# .99999999 equal to 1 ??? **URL:** \ **Category:** Factual Questions\ **Created:** [August 15, 2001, 11:42pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468 "2001-08-15T23:42:36Z")\ **Posts on this page:** 20\ **Page:** 8
**Author:** ![kinoons](https://avatars.discourse-cdn.com/v4/letter/k/a698b9/32.png) [@kinoons](https://boards.straightdope.com/u/kinoons)\ **Post date:** [August 18, 2001, 3:47am UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/141 "2001-08-18T03:47:04Z")
like I said above – 1/3 is a specific value, 1/4 is a specific value. 0.25 is a specific value, it terminates. 0.333… is a non terminating repeating decimal, therefor how can it have a specific value if it never stops? ---
**Author:** ![hansel](https://avatars.discourse-cdn.com/v4/letter/h/6bbea6/32.png) [@hansel](https://boards.straightdope.com/u/hansel)\ **Post date:** [August 18, 2001, 4:11am UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/142 "2001-08-18T04:11:22Z")
> [@](#): > > \*Originally posted by kinoons \* > \*\*like I said above – 1/3 is a specific value, 1/4 is a specific value. 0.25 is a specific value, it terminates. 0.333… is a non terminating repeating decimal, therefor how can it have a specific value if it never stops? \*\* That’s exactly the point, Kinoons–it doesn’t have a specific value when represented as a repeating decimal. But it is equal–exactly equal, in your sense–to 1/3. When dealing with the repeating decimal, we refer to the definite value of it as a limit. The limit of 0.999… is 1. It really is just a difference of representation. Let me try one more proof: between any two distinct real numbers (meaning numbers with decimal expansions) there are an infinite number of real numbers (actually, a non-denumerable number of real numbers–an uncountable infinite). There are no real numbers between 0.999… and 1.000…, so they’re the same number. ---
**Author:** ![Wumpus](https://avatars.discourse-cdn.com/v4/letter/w/8c91f0/32.png) [@Wumpus](https://boards.straightdope.com/u/Wumpus)\ **Post date:** [August 18, 2001, 4:19am UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/143 "2001-08-18T04:19:37Z")
Since the anti-equaltarians have admitted they don’t care about pesky proofs, I suggest this be moved to Great Debates. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [August 18, 2001, 4:36am UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/144 "2001-08-18T04:36:45Z")
> [@](#): > > \*Originally posted by Wumpus \* > \*\*Since the anti-equaltarians have admitted they don’t care about pesky proofs, I suggest this be moved to Great Debates. \*\* I second that. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [August 18, 2001, 4:44am UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/145 "2001-08-18T04:44:09Z")
> [@](#): > > \*Originally posted by kinoons \* > \*\*like I said above – 1/3 is a specific value, 1/4 is a specific value. 0.25 is a specific value, it terminates. 0.333… is a non terminating repeating decimal, therefor how can it have a specific value if it never stops? \*\* All right, one more time with emphasis. Do you remember how to do long division by hand? Good. Now get out a piece of paper, and divide one by four. First you get a two, then a five, then a zero. Your teachers told you to stop when you got to zero. You know why? Cause you’d just keep getting more zeroes forever. Same way that you keep getting threes when you divide one by three. Not just similar, mind you. Not even just close. Exactly the same thing’s going on here. Just cause we don’t normally write the zeroes out doesn’t mean they aren’t there. .25 is equal to 1/4 in the limit the same way that .333… is equal to 1/3 in the limit, except that one series converges after a finite number of terms and the other doesn’t. This difference is absolutely inconsequential. You still haven’t explained why 1/2 = 2/4 doesn’t bother you. How about why 1/3 = 3/9? ---
**Author:** ![Boris\_B](https://avatars.discourse-cdn.com/v4/letter/b/fbc32d/32.png) [@Boris\_B](https://boards.straightdope.com/u/Boris_B)\ **Post date:** [August 18, 2001, 5:21am UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/146 "2001-08-18T05:21:27Z")
I’d like to offer an analogy (I wish I could call it a proof but it really isn’t one) to have one more go at convincing the anti-equalitarians. Influenced by Xeno, take a look: Take 1 + 1/2 + 1/4 + 1/8 + 1/16 and keeping going forever. When you are done, what is your sum? I mean _exactly_. If you can’t say exactly what the sum is, I haven’t gotten through to you. I will grant that this infinite series is a different _concept_ from 2 (oops, gave it away 🙂 ) but it is not a different _number_. Okay here is something, maybe closer to a proof. Humor me a little - I’m going to break some rules but I hope it will be worth it. First, let’s do some subtraction (and in all cases, my ellipses mean “continued to infinity”). 1.0… - 0.9… = 0.0…1 = Z (this is where you have to indulge me, I know you can’t really follow infinity with single number but if we temporarily forget that you’ll see where I’m going) Now let’s recall that 10A = A _only_ if A = 0 If A is non-zero it cannot equal a product of itself and non-one. So let’s return to Z. 0.0…1 x 10 = 0.0…10 Look at that. _0.0…10_ Drop the trailing decimal … Z = 0 QED But I bet I’ve broken so many rules with that that the real mathematicians are reaching for the Rolaids. ☹ ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [August 18, 2001, 5:25am UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/147 "2001-08-18T05:25:26Z")
> [@](#): > > \*Originally posted by Boris B \* > \*\*First, let’s do some subtraction (and in all cases, my ellipses mean “continued to infinity”). > 1.0… - 0.9… = 0.0…1 = Z (this is where you have to indulge me, I know you can’t really follow infinity with single number but if we temporarily forget that you’ll see where I’m going) > \*\* There were some issues with the dots being surrounded on either side earlier in this thread (basically, it’s a no no.). Other than that, it’s not bad. ---
**Author:** ![mangeorge](https://avatars.discourse-cdn.com/v4/letter/m/d78d45/32.png) [@mangeorge](https://boards.straightdope.com/u/mangeorge)\ **Post date:** [August 18, 2001, 6:41am UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/148 "2001-08-18T06:41:09Z")
All of the pro-equalitarians proofs say that the difference between 1 and .999… eventually gets so small that it doesn’t count? Because if you take away infinity you lose the possibility of equality. And if you keep infinity you’re stuck with infinite fractions. If you go out a gazillion fractions (9’s) you’re no closer to infinity than when you started. And the fractions still don’t equal zero. Or is infinity approachable? Do I need to revisit infinity? Peace, mangeorge ---
**Author:** ![MrDeath](https://avatars.discourse-cdn.com/v4/letter/m/22d042/32.png) [@MrDeath](https://boards.straightdope.com/u/MrDeath)\ **Post date:** [August 18, 2001, 7:09am UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/149 "2001-08-18T07:09:05Z")
> [@](#): > > \*Originally posted by ultrafilter \* > \*\* > > > [@](#): > > > > \*Originally posted by Boris B \* > > \*\*First, let’s do some subtraction (and in all cases, my ellipses mean “continued to infinity”). > > 1.0… - 0.9… = 0.0…1 = Z (this is where you have to indulge me, I know you can’t really follow infinity with single number but if we temporarily forget that you’ll see where I’m going) > > \*\* > > There were some issues with the dots being surrounded on either side earlier in this thread (basically, it’s a no no.). Other than that, it’s not bad. \*\* You got a problem with omega + 1? (This is in fact _exactly_ what happens when you work with surreals.) I’m thinking of starting a MPSIMS thread to discuss the Axiom of Infinity. Anyone interested? ---
**Author:** ![SPOOFE](https://avatars.discourse-cdn.com/v4/letter/s/c5a1d2/32.png) [@SPOOFE](https://boards.straightdope.com/u/SPOOFE)\ **Post date:** [August 18, 2001, 9:07am UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/150 "2001-08-18T09:07:40Z")
Here’s another way to think about it (utilizing the 1/3 comparison)… When rounded up, the decimal shorthand for 1/3 is often written as .334 (or somesuch). When rounded up, the decimal shorthand for 2/3 is written as .667. And, finally, when rounded up, the shorthand for 3/3 (the .99… that is giving us so much trouble) is written as 1. Why? Not because it’s rounded (necessarily), but because it’s easier for people to quanitify it that way. Because, ultimately, the .333… or .666… will ultimately become something MORE than .333 or .666. Same with .999… IF it continues on for infinity. ---
**Author:** ![zut](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/zut/32/2875_2.png) [@zut](https://boards.straightdope.com/u/zut)\ **Post date:** [August 18, 2001, 12:59pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/151 "2001-08-18T12:59:32Z")
> [@](#): > > \*Originally posted by mangeorge \* > If you go out a gazillion fractions (9’s) you’re no closer to infinity than when you started. And the fractions still don’t equal zero. > Or is infinity approachable? Do I need to revisit infinity? Hmmm, well maybe you do and maybe you don’t. Let’s take a look at 1/3 = 0.3rep as an example. As you say, if you add up the values of a gazillion decimal placeholders (3s in this case), you’re no closer to infinity than when you started. And the sum of the things you added up don’t equal 1/3. Yep, correct. However, where you’re going wrong is in thinking that you _have_ to add up the values of an infinite number of decimal placeholders _one at a time_. You might want to revisit the concept of an infinite series, where an infinite (and not just “large number”) of terms can be added together to produce a discrete value. This question is kind of like Zeno’s paradox in reverse. What you are saying is this: “If you move your finger on the number line from 0 to 1/3, first you have to cross 0.3. Then you have to cross 0.33, then 0.333 and 0.3333 and 0.33333 and so forth, infinitely. Since you have to do an infinite number of things in a finite length of time, your finger never reaches 1/3.” Which is of course ridiculous; it’s easy to touch 1/3 on a number line. Of course you can substitute 9s for 3s and 3/3 for 1/3 in the argument above, should you so desire. ---
**Author:** ![Joe\_Cool](https://avatars.discourse-cdn.com/v4/letter/j/ecb155/32.png) [@Joe\_Cool](https://boards.straightdope.com/u/Joe_Cool)\ **Post date:** [August 18, 2001, 1:19pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/152 "2001-08-18T13:19:50Z")
> [@](#): > > \*Originally posted by kinoons \* > **Because the mathematical proof is beside the point** > [@](#): > > \*Originally stated by Arthur Dent \* > **I don’t believe it. Prove it to me and I still won’t believe it.** Therefore, kinoons = Arthur Dent? 😃 And Dr. Matrix, you are the king of cool sig lines. ---
**Author:** ![23skidoo](https://avatars.discourse-cdn.com/v4/letter/2/f04885/32.png) [@23skidoo](https://boards.straightdope.com/u/23skidoo)\ **Post date:** [August 18, 2001, 2:39pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/153 "2001-08-18T14:39:47Z")
> [@](#): > > \*Originally posted by Evno \* > \*\* > > > [@](#): > > > > **posted by 23skidoo** > > When you perform the operation 1/3, what is the result? Is it 0.3…? > > this would be a reasonable point if, when dividing 3 into 1, you were able to finish the problem. But since you could never finish it, there is never a “result”. And 1/3 has a “result”, but it’s for use with finite numbers, or pies, or splitting the dinner bill. > > 1/3 and .333… are COMPLETELY different ideas. \*\* You can finish the problem quite easily. You have to realize that there is a pattern to the numbers in the answer. That pattern is: “All numbers are three.” No matter how long you keep up the long division, you will always get another 3. Just because the list is infinite doesn’t mean I don’t know what the next number will be, or what all the numbers are. 1/3 has a result. The result is that it is equal to a infinitely long decimal where every number after the decimal point is a “3”. ---
**Author:** ![Myrr21](https://avatars.discourse-cdn.com/v4/letter/m/e495f1/32.png) [@Myrr21](https://boards.straightdope.com/u/Myrr21)\ **Post date:** [August 18, 2001, 4:06pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/154 "2001-08-18T16:06:38Z")
> [@](#): > > okay, you want to prove to me that 0.999… is equal to 1.000… then do this for me… > > draw a line. Label one point on the line 0.9. one inch later on that line mark a point 1.000… (add as many 0’s as you would like. > > now, draw a line for 0.99. Now draw a line for 0.999. Now draw a line for 0.9999. Now one for 0.99999. Continue to keep drawing lines please. > > Call me when you draw an additional line on 1.000 You’ve just described a converging series, which was nicely proved to be equal to 1 quite a few posts up. Yes, this will only reach 1 in the infinite limit, but we’re dealing with infinity all over the place here anyway. Your disbelief doesn’t invalidate the multiple proofs. (also, the particle thing is a terrible analogy, as there is no correlation between the two, and particles can occupy the same space…after a fashion) ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [August 18, 2001, 4:40pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/155 "2001-08-18T16:40:15Z")
> [@](#): > > \*Originally posted by MrDeath \* > \*\* > > > [@](#): > > > > \*Originally posted by ultrafilter \* > > \*\* > > > > > [@](#): > > > > > > \*Originally posted by Boris B \* > > > \*\*First, let’s do some subtraction (and in all cases, my ellipses mean “continued to infinity”). > > > 1.0… - 0.9… = 0.0…1 = Z (this is where you have to indulge me, I know you can’t really follow infinity with single number but if we temporarily forget that you’ll see where I’m going) > > > \*\* > > > > There were some issues with the dots being surrounded on either side earlier in this thread (basically, it’s a no no.). Other than that, it’s not bad. \*\* > > You got a problem with omega + 1? (This is in fact _exactly_ what happens when you work with surreals.) > > I’m thinking of starting a MPSIMS thread to discuss the Axiom of Infinity. Anyone interested? \*\* Nah, Omega + 1 is fine by me. Other people had a problem with it, and I’m not sure it’s the clearest way to explain this. Go ahead and start that MPSIMS thread. ---
**Author:** ![Scott\_Dickerson](https://avatars.discourse-cdn.com/v4/letter/s/f04885/32.png) [@Scott\_Dickerson](https://boards.straightdope.com/u/Scott_Dickerson)\ **Post date:** [August 18, 2001, 9:10pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/156 "2001-08-18T21:10:48Z")
Fun stuff! It seems both the equalitarians and the anti-equalitarians have reached one point of agreement: that “.9rep” and “1” are understood as distinct (and thus different) concepts. It may be that even the anti-equalitarians are willing to concede that our system of mathematics mandates, beyond a shadow of a doubt, that the two symbols represent the exact same quantity. The disagreement at this point appears to be: Is “a certain number” a reference to (a) precisely one concept; or (b) precisely one quantity. If (a)–\> Kinoons and his followers have a good case. If (b)–\> They don’t. ---
**Author:** ![Evno](https://avatars.discourse-cdn.com/v4/letter/e/c68b51/32.png) [@Evno](https://boards.straightdope.com/u/Evno)\ **Post date:** [August 18, 2001, 9:31pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/157 "2001-08-18T21:31:07Z")
> [@](#): > > \*Originally posted by ultrafilter \* > **You still haven’t explained why 1/2 = 2/4 doesn’t bother you. How about why 1/3 = 3/9?** I’ve explained that. Perhaps you’d care to read these 150+ posts again… ---
**Author:** ![kinoons](https://avatars.discourse-cdn.com/v4/letter/k/a698b9/32.png) [@kinoons](https://boards.straightdope.com/u/kinoons)\ **Post date:** [August 18, 2001, 9:35pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/158 "2001-08-18T21:35:54Z")
> [@](#): > > If (a)–\> Kinoons and his followers have a good case. > If (b)–\> They don’t. 🙂 sounds like I’m a cult leader. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [August 18, 2001, 9:38pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/159 "2001-08-18T21:38:49Z")
> [@](#): > > \*Originally posted by Evno \* > \*\* > > > [@](#): > > > > \*Originally posted by ultrafilter \* > > **You still haven’t explained why 1/2 = 2/4 doesn’t bother you. How about why 1/3 = 3/9?** > > I’ve explained that. Perhaps you’d care to read these 150+ posts again… \*\* I did. I went back through and read all the posts. You have not even addressed anything remotely like this. If you have an explanation, share it. If you explained it before, quote it. But don’t count on my laziness to carry you. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [August 18, 2001, 9:42pm UTC](https://boards.straightdope.com/t/99999999-equal-to-1/76468/160 "2001-08-18T21:42:46Z")
> [@](#): > > \*Originally posted by Scott Dickerson \* > \*\*Fun stuff! > > It seems both the equalitarians and the anti-equalitarians have reached one point of agreement: that “.9rep” and “1” are understood as distinct (and thus different) concepts. > > It may be that even the anti-equalitarians are willing to concede that our system of mathematics mandates, beyond a shadow of a doubt, that the two symbols represent the exact same quantity. > > The disagreement at this point appears to be: > > Is “a certain number” a reference to (a) precisely one concept; or (b) precisely one quantity. > > If (a)–\> Kinoons and his followers have a good case. > If (b)–\> They don’t. \*\* I will not agree that .9… and 1 are different concepts. Both are infinite series of base 10 coefficients. They converge on the same quantity. “A certain number” is multiple concepts, depending on whether we’re talking about naturals, rationals, reals, complexes, or other things. There is also no single notion of “quantity” that applies to all these things, because the equality relationship denoted by “=” is different on each set. Assuming we restrict ourselves to the reals, it would be more accurate to describe “a certain number” as one quantity. [Previous page](https://boards.straightdope.com/t/99999999-equal-to-1/76468.md?page=7) [Next page](https://boards.straightdope.com/t/99999999-equal-to-1/76468.md?page=9)