# Is 0.999...=1 ? **URL:** \ **Category:** Factual Questions\ **Created:** [April 4, 2000, 2:32pm UTC](https://boards.straightdope.com/t/is-0-999-1/10990 "2000-04-04T14:32:00Z")\ **Posts on this page:** 20\ **Page:** 3
**Author:** ![MrKnowItAll](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mrknowitall/32/87_2.png) [@MrKnowItAll](https://boards.straightdope.com/u/MrKnowItAll)\ **Post date:** [April 5, 2000, 4:21am UTC](https://boards.straightdope.com/t/is-0-999-1/10990/41 "2000-04-05T04:21:00Z")
OK, NTG. I get your meaning, but here’s where I have a problem. Just because it’s impossible to write out doesn’t mean that that 1 at the end of all those zeros doesn’t exist. We’re dealing with the pure and inviolate realm of mathematics, which doesn’t mean have all that much to do with the real world. That “1” is a theoretical “1”, but it is there. Isn’t it? * * * _God is my co-pilot. Blame Him._ ---
**Author:** ![MrKnowItAll](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mrknowitall/32/87_2.png) [@MrKnowItAll](https://boards.straightdope.com/u/MrKnowItAll)\ **Post date:** [April 5, 2000, 4:30am UTC](https://boards.straightdope.com/t/is-0-999-1/10990/42 "2000-04-05T04:30:00Z")
Ok, quick follow up. First of all, strike the “mean” in my post. It’s what I get for trying to form coherent sentences when I should be in bed. Second, the idea that .999… equals 1 just because they’re infinitely close just doesn’t hold water with me. As my Grandpappy used to say, “Close only counts in hand grenades and horseshoes.” Thirdly, why the hell hasn’t Cecil done a column on this? It’s come up several times in GQ, and it seems to be right up his alley. How 'bout it Nick and manny? Can we get Unca Cece to weigh in on this? * * * _God is my co-pilot. Blame Him._ ---
**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:** [April 5, 2000, 4:45am UTC](https://boards.straightdope.com/t/is-0-999-1/10990/43 "2000-04-05T04:45:00Z")
0.000…1 is not a real number. If you want to stray into non-standard analysis and call it dx, OK, but first you had better have a solid grip on standard analysis where 0.9999… is exactly equal to 1. * * * Virtually yours, DrMatrix “Feynman was wrong. I understand Quantum Physics completely. Anybody seen my drugs?” - **WallyM7** ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [April 5, 2000, 4:46am UTC](https://boards.straightdope.com/t/is-0-999-1/10990/44 "2000-04-05T04:46:00Z")
Silo’s point was a proof that .99… = 1 using the sum of an infinite geometric series. The sum of such a series is a/(1-r), where a is the first term and r is the common ratio. .999… = .9 + .09 + .009 + .0009 + … which is an infinite geometric series with first term .9 and common ratio .1. So .999… = .9/(1 - .1) = 1. ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [April 5, 2000, 4:48am UTC](https://boards.straightdope.com/t/is-0-999-1/10990/45 "2000-04-05T04:48:00Z")
Silo’s point was a proof that .99… = 1 using the sum of an infinite geometric series. The sum of such a series is a/(1-r), where a is the first term and r is the common ratio. .999… = .9 + .09 + .009 + .0009 + … which is an infinite geometric series with first term .9 and common ratio .1. So .999… = .9/(1 - .1) = 1. * * * …ebius sig. This is a moebius sig. This is a mo… (sig line courtesy of WallyM7) ---
**Author:** ![tony1234](https://avatars.discourse-cdn.com/v4/letter/t/4491bb/32.png) [@tony1234](https://boards.straightdope.com/u/tony1234)\ **Post date:** [April 5, 2000, 4:58am UTC](https://boards.straightdope.com/t/is-0-999-1/10990/46 "2000-04-05T04:58:00Z")
Briefly, the answer it: it depends. Math is a system made up by people. Different ssytems have been made up. In some the answer is yes, in others no. That is to say, either answer will be consistent with the rest of mathematics. i’m not sure what the answer is for the standard model, but I’m sure that there are “nonstandard” models with the alternative answer. Hope this helps. [tony12345678@yahoo.com](mailto:tony12345678@yahoo.com) ---
**Author:** ![Steve-o](https://avatars.discourse-cdn.com/v4/letter/s/ecc23a/32.png) [@Steve-o](https://boards.straightdope.com/u/Steve-o)\ **Post date:** [April 5, 2000, 5:28am UTC](https://boards.straightdope.com/t/is-0-999-1/10990/47 "2000-04-05T05:28:00Z")
> [@](#): > > Originally posted by KarlGauss: > \*\* > The problem with this is that whereas 0.999… is readily definable and calculatable (just add 0.9 to 0.09 to 0.009 etc), the same can’t be said about 0.000…1 Other than saying “first write down a decimal point followed by an infinite number of 0’s then put a 1”, how else could you give me (or a machine) explicit instructions on how to define or calculate it? \*\* Actually, I see telling a machine to calculate 0.999… as being more complicated. A = 0.999… = 9/(10^n) Sum for each successive “n” value (starting with 1 and increasing by units) B = 0.0000…001 = 1/(10^n) For each successive “n” value (starting with 1 and increasing by units), calculate without regard to previous calculations In any case, you could get a machine to tell you B+A=1 for any value “n”, so both numbers are definable. * * * Things are random only insofar as we don’t understand them. ---
**Author:** ![Steve-o](https://avatars.discourse-cdn.com/v4/letter/s/ecc23a/32.png) [@Steve-o](https://boards.straightdope.com/u/Steve-o)\ **Post date:** [April 5, 2000, 5:34am UTC](https://boards.straightdope.com/t/is-0-999-1/10990/48 "2000-04-05T05:34:00Z")
> [@](#): > > Originally posted by Steve-o: > \*\* …so both numbers are definable. > \*\* Oops, I should say that one is AS definable as the other. We’re talking about infinity here, so neither 0.9999… or 0.000…01 are truly definable. They are both concepts. * * * Things are random only insofar as we don’t understand them. ---
**Author:** ![Surgoshan](https://avatars.discourse-cdn.com/v4/letter/s/e95f7d/32.png) [@Surgoshan](https://boards.straightdope.com/u/Surgoshan)\ **Post date:** [April 5, 2000, 5:50am UTC](https://boards.straightdope.com/t/is-0-999-1/10990/49 "2000-04-05T05:50:00Z")
Oh, and I saw references to rational v. irrational earlier. An irrational number is one that goes on infinitely without repeating. Thus pi, r 3.14159 is irrational. .999… is rational. * * * I sold my soul to Satan for a dollar. I got it in the mail. ---
**Author:** ![Glyph](https://avatars.discourse-cdn.com/v4/letter/g/91b2a8/32.png) [@Glyph](https://boards.straightdope.com/u/Glyph)\ **Post date:** [April 5, 2000, 6:13am UTC](https://boards.straightdope.com/t/is-0-999-1/10990/50 "2000-04-05T06:13:00Z")
What it comes down to is: .999… basically, might as well =1 since they are so close. but realistically in absolutes 1 = 1 and .999… = .999… In math 1 = 1 and .999… = .999… unless you want to round up at a certain point then it could equal 1 but does not unless you set a rule (such as, lets round to the tenth). If you do **not** set a rule on which you round up .999… = .999… But again we say, “well close enough.” When it comes to absolutes, close do not count. ---
**Author:** ![Glyph](https://avatars.discourse-cdn.com/v4/letter/g/91b2a8/32.png) [@Glyph](https://boards.straightdope.com/u/Glyph)\ **Post date:** [April 5, 2000, 6:15am UTC](https://boards.straightdope.com/t/is-0-999-1/10990/51 "2000-04-05T06:15:00Z")
> [@](#): > > But again we say, “well close enough.” > When it comes to absolutes, close do not count. That is supposed to be, “close **does** not count.” I know how to talk, really. ---
**Author:** ![kinoons](https://avatars.discourse-cdn.com/v4/letter/k/a698b9/32.png) [@kinoons](https://boards.straightdope.com/u/kinoons)\ **Post date:** [April 5, 2000, 6:22am UTC](https://boards.straightdope.com/t/is-0-999-1/10990/52 "2000-04-05T06:22:00Z")
> [@](#): > > Originally posted by MrKnowItAll: > \*\*Ok, quick follow up. First of all, strike the “mean” in my post. It’s what I get for trying to form coherent sentences when I should be in bed. > > Second, the idea that .999… equals 1 just because they’re infinitely close just doesn’t hold water with me. As my Grandpappy used to say, “Close only counts in hand grenades and horseshoes.” > > Thirdly, why the hell hasn’t Cecil done a column on this? It’s come up several times in GQ, and it seems to be right up his alley. > > How 'bout it Nick and manny? Can we get Unca Cece to weigh in on this? > \*\* Close only counts in hourseshoes, hand grenades, and nuclear weps * * * “Bones, help that man!” – “Damnit Jim, I’m a doctor not a doc…I’ll get right on it.” ---
**Author:** ![Tinker\_Grey](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tinker_grey/32/8136_2.png) [@Tinker\_Grey](https://boards.straightdope.com/u/Tinker_Grey)\ **Post date:** [April 5, 2000, 6:42pm UTC](https://boards.straightdope.com/t/is-0-999-1/10990/53 "2000-04-05T18:42:00Z")
The concept of infinity, in this case, allows for consistency. Since we know that 1/3 + 1/3 + 1/3 = 1 (as stated above), then either 3 X (.333…) = 1, OR you simply concede that 1/3 can NOT be represented as a decimal in base 10. To restate, infinity is a convenient idea that allows us to do math with consistent results. Tinker ---
**Author:** ![RM\_Mentock](https://avatars.discourse-cdn.com/v4/letter/r/e274bd/32.png) [@RM\_Mentock](https://boards.straightdope.com/u/RM_Mentock)\ **Post date:** [April 5, 2000, 11:33pm UTC](https://boards.straightdope.com/t/is-0-999-1/10990/54 "2000-04-05T23:33:00Z")
Tinker I think the holdouts may concede that 3 x .333… is equal to 1, but they may not concede that it is equal to .999… I’d still like to know what they think 9.999… minus 0.999… is. * * * rocks ---
**Author:** ![MrKnowItAll](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mrknowitall/32/87_2.png) [@MrKnowItAll](https://boards.straightdope.com/u/MrKnowItAll)\ **Post date:** [April 6, 2000, 3:37am UTC](https://boards.straightdope.com/t/is-0-999-1/10990/55 "2000-04-06T03:37:00Z")
Well, OK. I’ve been pigheaded. Go back over the previous posts, I spotted AWB’s proof from early on. I had seen it before and forgotten it. What can I say? I haven’t been getting enough sleep lately. As counter-intuitive as it may seem, I hereby concede that 0.999… equals 1. (Now I’ve going to get a headache figuring out why…) * * * _God is my co-pilot. Blame Him._ ---
**Author:** ![tony1234](https://avatars.discourse-cdn.com/v4/letter/t/4491bb/32.png) [@tony1234](https://boards.straightdope.com/u/tony1234)\ **Post date:** [April 6, 2000, 4:16am UTC](https://boards.straightdope.com/t/is-0-999-1/10990/56 "2000-04-06T04:16:00Z")
The brief answer is that it depends. Standard systems of real numbers equate 1 and .999… On the other hand, there are nonstandard systems that permit infinitesimals and that who distinguish them. Both system are consistent. There is not “true” answer to whether they are equal any more than there is a true answer to where there are 360 degrees in a triangle. it depends on whether you are using euclidean or noneuclidean geometry. Tony ---
**Author:** ![matt\_mcl](https://avatars.discourse-cdn.com/v4/letter/m/7ab992/32.png) [@matt\_mcl](https://boards.straightdope.com/u/matt_mcl)\ **Post date:** [April 6, 2000, 4:31am UTC](https://boards.straightdope.com/t/is-0-999-1/10990/57 "2000-04-06T04:31:00Z")
Oh, come on now. I’m a linguistics major who dropped out of sciences two years ago just so I wouldn’t have to do any more calculus, and I STILL know the answer to this one. There is no such thing as infinity minus one, so there can’t be infinity minus one zeros, followed by one (0.000…01). INFINITY HAS NO END (duh) so you can’t tag a 1 onto the end of it, nor can you deduct a zero from the end of it and replace it with a 1. So you can’t make 0.000…01, so you can’t subtract it from 1, so you can’t come up with 9.999… that is different from 1. So there. Jeez. I thought after I went into the humanities I wouldn’t have to deal with this shit anymore. ---
**Author:** ![matt\_mcl](https://avatars.discourse-cdn.com/v4/letter/m/7ab992/32.png) [@matt\_mcl](https://boards.straightdope.com/u/matt_mcl)\ **Post date:** [April 6, 2000, 4:35am UTC](https://boards.straightdope.com/t/is-0-999-1/10990/58 "2000-04-06T04:35:00Z")
Oh, come on now. I’m a linguistics major who dropped out of sciences two years ago just so I wouldn’t have to do any more calculus, and I STILL know the answer to this one. There is no such thing as infinity minus one, so there can’t be infinity minus one zero, followed by one (0.000…01). INFINITY HAS NO END (duh) so you can’t tag a 1 onto the end of it, nor can you deduct a zero from the end of it and replace it with a 1. So you can’t make 0.000…01, so you can’t subtract it from 1, so you can’t come up with 9.999… that is different from 1. So there. Also, 9.999… is not an irrational number because it is the ratio of two numbers: 3 and 3. Jeez. I thought after I went into the humanities I wouldn’t have to deal with this shit anymore. ---
**Author:** ![RM\_Mentock](https://avatars.discourse-cdn.com/v4/letter/r/e274bd/32.png) [@RM\_Mentock](https://boards.straightdope.com/u/RM_Mentock)\ **Post date:** [April 6, 2000, 4:51am UTC](https://boards.straightdope.com/t/is-0-999-1/10990/59 "2000-04-06T04:51:00Z")
Most would agree that we are at least _trying_ to use standard analysis. > [@](#): > > Originally posted by MrKnowItAll: > \*\*As counter-intuitive as it may seem, I hereby concede that 0.999… equals 1. > > (Now I’ve going to get a headache figuring out why…) > \*\* That leaves KarmaComa as a holdout, maybe, but they haven’t posted in a while. MrKnowItAll, here is something that you may find interesting. Consider the sum of 1/2 + 1/4 + 1/8 + 1/16 + … This is the basis of the old Zeno’s Paradox: to get somewhere, you have to traverse half the distance, then half the remaining distance, then half what’s left, then… how do you ever _get there_? You do get there eventually, and the sum of all those halves is one. So, obviously, the sum of all those fractions is 1. By a quirk of algebra, A/(1-r) equals A + Ar + Ar^2 + Ar^3 + Ar^4… Go ahead, try the polynomial division. IF r is less than one, that infinite sum actually behaves quite nicely, and A/(1-r) actually equals the infinite sum. In the case of the halves, A equals 1/2, and so does r, and A/(1-r) equals one, as above. This is a handy formula. If you have the repeating decimal .181818… you can figure out what the fraction is by using that formula. For .181818…, A=.18 and r=.01. That is, .181818… = .18 + .18\*.01 + .18\*.01^2… Plug A and r into the formula, and you find that .181818 = .18/.99 = 2/11 That works for any repeating decimal. All repeating decimals are rational numbers, and that is the way to find the corresponding rational number. Now try it with .9999… * * * rocks ---
**Author:** ![matt\_mcl](https://avatars.discourse-cdn.com/v4/letter/m/7ab992/32.png) [@matt\_mcl](https://boards.straightdope.com/u/matt_mcl)\ **Post date:** [April 6, 2000, 5:03am UTC](https://boards.straightdope.com/t/is-0-999-1/10990/60 "2000-04-06T05:03:00Z")
Oh, come on now. I’m a linguistics major who dropped out of sciences two years ago just so I wouldn’t have to do any more calculus, and I STILL know the answer to this one. There is no such thing as infinity minus one, so there can’t be infinity minus one zero, followed by one (0.000…01). INFINITY HAS NO END (duh) so you can’t tag a 1 onto the end of it, nor can you deduct a zero from the end of it and replace it with a 1. So you can’t make 0.000…01, so you can’t subtract it from 1, so you can’t come up with 9.999… that is different from 1. So there. Also, 9.999… is not an irrational number because it is the ratio of two numbers: 3 and 3. Jeez. I thought after I went into the humanities I wouldn’t have to deal with this shit anymore. [Previous page](https://boards.straightdope.com/t/is-0-999-1/10990.md?page=2) [Next page](https://boards.straightdope.com/t/is-0-999-1/10990.md?page=4)