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The problem is the infinite or endless repeated digits of $9's$ after zero digit and the decimal notation,

Despite its apparent simplicity & the huge talk about it every where in mathematics or scientific community, one wonders if it is of all that importance,

What its background, Is it settled as being equal to one or meaningless, how can this problem affects mathematics,

How many proofs or disproofs for this puzzle?

My question is a bet different in the sense of its absolute legend truthiness, where one may easily spot the illusion in its absolute truthiness by means of approximation, limits, ambiguous use of infinity, convergence, famous cuts, ...etc,

where all these tools are good for calculating approximately (also in our own sense only), the area of a circle for example, but the exactness sense in mathematics doesn't require any kind of approximation, it doesn't consider terms being large as $10^n$ or being little as $10^-n$ when $n$ tends to infinity, it consider them existing, regardless of our own needs or sense

To illustrate further, if we define the accuracy of approximating $pi$ by how many digits can be obtained,(instead of area of a circle of radius one), then all the formulas of $ pi $ are useless

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marked as duplicate by Conifold, VicAche, Andrés E. Caicedo, HDE 226868 Jan 15 '16 at 0:22

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ This question is answered here, the earliest mention is by Lambert in 1758 hsm.stackexchange.com/questions/2740/… $\endgroup$ – Conifold Jan 11 '16 at 22:57
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    $\begingroup$ It's really hard to understand what you mean with this "question". $\endgroup$ – ch7kor Jan 13 '16 at 16:19
  • $\begingroup$ @ch7kor Thanks for downvote first, how would you get what do I mean as long as arguing on unequal ground, my answer was deleted by Logan M, (a moderator I suppose), but he may not be blamed since What I claim seems so ridiculous, who is going to believe or consider or even give any possibility to be skeptical of existence of infinitely many numbers well established in mathematics (without rigorous proofs) to be fake numbers that even impossible to exist or be fixed like any constructible number on the real line number, it is a kind of madness I suppose, it is always the case "refusing new... $\endgroup$ – bassam karzeddin Jan 13 '16 at 17:13
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    $\begingroup$ @bassamkarzeddin As has been said a few times already now: Your (peculiar) views on modern mathematics are just something that is off-topic on this site about the history of science and mathematics. This is the reason your posts have a tendency to be deleted. $\endgroup$ – Danu Jan 14 '16 at 8:08
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    $\begingroup$ Ok, @bassamkarzeddin I understand your question now, it is just a way to promote your pet idea. This 0.999...=1 stuff is not advanced math, any student is supposed to understand it (you don't) $\endgroup$ – ch7kor Jan 14 '16 at 13:25
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One of the earliest places where an infinite string of 9s is rounded off to a finite string is Euler's text on algebra around 1777. Here he gets the answer 9.999... and makes a comment that this is virtually indistinguishable from 10. That was one of the earliest places as I mentioned. Conifold found an earlier source (see comment above).

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  • $\begingroup$ I hope this would be the oldest reference, but do you know the problem for which he considered, $10 = 9.999...$, thanks $\endgroup$ – bassam karzeddin Jan 11 '16 at 16:38
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    $\begingroup$ As I recall he summed a series and got the answer 9.999... and then replaced this by 10. Check at wiki to see if they have more information about this. $\endgroup$ – Mikhail Katz Jan 11 '16 at 16:43
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    $\begingroup$ @katz Note the comment on the question (by Conifold): This does not seem to be the earliest occurrence. $\endgroup$ – Danu Jan 12 '16 at 17:14
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The infinite series $9 \cdot \sum_{n \ge 1} 10^{-n} = 9 \cdot 10^{-1} \cdot \frac{1}{1 - 10^{-1}} = 1$. Just a run of the mill geometric series. No mystery, no puzzle.

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    $\begingroup$ Please note that is not exactly my question, I do appreciate that (0.999...) of people can prove it, but my question was about its origin, who, when & why?, thanks $\endgroup$ – bassam karzeddin Jan 11 '16 at 14:09
  • $\begingroup$ This does not address the question, which is about who first noticed the equality, not whether it's elementary or simple to discover. $\endgroup$ – Danu Jan 12 '16 at 17:13
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    $\begingroup$ @Danu, "when" is in the far antiquity (infinite geometric series were summed by e.g. Archimedes), "who & why", check again Archimedes. Perhaps one of his less known/lost predecessors or contemporaries. "Puzzle" it isn't, just a somewhat quaint quirk of positional number systems used to write fractions. For a exhaustive answer on who first noted/described this rather boring phenomenon (and several much more interesting ones), see the comment by Lambert. $\endgroup$ – vonbrand Jan 13 '16 at 13:01
  • $\begingroup$ @vonbrand I guess that what I'm saying is: On this site, which is specifically about the historical aspects, your answer should include such information as what you wrote in the above comment. If your answer is something along the lines of "this was not significant enough for anyone to write it down explicitly" that's also okay, and on-topic, but as it stands your answer has no historical content at all. $\endgroup$ – Danu Jan 14 '16 at 8:05
  • $\begingroup$ @vonbrand I forgot to add a link to my deleted answer by Danu, please see this one page link: quora.com/Philosophy-of-Mathematics/… $\endgroup$ – bassam karzeddin Jan 25 '16 at 15:16

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