# .999 = 1? **URL:** \ **Category:** Factual Questions\ **Created:** [July 31, 2000, 6:28am UTC](https://boards.straightdope.com/t/999-1/27517 "2000-07-31T06:28:03Z")\ **Posts on this page:** 20\ **Page:** 21
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [August 6, 2012, 8:00pm UTC](https://boards.straightdope.com/t/999-1/27517/401 "2012-08-06T20:00:06Z")
> [@allotrope](#): > > Oh god, that was completely meaningless. You really went all out trying to camouflage the fact that n has to be evaluated AT infinity - which is precisely what I’ve been saying all along. But the epsilon delta stuff was a nice touch. Hey, I’m happy to evaluate n at infinity. It’s not what you will find in those calculus textbooks you accused me of ignoring, but I’m happy to do it. I’ve noted the coherence of doing so more than anyone else in this thread (e.g., [here](http://boards.straightdope.com/sdmb/showpost.php?p=15351065&postcount=176)). But then we will run up against this: > [@Indistinguishable](#): > > That image also concludes by using the rule that zero is the limit, as n goes to infinity, of 1/10^n. This is just a rule (or consequence of certain rule choices). One could pick a different one. Who says 1/10[sup]∞[/sup] is 0? Maybe 1/10[sup]∞[/sup] is just 1/10[sup]∞[/sup]. Maybe it’s just a very, very small, but positive, quantity. On the other hand, maybe it is 0. That’s the choice of rules I was pointing out. ---
**Author:** ![allotrope](https://avatars.discourse-cdn.com/v4/letter/a/edb3f5/32.png) [@allotrope](https://boards.straightdope.com/u/allotrope)\ **Post date:** [August 6, 2012, 8:01pm UTC](https://boards.straightdope.com/t/999-1/27517/402 "2012-08-06T20:01:26Z")
Whatever dude. We were both right all along - how’s that? ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [August 6, 2012, 8:06pm UTC](https://boards.straightdope.com/t/999-1/27517/403 "2012-08-06T20:06:42Z")
The original disagreement we had was like so: > [@allotrope](#): > > I’m sorry to jump in guys (I’ll be ignoring you, Erik), but AT infinity, there really is no infinitesimal. It’s not just a rule. > [@Indistinguishable](#): > > I have no idea what this first sentence means, but you’re wrong. Everything in math is just a rule. On this, I continue to be correct. Everything in math is just a rule, including the question of whether 1/10[sup]∞[/sup] is 0 or positive. Saying “We were both right all along” is only saying “Yup. It’s just a choice of rules”. Beyond that original disagreement, I never had any quarrel with you. You’re the one who suddenly went off on me. ---
**Author:** ![allotrope](https://avatars.discourse-cdn.com/v4/letter/a/edb3f5/32.png) [@allotrope](https://boards.straightdope.com/u/allotrope)\ **Post date:** [August 6, 2012, 8:09pm UTC](https://boards.straightdope.com/t/999-1/27517/404 "2012-08-06T20:09:45Z")
And as the equation we just looked at shows, when the decimal recurs ad infinitum and n = ∞, then it is not a rule but in actual fact there is no residual infinitesimal. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [August 6, 2012, 8:14pm UTC](https://boards.straightdope.com/t/999-1/27517/405 "2012-08-06T20:14:46Z")
The equation we looked at ([I shall link it again](http://imageshack.us/photo/my-images/213/capturenfo.png/)) does nothing to show that 1/10[sup]∞[/sup] is 0. It simply claims this (or rather, claims that 1 - the limit as n goes to ∞ of 1/10[sup]n[/sup] is 1) without argument. It _assumes_ this as one of its background rules. It is dependent upon those background rules. It is based on those rules. It is not a matter of external fact. We could work in a context of different background rules, and then the argument would no longer go through. We would have a different conclusion, relative to different rules. ---
**Author:** ![allotrope](https://avatars.discourse-cdn.com/v4/letter/a/edb3f5/32.png) [@allotrope](https://boards.straightdope.com/u/allotrope)\ **Post date:** [August 6, 2012, 8:18pm UTC](https://boards.straightdope.com/t/999-1/27517/406 "2012-08-06T20:18:34Z")
Yes, the background rules are basic logic. But since you’re the expert, which of the axioms are to blame for this travesty? I’m curious now. ---
**Author:** ![Senegoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/senegoid/32/6606_2.png) [@Senegoid](https://boards.straightdope.com/u/Senegoid)\ **Post date:** [August 6, 2012, 10:59pm UTC](https://boards.straightdope.com/t/999-1/27517/407 "2012-08-06T22:59:34Z")
Holy tamoley! Four pages of new posts, and that’s just TODAY! Before I even start to skim them, I’m already wondering (as I already did yesterday, but that was _four pages_ ago [ETA: Post #205] ): Is there anything new still to be said? Does **erik150x** have a new question to ask or a new argument to make, that he hasn’t already written? Does anyone else have any new and different answers, different from what’s already been written? This thread is rapidly overtaking the [Problems With Relativity](http://www.boards.straightdope.com/sdmb/showthread.php?t=404854) thread! Relativity: 598 posts as of 07/04/2012. Value of .999… : 405.999999999999… and counting! Okay, okay, now I’ll go skim all the new stuff and see if there’s anything new there. . . . ---
**Author:** ![Monimonika](https://avatars.discourse-cdn.com/v4/letter/m/85e7bf/32.png) [@Monimonika](https://boards.straightdope.com/u/Monimonika)\ **Post date:** [August 6, 2012, 11:32pm UTC](https://boards.straightdope.com/t/999-1/27517/408 "2012-08-06T23:32:31Z")
> [@erik150x](#): > > Yeah, I don’t follow what point your tryign to make. You posted something about adding .999… to itself 10 times to simulate multiplication by 10. Which you can’t really do without limits. I think you need to go back to the post of yours I was repsonding to. And also the fact that in either case they are on shakey ground. But you want to just ignore all the decimal places beyond some unamed limit? > > Maybe we are just not connecting with each other here. I don’t know why there is need to bring the construction of reals or divison into this? We were simply tlaking about adding already existing numbers. Namely .999… and 10 of them. Okay, my main problem with you is that you seem to think that multiplying by 10 means adding a 0 at the “end” of an infinite string of repeating digits and shifting the decimal point to the right. (Also, that you seem to think that there’s an “end” to as infinite string of digits.) Let’s go with that idea of yours for a bit and see what happens when you try to do the reverse. Namely, divide by 10. I will denote “divide by 10” as “x 0.1” in the following (you agree that 1/10 = 0.1, right?). When trying to solve for: 0.3 x 0.1 x 0.829 x 10 = ? we usually first resolve the multiplication of “0.1” and “10” to just “1” and don’t bother with multiplying in strict order from left to right. But for the following examples we’re not going to use this shortcut. Let’s use your version of decimal notation to see how this would work with repeating decimal numbers: 0.333…3 x 10 x 0.1 = 3.333…0 x 0.1 = 0.333…3 Nicely resolves to the same answer as 0.333…3 x 1, doesn’t it? Try this then: 0.333…3 x 0.1 x 10 = 0.033…3 (–\>3?) x 10 Whoa. What am I supposed to do with that 3 on the right there that just got shoved out of the “end” position to the right? I mean, it’s not like there are any more decimal places to the right for that 3 to go to, is there? The end is the end is the end. I guess we’ll just have to drop it into the void. Bye-bye, 3 that was formerly at the end of 0.333…3! Continuing: 0.033…3 x 10 = 0.333…0 …why is this answer different from the previous answer? Would you argue that the order of the numbers matter in multiplication? That multiplying by “0.1 x 10” is not the same as multiplying by “10 x 0.1” or just by “1”? Or would you say that I should not have dropped that dangling 3 when I multiplied by 0.1? Should it have kept its “end” position and the number of decimal places just be increased to accommodate it? But then why can’t the same be done for multiplying by 10? Keep the 3 at the “end”, don’t add 0, and just decrease the number of infinite decimal places. ((The stuff below here is not as important, so you can just skip replying to it. I’d rather you answer to the above than get sidetracked. Plus, I purposely insult your intelligence below anyway.)) “But you must add 0!” you might or might not say. Look, it’s most likely been pointed out before (can’t recall), but that 0 only seems like it’s added because you’ve only been multiplying 10 to terminating rational numbers. Terminating rational numbers are numbers like 0.978 (which you apparently think only has three available decimal places to the right) (and yeah, that’s what your logic leads to). What’s 0.978 also equal to? 0.9780000 and 0.9780000000000 and 0.978000 and 0.9780. There’s actually an infinite number of the decimal places to the right. They’re just all filled by 0s. When multiplying by 10, the decimal point shifts to the right and the zero to the right of 0.978 becomes visible and you get 9.780, which can also be written at 9.78 (Oh no, I’m down to just two decimal places on the right now!). ((END)) ---
**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:** [August 6, 2012, 11:35pm UTC](https://boards.straightdope.com/t/999-1/27517/409 "2012-08-06T23:35:51Z")
> [@Senegoid](#): > > Holy tamoley! Four pages of new posts, and that’s just TODAY! We’ve had the ability to set the default posts per page to 200 for at least five years. How is it that so many people don’t know it? > [@](#): > > Does **erik150x** have a new question to ask or a new argument to make, that he hasn’t already written? No, he still thinks that when you subtract .999~ from 1 you can put a 1 at the end of the zeros. ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [August 6, 2012, 11:55pm UTC](https://boards.straightdope.com/t/999-1/27517/410 "2012-08-06T23:55:11Z")
> [@Senegoid](#): > > Holy tamoley! > … Is there anything new still to be said? But isn’t there an elegant relationship between question and thread? OP asks about 0.99999999… going on forever, even though it should come to a simple clean end. And the thread mimics its namesake, 0.999999… ---
**Author:** ![bobot](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bobot/32/21622_2.png) [@bobot](https://boards.straightdope.com/u/bobot)\ **Post date:** [August 7, 2012, 12:18am UTC](https://boards.straightdope.com/t/999-1/27517/411 "2012-08-07T00:18:32Z")
> [@erik150x](#): > > Even after infinity 9s I tend to think we are only infinitely close to 1. Ahhh, finally, The Truth! Let’s just call it a tolerance, or admit that the number is rounded to make our lives easier. Let’s have beer and pizza, huh? ---
**Author:** ![Monimonika](https://avatars.discourse-cdn.com/v4/letter/m/85e7bf/32.png) [@Monimonika](https://boards.straightdope.com/u/Monimonika)\ **Post date:** [August 7, 2012, 1:03am UTC](https://boards.straightdope.com/t/999-1/27517/412 "2012-08-07T01:03:10Z")
> [@erik150x](#): > > > [@allotrope](#): > > > > No, what I’m saying is that since there is no tail, there is no difference to find. > > > > I was using the subtraction example to make it feel more concrete. > > > > But the point really is that to find that tiny difference you have to keep hopping from one 9 to the next ad infinitum. Therefore, if you can never actually FIND the difference, no difference exists. > > **I dont think the numbers care if we can find them. They do their thing regardless 😉** > > I think 1 + 1 = 2 before we ever convcieved of a number. You know what, screw this. I went up to catch up on the other comments but then I see this kind of glib response that erik150 gives out and realize what I was suspecting all along. erik150 just ignores the majority of arguments and explanations on the contradictions and flaws in his (or her) ideas. See above. Did erik150 even consider what was written and what it means that a difference between two numbers cannot be found? Nope. Just makes a non sequitur about what numbers supposedly don’t “care” about, as if that makes a difference in how numbers are defined or manipulated. With this kind of attitude, no wonder erik150 claims in the resurrection post (yes, the very post that brought back a dead thread from almost 12 years ago!): > [@erik150x](#): > > “Every time I begin to think I am the smartest person in the world, some udiot comes along to prove me even smarter. ;-p” Yeah, says the one who _still_ can’t figure out where the “0” actually comes from when 0.978 is multiplied by 10 to get 9.780 and thinks mathematical assumptions/shortcuts learned in grade school for simple numbers are more important than learning about the reality behind those assumptions. I didn’t do much, and erik150 will most likely make fun of me for it (I don’t care), but I will leave now since reading erik150’s posts are like reading FSTDT, bad for the heart and facepalm-inducing. :smack: ---
**Author:** ![LinusK](https://avatars.discourse-cdn.com/v4/letter/l/258eb7/32.png) [@LinusK](https://boards.straightdope.com/u/LinusK)\ **Post date:** [August 7, 2012, 1:03am UTC](https://boards.straightdope.com/t/999-1/27517/413 "2012-08-07T01:03:43Z")
> [@Exapno\_Mapcase](#): > > We’ve had the ability to set the default posts per page to 200 for at least five years. How is it that so many people don’t know it? > No, he still thinks that when you subtract .999~ from 1 you can put a 1 at the end of the zeros. No, I’d say he thinks the difference is not that you put a 1 at the end of the zeroes, but that the difference is infinitely small. ---
**Author:** ![Senegoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/senegoid/32/6606_2.png) [@Senegoid](https://boards.straightdope.com/u/Senegoid)\ **Post date:** [August 7, 2012, 1:24am UTC](https://boards.straightdope.com/t/999-1/27517/414 "2012-08-07T01:24:06Z")
> [@septimus](#): > > But isn’t there an elegant relationship between question and thread? > > **OP asks about 0.99999999… going on forever, even though it should come to a simple clean end. > And the thread mimics its namesake, 0.999999…** As well as getting increasingly irrational . . . :o But when **erik150x** gets to the _last post_, a pot of enlightenment will be there. [sub]Oh, wait… There’s still infinitely more posts to go![/sub] ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [August 7, 2012, 1:36am UTC](https://boards.straightdope.com/t/999-1/27517/415 "2012-08-07T01:36:51Z")
Let me put this a different way with Erik’s assertion that .999… \* 10 = 9.999…0. Define an algorithm for multiplying by 10, such that each item is moved left one place and a 0 is appended to the end.\* .999 \* 10 = 9.990 Now try this with .999… \* 10 So for each nine, shift it to the left one place, when there are no more nines, append a zero. But wait… there are infinitely many nines. This algorithm CANNOT HALT. In fact, it can’t even get to step 2 (add a zero). Do you see why .999… \* 10 = 9.99…0 is problematic now? ---
**Author:** ![Leo\_Bloom](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/leo_bloom/32/10377_2.png) [@Leo\_Bloom](https://boards.straightdope.com/u/Leo_Bloom)\ **Post date:** [August 7, 2012, 1:45am UTC](https://boards.straightdope.com/t/999-1/27517/416 "2012-08-07T01:45:01Z")
> [@Senegoid](#): > > This thread is rapidly overtaking the [Problems With Relativity](http://www.boards.straightdope.com/sdmb/showthread.php?t=404854) thread!.. I thought the same thing a few pages ago. Go SDMB! ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 7, 2012, 1:45am UTC](https://boards.straightdope.com/t/999-1/27517/417 "2012-08-07T01:45:24Z")
You all may be right, but I don’t see how you have found the flaw in my anlogy of mapping the halfway points from Zeno’s Paradox of getting from point A to B to the infinite decimal expansion of what what ever repeating decimal. If I travel from point A to B, have I not crossed an infinite number of half way points? Is this infinity some how smaller than the repeating decimals? If so explain how? If not, then if I move form point A to point B have I not just reach end of af infinitie decimal expansion? If you want to make the same darn statement over and over and over, that there is no end, fine then tell me how I ever reach point B in Zeno’s paradox, because that is an infinite amount of points but I clearly reach the end. I see no one explaining the flaw. Simply saying there is no end, does not permit me to reach poitn B in Zeno’s Paradox, so then we have another paradox? ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [August 7, 2012, 1:58am UTC](https://boards.straightdope.com/t/999-1/27517/418 "2012-08-07T01:58:10Z")
> [@erik150x](#): > > If you want to make the same darn statement over and over and over, that there is no end, fine then tell me how I ever reach point B in Zeno’s paradox, because that is an infinite amount of points but I clearly reach the end. Yes, you have clearly restated Zeno’s paradox. The answer to it involves limits. You refuse to accept the concept of limits, so it’s kind of difficult to answer anything because you say “uh uh, without limits!” ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 7, 2012, 1:58am UTC](https://boards.straightdope.com/t/999-1/27517/419 "2012-08-07T01:58:29Z")
> [@erik150x](#): > > You all may be right, but I don’t see how you have found the flaw in my anlogy of mapping the halfway points from Zeno’s Paradox of getting from point A to B to the infinite decimal expansion of what what ever repeating decimal. > > If I travel from point A to B, have I not crossed an infinite number of half way points? > > Is this infinity some how smaller than the repeating decimals? > > If so explain how? > > If not, then if I move form point A to point B have I not just reach end of af infinitie decimal expansion? > > If you want to make the same darn statement over and over and over, that there is no end, fine then tell me how I ever reach point B in Zeno’s paradox, because that is an infinite amount of points but I clearly reach the end. > > I see no one explaining the flaw. Simply saying there is no end, does not permit me to reach poitn B in Zeno’s Paradox, so then we have another paradox? To clarify more… I am not contending that Zeno’s Pradox is a pradox, we can reach the end of an infinite series, thanks to limits… 1st half of the distance in half the time 1/2 the remaining in 1/4 the time and so on… taking the limit we have our answer. I am saying rather is that you can reach the end of an infiinite series as clearly as you do in the resolution to Zeno’s Pardox. ---
**Author:** ![erik150x](https://avatars.discourse-cdn.com/v4/letter/e/c37758/32.png) [@erik150x](https://boards.straightdope.com/u/erik150x)\ **Post date:** [August 7, 2012, 1:59am UTC](https://boards.straightdope.com/t/999-1/27517/420 "2012-08-07T01:59:41Z")
> [@Jragon](#): > > Yes, you have clearly restated Zeno’s paradox. The answer to it involves limits. You refuse to accept the concept of limits, so it’s kind of difficult to answer anything because you say “uh uh, without limits!” I said about 300 post ago let assume Limits are valid! [Previous page](https://boards.straightdope.com/t/999-1/27517.md?page=20) [Next page](https://boards.straightdope.com/t/999-1/27517.md?page=22)