# .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:** 71
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 26, 2014, 8:59pm UTC](https://boards.straightdope.com/t/999-1/27517/1406 "2014-07-26T20:59:25Z")
(In case my last paragraph gives the wrong impression, I don’t particularly have any problem with anything you’ve done in trying to bring posters such as **7777777** to understanding, **The Hamster King**. I’m just defending my choice to engage in the more sympathetic exercises I’ve engaged in, on top of all the other posts.) ---
**Author:** ![Bryan\_Ekers](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bryan_ekers/32/183_2.png) [@Bryan\_Ekers](https://boards.straightdope.com/u/Bryan_Ekers)\ **Post date:** [July 26, 2014, 10:16pm UTC](https://boards.straightdope.com/t/999-1/27517/1407 "2014-07-26T22:16:44Z")
> [@7777777](#): > > If you don’t see why, you must be blind: > …99999 = -1 is true Even your own cite points out the problem in this idea. Personally, I consider it a proof by contradiction: 1. Let’s say …99999 = _E_ 2. Multiply _E_ by 10 to get 10_E_, which would be …999990. 3. 10_E_ - _E_ = 9, clearly. One could also say _E_ - 10_E_ = 9. 4. The first equation resolves to _E_ = 1. The second to _E_ = -1. Both these results follow from the premises. The problem is treating an infinite value (in this case symbolized by _E_) like a conventional algebraic variable. It’s the same as defining _B_ = 0, slipping a “now divide both sides by _B_” into the sequence, and getting a nonsensical or contradictory result. James Tanton clearly recognizes this. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 26, 2014, 10:25pm UTC](https://boards.straightdope.com/t/999-1/27517/1408 "2014-07-26T22:25:06Z")
> [@Bryan\_Ekers](#): > > 1. 10_E_ - _E_ = 9, clearly No, you’ve made a sign error. 10E - E = -9, clearly, just as expected. We can make good sense of …999 = -1, if we want to, as I’ve noted above; for example, using the method outlined in [this post](http://boards.straightdope.com/sdmb/showpost.php?p=17582338&postcount=1367) to interpret infinitary decimal notation into the real numbers, or in terms of [10-adic numbers](https://en.wikipedia.org/wiki/P-adic_number), or various other things. Which isn’t to contradict that it is also frequently useful to instead think of …999 as a positively infinite value, as you say. ---
**Author:** ![The\_Great\_Unwashed](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/the_great_unwashed/32/4791_2.png) [@The\_Great\_Unwashed](https://boards.straightdope.com/u/The_Great_Unwashed)\ **Post date:** [July 26, 2014, 10:31pm UTC](https://boards.straightdope.com/t/999-1/27517/1409 "2014-07-26T22:31:57Z")
> [@Indistinguishable](#): > > But I’m not very fond of slapping people’s hands for wanting to explore on their own (the very thing to be encouraged in mathematics!) I agreed with pretty much your entire post, including the quote above. Unfortunately, I’m not sure this is a case of someone willing to explore but cowed by more sophisticated and cynical individuals, I think it is the opposite. Self-motivated exploration is not a quality usually associated with wanton stubborness to engage. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 26, 2014, 10:35pm UTC](https://boards.straightdope.com/t/999-1/27517/1410 "2014-07-26T22:35:43Z")
Oh, I agree. **7777777** is a lost cause. But it’s the general principle of the thing, for the sake of anyone else reading the thread. ---
**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:** [July 26, 2014, 10:42pm UTC](https://boards.straightdope.com/t/999-1/27517/1411 "2014-07-26T22:42:56Z")
> [@EdwinAmi](#): > > Indeed. That seems to be the final say on these weird maths. On the other thread, somebody said “if it’s defined a certain way…”, which is kind of what you’re saying here. > > Yes, if you have very specific definitions and conceputalizations of what you’re doing, you can use these maths. What laymen like us are saying is that it doesn’t directly correlate to anything you can do in the real world that forms intuitive math. Everything you can do in rational-number arithmetic can be “done” in real life. Any child can pick up multiple rocks, thereby “adding” them. You can easily draw standardized lengths and “add” or “multiply” fractions of them, making larger lengths. > > But whatever the hell these supposed sums of non-convergent series are, it doesn’t directly correlate to anything in real life, though the calculations may be ultimately useful. This is similar to imaginary numbers. The square root of a negative number is intuitively impossible, as well as _by definition_ (of what a square root is) impossible by the “normal” math that people are used to using. Nonetheless, said calculations are useful, because they create a two-dimensional number line that follows certain rules. “Traditional” or “intuitive” math would say it’s impossible, but you’re just ignoring that and creating a new math. This goes back to what I said earlier. People who don’t like the fact that 0.9999… = 1 want everything to be arithmetic. Your argument is conceptually equivalent. You are insisting that numbers are arithmetic and anything not learned in high school is meaningless, useless, or just plain insulting to normal people. Math is math. It includes arithmetic but that’s like saying that a human includes nostril hair. Nostril hair is useful but is such a ridiculously trivial part of being human that inflating its importance is laughable. If you’re a layman, then here’s what you do when the topic is math. Shut up and listen. Maybe even learn. You say the same thing more explicitly in your next post so I’ll just concentrate on a different aspect of it. > [@EdwinAmi](#): > > **But I would completely disagree and propose what perhaps we might call a “Chomsky” theory of math and mathematical understanding** > Anybody familiar with Chomsky’s proven theories on linguistics? Basically, people DO indeed have some innate, pre-programmed language skills. This should be obvious when viewed through a biological lense, but for whatever reason it was a subject of debate. > > What I’m saying is math is the same way. People have some innate, biologically pre-programmed sense of numbers, or the concept of scaling and counting, and tend to think of such things in terms of what they can directly do in reality; that is, our understanding evolved based on what we could actively do in the real world. I’m under the impression that most of Chomsky wrote on linguistics has been discredited to the point where only a few diehards adhere to his theories, similar to the fate of Freud. Whether humans have an innate sense of counting numbers is debatable. Its truth doesn’t matter because - again - arithmetic is not math. ---
**Author:** ![EdwinAmi](https://avatars.discourse-cdn.com/v4/letter/e/5fc32e/32.png) [@EdwinAmi](https://boards.straightdope.com/u/EdwinAmi)\ **Post date:** [July 26, 2014, 10:55pm UTC](https://boards.straightdope.com/t/999-1/27517/1412 "2014-07-26T22:55:05Z")
> [@](#): > > in high school is meaningless, useless, or just plain insulting to normal Not at ALL what I said Note that I actually specifically said that are “useful” > [@](#): > > Whether humans have an innate sense of counting numbers is debatable Also not what I said. I said an innate ability to learn math, and a species-wide tendency to develop it (note that numerous societies independently figured out the same mathematical principles). I also pointed to a proven innate sense of geometry that was proven in a study You clearly weren’t getting what I was writing, so feel free to go back, read it carefully, don’t add anything into it or put your own spin on it and instead actually try to listen and absorb it, and then maybe we can talk ---
**Author:** ![Bryan\_Ekers](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bryan_ekers/32/183_2.png) [@Bryan\_Ekers](https://boards.straightdope.com/u/Bryan_Ekers)\ **Post date:** [July 26, 2014, 10:59pm UTC](https://boards.straightdope.com/t/999-1/27517/1413 "2014-07-26T22:59:01Z")
> [@Indistinguishable](#): > > No, you’ve made a sign error. 10E - E = -9, clearly, just as expected. Nope. If _E_ is …999999 and 10_E_ is …999990 : ```auto *E*: ........99999999 10*E*: - ........99999990 ============== 9 ``` Why not? They both go infinitely to the left, after all. Similarly: ```auto 10*E*: ........99999990 *E*: - ........99999999 ============== -9 ``` …or 9, if we assume _E_ can be treated like a conventional algebraic variable. My point is that it cannot. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 26, 2014, 11:00pm UTC](https://boards.straightdope.com/t/999-1/27517/1414 "2014-07-26T23:00:16Z")
What is your argument that 10E - E = 9 by treating E like a conventional algebraic variable? I’m not seeing it. (I assure you, you _can_ treat E like a conventional algebraic variable in a consistent manner which determines it to be -1 and not 1. I’ve noted several natural contexts in which such reasoning applies.) ---
**Author:** ![Budget\_Player\_Cadet](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/budget_player_cadet/32/205_2.png) [@Budget\_Player\_Cadet](https://boards.straightdope.com/u/Budget_Player_Cadet)\ **Post date:** [July 26, 2014, 11:02pm UTC](https://boards.straightdope.com/t/999-1/27517/1415 "2014-07-26T23:02:44Z")
> [@Bryan\_Ekers](#): > > Even your own cite points out the problem in this idea. Personally, I consider it a proof by contradiction: > > 1. Let’s say …99999 = _E_ > 2. Multiply _E_ by 10 to get 10_E_, which would be …999990. > 3. 10_E_ - _E_ = 9, clearly. One could also say _E_ - 10_E_ = 9. > 4. The first equation resolves to _E_ = 1. The second to _E_ = -1. Both these results follow from the premises. > > The problem is treating an infinite value (in this case symbolized by _E_) like a conventional algebraic variable. It’s the same as defining _B_ = 0, slipping a “now divide both sides by _B_” into the sequence, and getting a nonsensical or contradictory result. > > James Tanton clearly recognizes this. [Relevant?](https://www.math.toronto.edu/mathnet/falseProofs/first1eq2.html) ---
**Author:** ![sich\_hinaufwinden](https://avatars.discourse-cdn.com/v4/letter/s/c5a1d2/32.png) [@sich\_hinaufwinden](https://boards.straightdope.com/u/sich_hinaufwinden)\ **Post date:** [July 26, 2014, 11:03pm UTC](https://boards.straightdope.com/t/999-1/27517/1416 "2014-07-26T23:03:51Z")
> [@Indistinguishable](#): > > No, you’ve made a sign error. I don’t think he made a sign error. If I had to guess, I’d say he thinks 10\*E is like E with the same number of 9s but now with a zero at the end and thus bigger than E. But an infinite number of 9s shifted left with a zero at the end is still an infinite number of 9s. And so 10E and E both have the same infinite number of digits. But since 10E has a zero in the tens place rather than a 9 it is smaller than E, leading to the -1 result. You never get out to infinity where 10E has 1 more digit than E. Which is non-intuitive for sure if I’m understanding that line of reasoning correctly. Apologies if I’ve gotten it wrong and muddied the waters even more and hopefully you’ll set me straight if so. ---
**Author:** ![Bryan\_Ekers](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bryan_ekers/32/183_2.png) [@Bryan\_Ekers](https://boards.straightdope.com/u/Bryan_Ekers)\ **Post date:** [July 26, 2014, 11:12pm UTC](https://boards.straightdope.com/t/999-1/27517/1417 "2014-07-26T23:12:12Z")
> [@Indistinguishable](#): > > What is your argument that 10E - E = 9 by treating E like a conventional algebraic variable? I’m not seeing it. Because you’re giving _E_ a coefficient and treating it as if it was meaningful. It’s like asking “what’s 10 times ∞?” The answer is “∞”, not “10∞”. You can _get_ an answer if you decide on certain conventions and apply them consistently, but not necessarily meaningfully. > [@](#): > > (I assure you, you _can_ treat E like a conventional algebraic variable in a consistent manner which determines it to be -1 and not 1. I’ve noted several natural contexts in which such reasoning applies.) Well, great, I’m confident you can get a whole _bunch_ of answers that seem perfectly logical. Somewhere along the way, though, there’s going to be an assumption made about _E_ that may not quite be supportable. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 26, 2014, 11:13pm UTC](https://boards.straightdope.com/t/999-1/27517/1418 "2014-07-26T23:13:57Z")
> [@sich\_hinaufwinden](#): > > I don’t think he made a sign error. If I had to guess, I’d say he thinks 10\*E is like E with the same number of 9s but now with a zero at the end and thus bigger than E. > > But an infinite number of 9s shifted left with a zero at the end is still an infinite number of 9s. And so 10E and E both have the same infinite number of digits. But since 10E has a zero in the tens place rather than a 9 it is smaller than E, leading to the -1 result. You never get out to infinity where 10E has 1 more digit than E. 10E _is_ like E with the same number of 9s but now with a zero at the end. And an infinite number 9s shifted left with a zero at the end _is_ still an infinite number of 9s. And since 10E has a zero in the (minor correction) units place rather than a 9, you do get, upon subtraction, 10E - E = -9, or E - 10E = 9, both amounting to E = -1. You _do_ (on this account of things) never get out to infinity where 10E has 1 more digit than E. All of that is true. And all of that leads to E = -1. I’m pointing out that **Bryan Ekers** was mistaken in supposing there to be a clear algebraic derivation of E = 1. There is not; there are well-studied mathematical contexts in which all the usual rules of algebra (not dealing with ordering) apply, such that E unambiguously equals -1. > [@](#): > > Which is non-intuitive for sure if I’m understanding that line of reasoning correctly. Apologies if I’ve gotten it wrong and muddied the waters even more and hopefully you’ll set me straight if so. Yes, the non-intuitive part is that this clashes with usual rules of ordering like “Adding positive values makes you bigger, not smaller” and “-1 is smaller than 0, not bigger”. So all of this can only be interpreted in a context where we either weaken those rules, or give up on the idea of ordering entirely. [Just as when we consider a numeric system in which there is a square root of -1, we do not impose the usual ordering structure upon it] ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 26, 2014, 11:18pm UTC](https://boards.straightdope.com/t/999-1/27517/1419 "2014-07-26T23:18:36Z")
> [@Bryan\_Ekers](#): > > Even your own cite points out the problem in this idea. Personally, I consider it a proof by contradiction: > > 1. Let’s say …99999 = _E_ > 2. Multiply _E_ by 10 to get 10_E_, which would be …999990. > 3. 10_E_ - _E_ = 9, clearly. Let’s go back to this. Why do you say “clearly” here? What makes it clear that 10E - E is 9? You later say: > [@](#): > > One could also say _E_ - 10_E_ = 9. That part _is_ clear. Digit-by-digit subtraction gives us E - 10E = …0009. But I am confused as to why you considered the first part clear. What is most clear is that 10 - E = -9, by digit-by-digit subtraction, as you yourself demonstrated in a subsequent post. > [@Bryan\_Ekers](#): > > You can _get_ an answer if you decide on certain conventions and apply them consistently, but not necessarily meaningfully. Yes meaningfully. I’ve spelt out such meanings! There’s nothing unmeaningful about, e.g., the 10-adics; it’s just a meaning you are unfamiliar with. ---
**Author:** ![sich\_hinaufwinden](https://avatars.discourse-cdn.com/v4/letter/s/c5a1d2/32.png) [@sich\_hinaufwinden](https://boards.straightdope.com/u/sich_hinaufwinden)\ **Post date:** [July 26, 2014, 11:19pm UTC](https://boards.straightdope.com/t/999-1/27517/1420 "2014-07-26T23:19:08Z")
> [@Bryan\_Ekers](#): > > Nope. If _E_ is …999999 and 10_E_ is …999990 : > > ```auto > > *E*: ........99999999 > 10*E*: - ........99999990 > ============== > 9 > > ``` > > Why not? They both go infinitely to the left, after all. Similarly: > > ```auto > > 10*E*: ........99999990 > *E*: - ........99999999 > ============== > -9 > > ``` > > …or 9, if we assume _E_ can be treated like a conventional algebraic variable. My point is that it cannot. If E - 10E = 9 then E = -1 If 10E - E = -9 then E=-1 Not what you wrote in your previous post where you said E could resolve to 1 or -1. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 26, 2014, 11:23pm UTC](https://boards.straightdope.com/t/999-1/27517/1421 "2014-07-26T23:23:41Z")
> [@Indistinguishable](#): > > 10 **E** - E = -9, by digit-by-digit subtraction, as you yourself demonstrated in a subsequent post. Typo corrected in bold. Anyway, **sich\_hinaufwinden** has said more clearly and succinctly what I’ve been trying to point out. Still, let me point it out again: > [@Bryan\_Ekers](#): > > ```auto > > 10*E*: ........99999990 > *E*: - ........99999999 > ============== > -9 > > ``` > > …or 9, if we assume _E_ can be treated like a conventional algebraic variable. Why do you say “or 9”, when you just finished showing me that the calculation actually comes out to -9? What justifies flipping the sign? ---
**Author:** ![Bryan\_Ekers](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bryan_ekers/32/183_2.png) [@Bryan\_Ekers](https://boards.straightdope.com/u/Bryan_Ekers)\ **Post date:** [July 26, 2014, 11:33pm UTC](https://boards.straightdope.com/t/999-1/27517/1422 "2014-07-26T23:33:22Z")
> [@Indistinguishable](#): > > Let’s go back to this. Why do you say “clearly” here? What makes it clear that 10E - E is 9? Because if we were going to treat _E_ like a conventional algebraic variable that happens to have 9 as a final digit: 10_E_ is _E_ shifted one digit to the left and has 0 as a final digit. 10_E_ - _E_ = 9. Of course, _E_ being infinite in length, this doesn’t work. Now, I’m currently reading the wiki page on _p-adic_ numbering systems, a mathematical concept I was not previously aware of, and I’m willing to entertain the idea that once I get a handle on how it works, I could end up saying _E_ = -1 _in a p-adic system_. I’m pretty sure the qualifier will be necessary. > [@](#): > > Yes meaningfully. I’ve spelt out such meanings! There’s nothing unmeaningful about, e.g., the 10-adics; it’s just a meaning you are unfamiliar with. Hey, no argument here. My initial comment was directed at 7777777, who (as far as I could tell) was citing James Tanton who wrote that …9999 = -1 _while using conventional arithmetic_ and pointing out the limitations of doing so. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 26, 2014, 11:36pm UTC](https://boards.straightdope.com/t/999-1/27517/1423 "2014-07-26T23:36:25Z")
> [@Bryan\_Ekers](#): > > Because if we were going to treat _E_ like a conventional algebraic variable that happens to have 9 as a final digit: > > 10E is E shifted one digit to the left and has 0 as a final digit. > 10E - E = 9. No, 10E - E = negative 9. You carried out the calculation yourself, above. In the units place, 10E has a 0 and E has a 9, so we get 0 - 9 = negative 9 in the units place. In every other place, 10E and E have the same digit 9, so we get no contribution. Thus, we ultimately get negative 9. Let me put it this way: What is 99990 - 99999? ---
**Author:** ![Bryan\_Ekers](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bryan_ekers/32/183_2.png) [@Bryan\_Ekers](https://boards.straightdope.com/u/Bryan_Ekers)\ **Post date:** [July 26, 2014, 11:42pm UTC](https://boards.straightdope.com/t/999-1/27517/1424 "2014-07-26T23:42:47Z")
> [@Indistinguishable](#): > > Let me put it this way: What is 99990 - 99999? It doesn’t matter because you’re using numbers of finite size, abandoning the essential concept of _E_, namely that it is of infinite size. I’m cheerfully prepared to admit that there are approaches to this issue outside my current sphere of knowledge, but I’m not about to get worked up about it. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [July 26, 2014, 11:49pm UTC](https://boards.straightdope.com/t/999-1/27517/1425 "2014-07-26T23:49:13Z")
> [@Bryan\_Ekers](#): > > It doesn’t matter because you’re using numbers of finite size, abandoning the essential concept of _E_, namely that it is of infinite size. Why would you say 0 - 9 = negative 9, 90 - 99 = negative 9, 990 - 999 = negative 9, etc., but applying the same treatment to …990 - …999 would give us positive 9? I agree it is an essential fact about E that it is of infinite size. But regardless of that essential fact, I don’t see why you would look at a subtraction where the first value has a units place of 0 and second value has a units place of 9, and conclude that this means the result “clearly” gets a contribution of +9 in the units place rather than -9. You say this is what conventional algebraic rules would demand, but the conventional rules are that 0 - 9 = -9, not 0 - 9 = +9. > [@](#): > > I’m cheerfully prepared to admit that there are approaches to this issue outside my current sphere of knowledge, but I’m not about to get worked up about it. I’m glad. However, I still feel you would benefit from re-examining your initial argument that 10E - E clearly equals 9. Without further explanation for how you came to that conclusion, this still seems to me a sign error; a calculational slip, like the child who mistakenly supposes 42 - 18 = 36. [Previous page](https://boards.straightdope.com/t/999-1/27517.md?page=70) [Next page](https://boards.straightdope.com/t/999-1/27517.md?page=72)