am UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/58 "2003-04-25T00:08:16Z")
> [@](#): > > \*Originally posted by kanicbird \* > \*\*2/9 = 0.22222(repeating) and 2/9th at the end as a fraction of the last digit in the repeating series as the number of 2’s goes to infinity \*\* Say this to yourself over and over. Infinity means unending. There is no last digit. Infinity means unending. There is no last digit. Infinity means unending. There is no last digit. Infinity means unending. There is no last digit. Infinity means unending. There is no last digit. Infinity means unending. There is no last digit. … ---
**Author:** ![Orbifold](https://avatars.discourse-cdn.com/v4/letter/o/779978/32.png) [@Orbifold](https://boards.straightdope.com/u/Orbifold)\ **Post date:** [April 25, 2003, 1:47am UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/59 "2003-04-25T01:47:52Z")
> [@](#): > > \*Originally posted by MilTan \* > But then I figured that p-adic numbers probably came first, and so twos-complement notation uses the properties of 2-adic numbers to do its magic, instead of vice-versa. I always wondered how twos-complement worked (it always seemed sort of like magic), and this gives me at least some basis for understanding. I think twos-complement notation owes more to plain old modular arithmetic than to p-adic arithmetic. A 8-bit integer register (for example) is just performing arithmetic modulo 2[sup]8[/sup], which is why -1 is represented as 2[sup]8[/sup]-1, or 11111111 in binary. There is probably some way in which 2-adic arithmetic can be thought of as the “limit” of arithmetic modulo 2[sup]n[/sup] as n goes to infinity, and a number theorist could probably explain it detail, but I can’t. It’s certainly an interesting correspondence, though. ---
**Author:** ![John\_Mace](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/john_mace/32/185_2.png) [@John\_Mace](https://boards.straightdope.com/u/John_Mace)\ **Post date:** [April 25, 2003, 1:59am UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/60 "2003-04-25T01:59:30Z")
Is there any fancy mathematical theorum needed to justify: .9999… - .9999… = 0? Or is .9999… just another number? It’s been quite awhile since that kind of stuff was second nature to me. [Previous page](https://boards.straightdope.com/t/does-99-repeating-1/170821.md?page=2) [Next page](https://boards.straightdope.com/t/does-99-repeating-1/170821.md?page=4)