# Why doesn't .9999~ = 1? **URL:** \ **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:** 12
**Author:** ![StephanieG](https://avatars.discourse-cdn.com/v4/letter/s/c77e96/32.png) [@StephanieG](https://boards.straightdope.com/u/StephanieG)\ **Post date:** [June 3, 2003, 11:54pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/221 "2003-06-03T23:54:02Z")
> [@](#): > > \*Originally posted by ultrafilter \* > \*\*This is incorrect. An infinite series is an infinite series. It has more terms than any finite series. The “proof” that you quoted is incorrect because laws of arithmetic only work for convergent series, but there are contexts in which it is a valid argument: add all the powers of 2, and you get -1. \*\* Unfortunately, I could not edit my post to reflect additional information. An infinite series either converges or diverges: Converge = to approach a limit Diverge = fails to approach a limit X = 1 + 2 + 4 + 8 + … diverges x(n) = Sum (2^i) i=0 to n (Whole numbers = positive Integers) = 2^(n+1) - 1 x(n) diverges because there is no maximum Whole number in the mathematical definition of Whole numbers, 2^(infinity) = infinity ---
**Author:** ![StephanieG](https://avatars.discourse-cdn.com/v4/letter/s/c77e96/32.png) [@StephanieG](https://boards.straightdope.com/u/StephanieG)\ **Post date:** [June 3, 2003, 11:57pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/222 "2003-06-03T23:57:34Z")
An infinte series either converges or diverges. Converge = to approach a limit Diverge = fails to approach a limit We’re asking if .999999999… converges or diverges. If .99999999… diverges, then it doesn’t approach a specific value. Let X = 1 - .99999999999999999… be an element of the mathematical definition of real numbers x(n) = 10^(-n) x = 1/1000000000000000… 0 \< x(n) \< 1 x(n) approaches, but never passes 0, therefore it converges. 0 \<= x \< x(n) Suppose there exists y, such that 0 \< y = limit x(n) Certainly 1/y is big, but there’s always a big N, such that 10^N is bigger than y, but that means 10^(-N) \< x, which is a contradiction to the last assumption. Therefore the limit x(n) is not greater than 0, and is not less than 0, so it must be equal to 0. Therefore 0 = limit x(n), and 0 = x = 1 - .9999999…, and 1 = .999999999… ---
**Author:** ![William\_Ashbless](https://avatars.discourse-cdn.com/v4/letter/w/a87d85/32.png) [@William\_Ashbless](https://boards.straightdope.com/u/William_Ashbless)\ **Post date:** [June 3, 2003, 11:59pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/223 "2003-06-03T23:59:41Z")
I love proof-by-contradiction. Such a powerful tool… ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [June 4, 2003, 12:00am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/224 "2003-06-04T00:00:50Z")
> [@](#): > > \*Originally posted by StephanieG \* > **Unfortunately, I could not edit my post to reflect additional information.** We don’t allow that here, because not everyone can be trusted to use it honestly. The preview button is your new best friend. Welcome to the boards, by the way. ---
**Author:** ![FranticMad](https://avatars.discourse-cdn.com/v4/letter/f/c68b51/32.png) [@FranticMad](https://boards.straightdope.com/u/FranticMad)\ **Post date:** [June 4, 2003, 12:17am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/225 "2003-06-04T00:17:40Z")
> [@](#): > > \*Originally posted by Phage \* > \*\* but as infinity cannot actually be reached in reality 0.999… will never equal 1. \*\* You are confusing two things: our ability to write down a number, and the number itself. 0.999… is a number. And yes infinity **is** reached. Just because you cannot write down an infinite number of 9s, or “compute” them in finite time doesn’t mean a thing. “0.999…” is just a way to represent a number. In the same way, 2/2 represents 1 just as much as “1” does. Same with 0.5\*2. In other words, 0.999… does not have to arrive at an infinite number of decimals, it is already there. That’s what the “…” means. It represents the concept of the number. You have confused how you visualize the creation of the number with the number itself. The number exists, infinite 9’s are there by definition, and the number behaves like 1. ---
**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:** [June 4, 2003, 12:42am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/226 "2003-06-04T00:42:12Z")
> [@](#): > > \*Originally posted by \*\*\*Pleonast \*\* > > Is there still a GQ here? I humbly suggest a moderator either close this thread or move it to GD. > [@](#): > > \*Originally posted by \*\*\*Jabba \*\* > > While I understand your frustration, I think many of us would be worried by an official statement that mathematical quandaries were matters of debate rather than factual questions. That is _exactly_ why I will not move this to Great Debates. This is a question of fact. 0.999… either equals 1.0 or it does not. The answer, of course, is that it does. Yes, the General Question has been answered several times. Yes, this is frustrating. There are several concepts in mathematics (like 1=0.999…, recursion, infinity and many others) that when you first understand them give you a wonderful sense of A-Ha! I’m leaving this thread open in the (perhaps vain) hope that someone will understand why 1=0.999… and say “A-Ha!” **DrMatrix** - GQ Moderator ---
**Author:** ![Una\_Persson](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/una_persson/32/346_2.png) [@Una\_Persson](https://boards.straightdope.com/u/Una_Persson)\ **Post date:** [June 4, 2003, 12:42am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/227 "2003-06-04T00:42:17Z")
> [@](#): > > \*Originally posted by ultrafilter \* > \*\*And have I ever told you that the integer 1 is not the same as the real number 1? \*\* What is the difference between them, out of curiosity? Una ---
**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 4, 2003, 1:08am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/228 "2003-06-04T01:08:37Z")
> [@](#): > > \*originally posted by \ ***DrMatrix** I’m leaving this thread open in the (perhaps vain) hope that someone will understand why 1=0.999… and say “A-Ha!” I’ve said “A-Ha!” I’ve been buggin’ out at work over this all day. Zeno’s (supposed) paradox is buggin’ me out also. I love it! ---
**Author:** ![ragerdude](https://avatars.discourse-cdn.com/v4/letter/r/d07c76/32.png) [@ragerdude](https://boards.straightdope.com/u/ragerdude)\ **Post date:** [June 4, 2003, 1:28am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/229 "2003-06-04T01:28:58Z")
How about this proof: 1.000… does NOT equal 1. 1. The number 1.000… has an infinite number of 0s on the end. 2. Infinity can never be reached. 3. As more 0s in 1.000… are considered, we have a sequence. The LIMIT equals one, but we cannot ever reach the end. We don’t know how it will actually turn out in the end! There might be a surprise! 4. The only way for 1.000… to be equal to 1, is after an infinite number of 0s have been placed on the end by a human, to see how it ends up. (Mathematics is based on observation.) This cannot be done. Thus 1.000… will never equal 1. 5. Therefore, 1.000… is the number closest to 1 from above. [Note: this is a satirical parallel, intended to demonstrate the faulty reasoning therein.] ---
**Author:** ![netscape\_6](https://avatars.discourse-cdn.com/v4/letter/n/8c91f0/32.png) [@netscape\_6](https://boards.straightdope.com/u/netscape_6)\ **Post date:** [June 4, 2003, 1:31am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/230 "2003-06-04T01:31:25Z")
> [@](#): > > \*Originally posted by William\_Ashbless \* > \*\*Er, so Amethyst’s statement > > “I can accept that .9999… = 1 if I can accept that an asymptotic function will ultimately equal its asymptote as the function extends to infinity.” > > means that if an asymptotic function (like f(x)=1/x) can be said to be at the asymptote (0, in my example) if you allow the range (x) to go up to infinity, then Amethyst will accept 0.999… =1. > > This is, of course, is what allows me to say > > lim 1/x = 0 > x-\>infinity \*\* OK I think I got it. So Amethyst meant as the number of decimals you go out in 9999, increases to infinity the differance between 99999, decreases to zero? ---
**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 4, 2003, 1:32am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/231 "2003-06-04T01:32:15Z")
> [@](#): > > \*Originally posted by Anthracite \* > \*\*What is the difference between them, out of curiosity? > > Una \*\* I think he means the difference of the definition of what a real number is versus what an integer is, thought I’m not quite sure how those would be fundamentally different. the value, 1, obviously cannot change. 1[sub]real[/sub] = 1[sub]integer[/sub], AFAIK. ---
**Author:** ![ragerdude](https://avatars.discourse-cdn.com/v4/letter/r/d07c76/32.png) [@ragerdude](https://boards.straightdope.com/u/ragerdude)\ **Post date:** [June 4, 2003, 1:33am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/232 "2003-06-04T01:33:22Z")
**ultrafilter** , are you considering the distinction to be the different roles in various algebraic structures; namely, 1 defined as an element of an integral domain, and 1 as and element of a field of real numbers? ---
**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 4, 2003, 1:34am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/233 "2003-06-04T01:34:30Z")
> [@](#): > > \*Originally posted by netscape 6 \* > \*\*OK I think I got it. So Amethyst meant as the number of decimals you go out in 9999, increases to infinity the differance between 99999, decreases to zero? \*\* If you meant the difference between **1** and .999… decreases to zero, then, yes, you have it exactly. ---
**Author:** ![Joe\_Random](https://avatars.discourse-cdn.com/v4/letter/j/e8c25b/32.png) [@Joe\_Random](https://boards.straightdope.com/u/Joe_Random)\ **Post date:** [June 4, 2003, 1:35am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/234 "2003-06-04T01:35:33Z")
> [@](#): > > \*Originally posted by netscape 6 \* > \*\*OK I think I got it. So Amethyst meant as the number of decimals you go out in 9999, increases to infinity the differance between 99999, decreases to zero? \*\* YOu can’t make the number of decimals in 0.999… increase to infinity, because _it’s already there_. That’s what the notation means. ---
**Author:** ![netscape\_6](https://avatars.discourse-cdn.com/v4/letter/n/8c91f0/32.png) [@netscape\_6](https://boards.straightdope.com/u/netscape_6)\ **Post date:** [June 4, 2003, 2:17am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/235 "2003-06-04T02:17:02Z")
> [@](#): > > \*Originally posted by Q.E.D. \* > \*\*If you meant the difference between **1** and .999… decreases to zero, then, yes, you have it exactly. \*\* thats what i meant. > [@](#): > > \*Originally posted by Joe Random \* > \*\*YOu can’t make the number of decimals in 0.999… increase to infinity, because _it’s already there_. That’s what the notation means. \*\* maybe increase was the wrong word. i meant as you travel along .9999, to the right the difference between what you have seen so far and 1 decreases to zero as the decimal places you have traveled increase to infinity. thanks to **William\_Ashbless** for explaing it to me by the way ---
**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 4, 2003, 2:20am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/236 "2003-06-04T02:20:10Z")
What you meant to say (and what I meant to say) is that as you add 9s to the end of .9999 (no elipsis), you approach .999… and therefore, 1. The elipsis already means an indefinitely continuing series. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [June 4, 2003, 4:12am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/237 "2003-06-04T04:12:56Z")
As for the difference between 1[sub] **R** [/sub] and 1[sub] **Z** [/sub], it has to do with constructions from set theory. Around the beginning of the 20th century, there was a push to base all of mathematics on sets, and to come up with a good set theory to be the basis of everything else. One of the outcomes was a very rigorous set of definitions of **N** , **Z** , **Q** , and **R**. The first three are simple:[ul][li]A set S is called a successor set if for some operator ‘, [symbol]Æ[/symbol] is in S, and whenever X is in S, X’ is in S. **N** is defined as the smallest successor set. In the standard model, X’ is defined as X [symbol]È[/symbol] {X}.[_] **Z** is defined as a partition of **N** [sup]2[/sup], with (a, b) and (c, d) in the same class iff a + d = b + c.[_] **Q** is defined as a partition of **Z** [sup]2[/sup], with (a, b) and (c, d) in the same class iff ad = bc.[/ul] **R** is defined using Dedekind cuts, and there are many good explanations of those out there, such as [this one](http://mathforum.org/library/drmath/view/52511.html).[/li] Since 1[sub] **R** [/sub] and 1[sub] **Z** [/sub] aren’t even in the same set, or two sets of objects with the same type, they can’t be the same. Fortunately, there’s a subset of **R** that is indistinguishable in behavior from **N** , so we can regard them as the same in ordinary math. ---
**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:** [June 4, 2003, 4:34am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/238 "2003-06-04T04:34:48Z")
> [@](#): > > _Originally posted by_\*\* ultrafilter \*\* > And have I ever told you that the integer 1 is not the same as the real number 1? > [@](#): > > \*Originally posted by \*\*\*Anthracite \*\* > What is the difference between them, out of curiosity? I can think of a couple of things **ultrafilter** might have meant. 1. He might have meant that the structure of integers is different from the structure of real numbers. Every integer has a successor; 1’s successor is 2. There are no integers between 1 and 2. The real number 1 does not have a successor and between any pair of distinct real numbers you can find another real number. So the integers have different properties than the real numbers. or 1. You can construct the rational numbers from the integers and then construct the real numbers from the rationals. Every rational number is an ordered pair of integers (a,b) b not zero. We usually write this as a/b. The rational number 1 is an ordered pair of integers of the form (a,a). An ordered pair of integers is distinct from a single integer. The reals are constructed as “cuts” of integers – an ordered pair of sets of reals (A,B) where every element of A is less than every element of B. So the real number 1 is an ordered pair of sets of rational numbers and is not the same thing as the integer 1. Either way, the integers are isomorphic to a subset of the reals and so usually the integers are identified with that subset. > [@](#): > > \*Originally posted by \*\*\*x-ray vision \*\* > I’ve said “A-Ha!” I’ve been buggin’ out at work over this all day. Zeno’s (supposed) paradox is buggin’ me out also. I love it! Thank you for saying that. ---
**Author:** ![MonkeyMensch](https://avatars.discourse-cdn.com/v4/letter/m/82dd89/32.png) [@MonkeyMensch](https://boards.straightdope.com/u/MonkeyMensch)\ **Post date:** [June 4, 2003, 5:36am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/239 "2003-06-04T05:36:59Z")
> [@](#): > > \*Originally posted by DrMatrix \* > \*\* > 2) You can construct the rational numbers from the integers and then construct the real numbers from the rationals. Every rational number is an ordered pair of integers (a,b) b not zero. We usually write this as a/b. The rational number 1 is an ordered pair of integers of the form (a,a). An ordered pair of integers is distinct from a single integer. The reals are constructed as “cuts” of integers – an ordered pair of sets of reals (A,B) where every element of A is less than every element of B. \*\* Can you elaborate on that, **DM**? Is there a one-to-one relationship between ordered pairs of sets of reals and reals? Do tell… ---
**Author:** ![kabbes](https://avatars.discourse-cdn.com/v4/letter/k/e99b99/32.png) [@kabbes](https://boards.straightdope.com/u/kabbes)\ **Post date:** [June 4, 2003, 9:18am UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/240 "2003-06-04T09:18:32Z")
One problem that strikes me as I read through these posts (and I think maybe this is the problem **Archenar** refers to) is the consistent use by some of the phrase “an infinite number of…” This is a phrase that has some intuitive feel colloquially but has all kinds of problems in a rigorous mathematical sense. Basically, there is no “infinite number”. There are cardinal numbers, which have their own arithmetic, but it is an error to attempt to treat these as “normal” numbers. When you say that 0.999… is “zero dot with an infinite number of nines”, you’re really using mathematical shorthand for the concept of an infinite _sum_, which comes back to the point that **ultrafilter, Archernar** et al have been making all along. _That’s_ why you can define the limit to the sum and _that’s_ why it makes sense to define 0.999… in terms of that limit. (As an aside, **RealityChuck** , therein also lies the problem with the “10x - x = 9” proof. Because 9.999… is an infinite sum, you have to prove that it converges before you’re allowed to do things like multiply and subtract it… but once you’ve proved it converges _you’ve already shown that it equals its limit and don’t need to do the multiplication-and-sutraction thing anyway_! Therefore whilst the “proof” is definitely useful for giving an intuitive feel for the reason the solution works, it is actually begging the question. Sadly, it isn’t as simple as saying “multiplication just moves the decimal place” because multiplciation on the reals is a field operation that doesn’t work on infinite sums unless they converge, leading you back once again to the same starting place) Back to the point: next time you refer to an “infinite number” of something, stop and ask yourself what you really mean by that and remember that infinity is not a number, at least in the sense that you are using the word. One != 1? I’ve come back into the debate too late to address the 1[sub] **Z** [/sub] != 1[sub] **R** [/sub] issue, so I’ll just say that my education concurs exactly with **ultrafilter’s** explanation. The set of reals is a completely different species to the set of integers and it is only by clever mathematical construction that we are able to shift between the two. Think of it in programming terms: you’ve defined parameter _s_ as a string and parameter _v_ as a variable. You set _s_ = “one” and _v_ = 1. Now what will the program do if you ask it to compute _s_ + _v_? So what _does_ “1” mean? In set-theoretic terms, “1” is the identity, defined such that under the multiplication operation 1._a_ = _a_. (Under a ring structure such as the reals we then use the addition operation in order to define things like “2” = “1” + “1”) Because the identity is such a universal concept in sets with the multiplication operation defined, it looks very similar in the various sets where it is defined. It’s the same idea in each, right? But you’re still comparing apples and oranges, or _s_ and _v_. That’s why group, ring and field theory should be left to those who have mad hair, smell slightly and live in basements. It’s too annoying for the rest of us. A digression on the nature of mathematicians In my university the pure mathematics department shared a building with the statistics department. It was noteworthy that the statisticians had the top two floors, which were air-conditioned and had a great view whilst the pure mathematicians had the bottom floor and the basement, where the car fumes entered and it was noisy. Also the statisticians had a car park with Aston Martins and Jaguars parked there whilst the pure mathematicians had a bicycle shed. I went into statistics. pan [Previous page](https://boards.straightdope.com/t/why-doesnt-9999-1/179019.md?page=11) [Next page](https://boards.straightdope.com/t/why-doesnt-9999-1/179019.md?page=13)