Timeline for Have any long-suspected irrational numbers turned out to be rational?
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15 events
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| Jan 20, 2022 at 1:18 | comment | added | tparker | Could you clarify what you mean by "doubtfully true"? Does that mean "unlikely to be true", or "possibly true but there is some doubt", or what? | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Dec 13, 2010 at 2:23 | comment | added | Joël | +1. I read this question and answers with months of delay, but for what it is worth, I think this one is the best answer. This example illustrates the point made in the question that it is often more surprising, more intersting and more fruitful to prove that some number is rational rather than the opposite. | |
| Sep 24, 2010 at 15:33 | comment | added | Max Lonysa Muller | @ Wadim Zudilin: that 'somebody' who "proved" that $log(2)$ is a rational multiple of $\pi^2$ was me. In the future, I'll try to ask such non-research questions to a professor (as soon as I have one) or math stackexchange. | |
| Aug 3, 2010 at 1:17 | comment | added | Gjergji Zaimi | I think this is in "the spirit" of the question. For one thing it shows that $\zeta (2)$ is a period, and periods are countable which means they are special and a random number is not a period with probability one. (Just like someone above suggested that substituting rational with algebraic is also in the spirit of the question...) | |
| Aug 3, 2010 at 1:10 | history | edited | Wadim Zudilin | CC BY-SA 2.5 |
added context; Post Made Community Wiki
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| Jul 23, 2010 at 3:39 | comment | added | Charles | I think a key difference here is that no one was asking about $\zeta(2)/\pi^2$, so you could have chosen any first- or second-tier mathematical constant to divide or multiply $\zeta(2)$ by. So instead of a single number which is surprisingly rational, it's a large finite class of numbers, one of which turns out to be rational. $\zeta(2)/e$, $\zeta(2)/(\pi+e)$, $\zeta(2)e^\gamma$, etc. | |
| Jul 23, 2010 at 3:18 | comment | added | Wadim Zudilin | Thanks, Kevin, this is exactly my feeling about the question. At least, I don't see myself alone in my subjectivity. | |
| Jul 23, 2010 at 3:08 | comment | added | Kevin Ventullo | Perhaps it's not a perfect example, but I think it's at least in the spirit of the question. The idea is that we had some description of a number which was believed to be the ONLY description of that number, but some clever person comes along and shows it's actually equal to something we're already familiar with. | |
| Jul 23, 2010 at 2:54 | comment | added | Wadim Zudilin | Keith, thanks for explaining the criticism. I take it. But not your conclusion of "this example does not seem to be in the spirit of the question", as it is really subjective. For example, I don't count Marty's response as "following the spirit", but Marty and some others do. | |
| Jul 23, 2010 at 2:47 | comment | added | KConrad | What lhf presumably means about this example not being fair is that you are rigging the terms to force a rational value. To emphasize the point, you could just as well have said that for a long time nobody knew 3*zeta(2)/pi^2 = 1/2 or 6*zeta(2)/pi^2 = 1, and these are not the way people usually think about this zeta evaluation. Moreover, if anybody had computed zeta(2)/pi^2 before anyone knew an exact formula for zeta(2), the estimate would be .16666... so the natural guess is that the ratio is 1/6. Thus this example does not seem to be in the spirit of the question. | |
| Jul 23, 2010 at 1:23 | comment | added | Wadim Zudilin | And a lot of thanks to an anonymous downvoter! Euler would have appreciated such evaluation of one of his famous results. :-) | |
| Jul 23, 2010 at 1:15 | comment | added | Wadim Zudilin | What I'm saying is that nobody expected that $\zeta(2)/\pi^2$ could be rational. I can't really catch what is counted by "not fair"... | |
| Jul 23, 2010 at 0:56 | comment | added | lhf | That's not a fair example because no one knew that $\zeta(2)$ involved $\pi^2$ until Euler computed it. | |
| Jul 23, 2010 at 0:36 | history | answered | Wadim Zudilin | CC BY-SA 2.5 |