# 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:** 14
**Author:** ![Terminus\_Est](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/terminus_est/32/3087_2.png) [@Terminus\_Est](https://boards.straightdope.com/u/Terminus_Est)\ **Post date:** [June 4, 2003, 1:42pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/261 "2003-06-04T13:42:45Z")
As if this thread wasn’t surreal enough, we have the mathematicians arguing amongst themselves about the concept of “1”, and a non-mathematician explaining to the mathematicians what they _really_ mean when they write “QED”. ---
**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, 1:58pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/262 "2003-06-04T13:58:05Z")
**Terminus Est** , quell wite! **23skiddoo** , no they all converge. But you can’t just assume that - you have to prove it. And the proof contains its own reward - you obtain your limit. > [@](#): > > \*Originally posted by Jabba \* > \*\*QED stands for _Quod Erat Demonstrandum_, “which was to be proved”. It was added to the end of the proofs in Euclid ( though obviously not by Euclid himself, as he wrote in Greek). > Similarly, demonstrations of constructions were ended with QEF, _Quod Erat Faciendum_, “Which was to be done”. \*\* So, **Tarantula** , the record now stands at three to one. “As it was to be shown”, “As it was to be demonstrated”, “As it was to be proved” - all equivalent formulations of _Quod Erat Demonstrandum_ and all meaning that something was to be proved and now it has been. So are you going to apologise for your pomposity and attempt to ensnare us in your oh-so-cunning wit at any point? Or are you going to carry on insisting that QED doesn’t stand for what we all mean it to stand for when we write it? pan ---
**Author:** ![DaveW](https://avatars.discourse-cdn.com/v4/letter/d/9d8465/32.png) [@DaveW](https://boards.straightdope.com/u/DaveW)\ **Post date:** [June 4, 2003, 1:58pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/263 "2003-06-04T13:58:40Z")
About this multiplying 0.999… by 10 business: If 0.999_2 is 1.998, and 0.9999_3 is 2.9997, and 0.99999\*4 is 3.99996, then it seems that for any number of 9s, n, after the decimal, when multiplied by an integer m, you wind up with a number which is (m-1)-numbers-with-n-decimals away from m. (I’m sure someone here more capable than I can come up with a generalized proof of this.) So, _obviously_ 😉 0.999… times 10 is precisely 9 numbers away from being 10. Since 9.999… has nothing but 9s in it, I wonder what the digits in the 9 numbers between it and 10 look like… (I said 😉 dammit) ---
**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, 2:00pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/264 "2003-06-04T14:00:00Z")
Hmm, I just had the bright idea of looking it up in [Merriam-Webster](http://www.m-w.com/cgi-bin/dictionary?book=Dictionary&va=QED): > [@](#): > > One entry found for QED. > > Main Entry: QED > Function: abbreviation > quantum electrodynamics, [Latin quod erat demonstrandum] which was to be demonstrated Well fancy that. pan ---
**Author:** ![zwaldd](https://avatars.discourse-cdn.com/v4/letter/z/a9adbd/32.png) [@zwaldd](https://boards.straightdope.com/u/zwaldd)\ **Post date:** [June 4, 2003, 2:08pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/265 "2003-06-04T14:08:32Z")
I’m sure it’s been stated somewhere in 6 pages, but those who hold that .999…!=1 do not understand infinity. They are still picturing a gap between those two numbers, as if the 9’s are still generating as we speak and have not reached their ‘limit’. The number of 9’s in .999… doesn’t approach infinity, it **is** infinity. So the concept of the gap getting smaller but never being gone does not apply. ---
**Author:** ![SCSimmons](https://avatars.discourse-cdn.com/v4/letter/s/e495f1/32.png) [@SCSimmons](https://boards.straightdope.com/u/SCSimmons)\ **Post date:** [June 4, 2003, 2:44pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/266 "2003-06-04T14:44:01Z")
Actually, the ‘10x-x=9’ proof is just fine-it doesn’t actually rely on the series converging. When you multiply by ten, you’re multiplying each element of the series by ten. When you subtract, you’re showing that each element of the series, when multiplied by ten, equals the previous element of the series-except, of course, for the first element. The elements of the original series and the elements (after the first) of the new one are shown to relate in a one-to-one onto identity relation; so the sums of the series’ can be expressed as another series in which every element after the first is zero. **23skidoo** , there aren’t any divergent decimal numbers. This is probably pretty obvious, but here’s a sketch of a formal proof … Any decimal number can be represented as d1_10^-1 + d2[/ssize]10^-2 + … + d[size=1]n10^-n + … , where 0 \<= dn \<= 9 for all n. Every element of this series is therefore \<= 9_10^-n, so the series sum is bounded above by the series we’ve all been fighting about. Basically, if any digit after the decimal point is less than nine, the sum is less than 1. ---
**Author:** ![SCSimmons](https://avatars.discourse-cdn.com/v4/letter/s/e495f1/32.png) [@SCSimmons](https://boards.straightdope.com/u/SCSimmons)\ **Post date:** [June 4, 2003, 2:48pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/267 "2003-06-04T14:48:11Z")
Actually, the ‘10x-x=9’ proof is just fine-it doesn’t actually rely on the series converging. When you multiply by ten, you’re multiplying each element of the series by ten. When you subtract, you’re showing that each element of the series, when multiplied by ten, equals the previous element of the series-except, of course, for the first element. The elements of the original series and the elements (after the first) of the new one are shown to relate in a one-to-one onto identity relation; so the sums of the series’ can be expressed as another series in which every element after the first is zero. **23skidoo** , there aren’t any divergent decimal numbers. This is probably pretty obvious, but here’s a sketch of a formal proof … Any decimal number can be represented as d1_10^-1 + d2_10^-2 + … + dn_10^-n + … , where 0 \<= dn \<= 9 for all n. Every element of this series is therefore \<= 9_10^-n, so the series sum is bounded above by the series we’ve all been fighting about. Basically, if any digit after the decimal point is less than nine, the sum is less than 1. ---
**Author:** ![Tarantula](https://avatars.discourse-cdn.com/v4/letter/t/e274bd/32.png) [@Tarantula](https://boards.straightdope.com/u/Tarantula)\ **Post date:** [June 4, 2003, 3:29pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/268 "2003-06-04T15:29:07Z")
Well \*\* kabbes\*\* not to be picky, but we’re not dealing with quantum electrodynamics. This is grammar. Again, unless you make a mistake and wish to draw the attention of others to it, you should use demonstr_atum_… Sorry dude. But still, maybe you’re right about the OP… ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [June 4, 2003, 3:40pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/269 "2003-06-04T15:40:32Z")
_quod_ is the neuter form of the relative pronoun _qui_, ‘who’, so means ‘[the thing] which’ _erat_ is the 3rd person singular imperfect tense form of _esse_, ‘to be’, and means ‘he/she/it was’. _demonstrandum_ is the gerundive of _demonstrare_, ‘to demonstrate’, and means ‘to be demonstrated/proved’. Thus _quod erat demonstrandum_ is ‘[Here is the result] which was to be proved’. ---
**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 4, 2003, 3:41pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/270 "2003-06-04T15:41:42Z")
> [@](#): > > \*Originally posted by lucwarm \* > \*\*Look, if f(x) = 1/x, then the limit of f(x) as x approaches infinity is zero. HOWEVER, at no point does f(x) actually equal zero. And it’s misleading to argue that f(infinity) = 0 because infinity is a concept, not a number. \*\* Fair. And I never said anything about f(infinity). I always referred to the limit of f(x) as x approaches infinity. And that equals 0. This enables us to talk about exact values that result from limits. The fact that I cannot choose a finite value of x such that f(x) is 0 is irrelevant because I’m not suggesting that x be selected as a finite value. I’m suggesting that x approaches infinity. x is not infinity. It is an arbitrarilily large value approaching infinity. Feel free to argue, but as I mentioned about Phage’s posts, you’d be inventing your own math. And I’ll reiterate – if you want a more pragmatic reason, the difference between 0 and f(x)=1/x as x approaches infinity can have no value difference from 0 in any equation ever since the error can be made arbitrarily small. ---
**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, 3:47pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/271 "2003-06-04T15:47:58Z")
I think you’ll find that Quantum Electrodynamics is another possible thing that QED is an abbreviation for, actually. QED as “that which was to be proved” is nothing to do with quantum electrodynamics. I don’t care what you think you know and I don’t care whether you could conduct this entire discussion in Latin. [Everyone](http://www.tiscali.co.uk/reference/dictionaries/difficultwords/data/d0010815.html) [agrees](http://algo.inria.fr/banderier/Recipro/node20.html) [it means](http://robin.hartshorne.net/QED.html) [what](http://www.swif.uniba.it/lei/foldop/foldoc.cgi?QED+-+quod+erat+demonstrandum) [I say](http://dictionary.reference.com/search?q=QED) That last definition is from [www.dictionary.com](http://www.dictionary.com) by the way. This is what its definition is, in its entirity: > [@](#): > > QED > _abbr._ > Latin. quod erat demonstrandum (which was to be demonstrated). At what point are you going to accept this and ADMIT THAT YOU ARE WRONG? pan ---
**Author:** ![Tarantula](https://avatars.discourse-cdn.com/v4/letter/t/e274bd/32.png) [@Tarantula](https://boards.straightdope.com/u/Tarantula)\ **Post date:** [June 4, 2003, 3:52pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/272 "2003-06-04T15:52:25Z")
Kabbes - I understand the Latin (thanks anyway). My point is that you are using it erroneously. That’s all. I have a much more pressing question in General Questions - please have a look at it and help there. No latin needed… ---
**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, 3:53pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/273 "2003-06-04T15:53:45Z")
[slight hijack] In the interest of truth, **kabbes** is right. Mathematicians mean “demonstrandum.” Demonstrandum is the nom. neut. sg. gerundive of demonstrare. The gerundive is a participle used to express the necessity or obligation of the action. Thus _quod erat demonstrandum_ = “which was _to be shown_.” Demonstratum is the perf. pass. part., nom. neut. sg. of demonstrare. _Quod erat demonstratum_ would then mean “which was/because it was shown.” (Quod did come to mean “because,” among other things, but originally it was the neut. of qui, meaning “which.”) **Tarantula** , you don’t seem to understand the rhetoric of a mathematical proof in the Euclidean style. Example: “Proposition: Socrates was mortal. “Proof: All men are mortal. Socrates was a man. Therefore, Socrates was mortal, _which claim is the very thing I was supposed to prove from the beginning._” The Q.E.D. does “restate the question,” and that is the whole point. The last line of the proof is exactly what I was expected to prove. Your theory does not fit the actual usage. There is no mistake involved, and “because it was shown” does not fit what the mathematician means. They mean that they end up with what they started to prove. Both my English (Heath) and Greek/Latin (Heiberg) editions of Euclid clearly have “(being) what it was required to prove” and “quod erat demonstrandum,” respectively. (The Greek text has “hoper edei deixai” meaning “the very thing it was necessary to show,” deixai being the future active infinitive of deiknumi. Greek has no gerundives, they use infinitives or verbal adjectives. Based on the clear Greek of Euclid’s text, there is no way that he meant “because it was shown.”) This is the pattern that many mathematicians have followed. [/slight hijack] ---
**Author:** ![Tarantula](https://avatars.discourse-cdn.com/v4/letter/t/e274bd/32.png) [@Tarantula](https://boards.straightdope.com/u/Tarantula)\ **Post date:** [June 4, 2003, 4:00pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/274 "2003-06-04T16:00:35Z")
Ragerdude, I am well versed in both Classical Greek and Latin. The gammar doesn’t bother me as we agree on the points in question. What bothers me is that the use of the Gerundive is not a effecient as the use of the perfect participle passive - “Because it was shown” is much better from a non mathematical point of view because it presses the necessity understanding the solution onto the reader. That’s all. ---
**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, 4:00pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/275 "2003-06-04T16:00:59Z")
Wow. This is even tougher to kill than the original OP. **Tarantula** , you were rather rude to me – and indeed implied that I was no mathematician – because I **correctly stated** what QED stands for in mathematics. Here’s what you said: > [@](#): > > ]\*Originally posted by Tarantula \* > \*\*Just for all the idiots that don’t understand humour… > > QED in Mathematics stands for “Quod Erat Demonstratum” - “Because it was shown”. > > Quod Erat Demonstrandum “As it was to have been shown” is a latinist joke… > > tsk ! > > Sometimes you guys take all the fun out of being a latinist… > > Although \*\* Kabbes \*\* was funny - - obviously, Kabbes, you’re not a mathematician then… > > Har har… \*\* I have now provided 6 cites that QED in the common usage means what I say it means and ONLY what I say it means. Your insistance that “QED in Mathematics stands for “Quod Erat Demonstratum” - “Because it was shown”” is, quite simply, wrong. It is ignorance. It is incorrect. How many other ways can I say this? It has a truth value of fucking zero. I expect some recognition by you of this fact, of the fact that I have been right from the beginning and the fact that you have been making yourself look like a prize idiot from the get-go on the whole issue. I also expect some form of apology. However, as a statisticial I know that actual rarely equals expected. I shall not hold my breath. pan ---
**Author:** ![Tarantula](https://avatars.discourse-cdn.com/v4/letter/t/e274bd/32.png) [@Tarantula](https://boards.straightdope.com/u/Tarantula)\ **Post date:** [June 4, 2003, 4:07pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/276 "2003-06-04T16:07:51Z")
Kabbes, Not being a mathematician I was unfamiliar with the Euclidian use of the abbreviation QED. I was, somewhat incorrectly, inferring that it stood for -atum. This was the basis of my joke. However, since you are simply copyiny what was used by Euclid, you are not at fault \*\* at all \*\*. I simply disagree with the choice of latin used by the translators. You shouldn’t get so hot-under-the-collar about these things… Did _this_ actual exceed the expected ? Hope I didn’t throw your stats way out… LOL ---
**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, 4:30pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/277 "2003-06-04T16:30:24Z")
Almost. You still have to cross the hurdle that you used an **assumption** to try to play a sly joke and insult those of us that took your words to mean what they actually meant. Then you refused to back away from your words even when faced with 6 cites, two classical scholors and your own knowledge that you were making an assumption all along. I’m still waiting for the apology for that. If I’m hot under the collar its because after 4 pages of explaining the same things repeatedly to the worst kind of idiots, you appeared to come in and still declare the same ignorance that we’d been fighting since page 1. That’s a kind of oil on troubled waters I could well have lived without. But thankyou for admitting that I am not at fault at all. After getting through this thread, it’s nice to see someone who can own up to being wrong when faced with some incontravertable evidence! pan ---
**Author:** ![MC\_Master\_of\_Ceremonies](https://avatars.discourse-cdn.com/v4/letter/m/bb73d2/32.png) [@MC\_Master\_of\_Ceremonies](https://boards.straightdope.com/u/MC_Master_of_Ceremonies)\ **Post date:** [June 4, 2003, 4:46pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/278 "2003-06-04T16:46:51Z")
To anyone who still thinks 0.9999… does not equal 1: what is the sqr rt of 0.99999…? ---
**Author:** ![Trinopus](https://avatars.discourse-cdn.com/v4/letter/t/2bfe46/32.png) [@Trinopus](https://boards.straightdope.com/u/Trinopus)\ **Post date:** [June 4, 2003, 4:55pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/279 "2003-06-04T16:55:35Z")
Tarantula: add the Thorndike Barnhart dictionary to those reference works that say _you are wrong._ Add me to the list of people who say you are rude and mule-headed. Trinopus ---
**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 4, 2003, 4:58pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/280 "2003-06-04T16:58:52Z")
lim sqrt(1 - 10^x) = sqrt(1) = 1. x -\> -infinity But then, I’m not one of MC’s candidates. 😃 [Previous page](https://boards.straightdope.com/t/why-doesnt-9999-1/179019.md?page=13) [Next page](https://boards.straightdope.com/t/why-doesnt-9999-1/179019.md?page=15)