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Some traditions say, based on a Gemmorah in Taanis 9a (also מדרש רבה נשא י' ו', זוהר פנחס רכ"א א, and more):

"ליכא מידי דלא רמיזא באורייתא"

"There is nothing that is not alluded to in the Torah"

That is, everything can be derived from the written Torah (the Pentateuch of 304,805 letters, and Gr"a adds from Parashat Bereshis alone etc). Many tried (and succeeded) to find certain integers, such as numbers and dates.

In what way can the irrational and infinite number Pi be derived from the written Torah?

(to clarify, the question is not about approximation of Pi, it is about the possibility to derive an irrational number from a list of letters)

robev
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Al Berko
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    Just because it can be done in principle doesn't mean anyone knows how to do it in practice. – Heshy Jul 30 '18 at 05:38
  • Are you accepting answers from Nach? In that case it’s trivial - the Gemara already discusses it. – DonielF Jul 30 '18 at 05:39
  • @DonielF What Nach do you mean? – Al Berko Jul 30 '18 at 05:41
  • Do you only want answers that correctly derive Pi? – Alex Jul 30 '18 at 05:43
  • @Heshy Ditto; like finding a cure for cancer in the written Torah. – Oliver Jul 30 '18 at 05:53
  • @AlBerko Melachim Aleph 7:23 says that the circumference of the Yam Shel Shlomo was 30 Amos, and the diameter was 10 Amos. The Gemara (Eruvin 14a) derives from here that halachically we can approximate π=3. – DonielF Jul 30 '18 at 05:54
  • @DonielF I edited the Q, thank you. I meant an irrational number vs integers. – Al Berko Jul 30 '18 at 05:59
  • @AlBerko So you’re looking for an exact value for a number that goes on infinitely without repeating, from a text with a finite amount of numbers? Am I getting this right? – DonielF Jul 30 '18 at 06:04
  • @DonielF Right, I'm looking for an idea how that can be done in theory. – Al Berko Jul 30 '18 at 06:17
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    @DonielF well, math textbooks define it in a finite number of characters. – Heshy Jul 30 '18 at 07:02
  • I'd suggest first defining the natural numbers. First you have to prove that they exist, which can probably be done using rules of drash on the Chanuka leining or the Musafin of Sukkos, where it says ביום הפלוני in a sequence. Then you derive properties of addition from the totals in Bamidbar and Pinchas and multiplication from זאת חנוכה. Rationals you can probably get from the numbers in Matos. Negative numbers are harder, but maybe you can get subtraction from the population of Shimon (Bamidbar - Balak ~= Pinchas). But I don't know how to make the jump to reals. – Heshy Jul 30 '18 at 07:09
  • Maybe once you get to a certain point you can just say "We've proved [in a drash sense, not a mathematical sense] that these things follow the axioms of natural numbers, so now just open the most convenient analysis textbook and define the reals from them." Then you can apply whatever proofs you like. – Heshy Jul 30 '18 at 07:12
  • If not a dup of https://judaism.stackexchange.com/a/16892/501 or https://judaism.stackexchange.com/questions/84337/501 then those are answers to this. – Danny Schoemann Jul 30 '18 at 08:46
  • Looking for an exact value for a number that goes on infinitely without repeating, from a text with a finite amount of numbers is a problem for https://math.stackexchange.com . – Avrohom Yitzchok Jul 30 '18 at 09:14
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    Gemara Eruvin. Pi was not irrational in theire mind. The discovery of rational-irrational concept is posterior – kouty Jul 30 '18 at 11:37
  • @AvrohomYitzchok any math textbook does exactly that! It defines pi in a finite number of characters. – Heshy Jul 30 '18 at 12:51
  • @kouty but the concept exists and is easy to define (well it took until Cantor to get a fully rigorous definition, but once you know about it it's easy to define). The question isn't "did Chazal know about it", it's "can you derive it from Chumash". Obviously Hashem knows about it, nobody is going to argue that it can't be there in principle, even if they argue on this understanding of that Gemara. – Heshy Jul 30 '18 at 13:16
  • @Heshy I understand you. To try find what we know now in torah is a strange, anyway this is not limud torah. To find what science will discover in future is also not torah but more interesting. nobody can make this – kouty Jul 30 '18 at 13:19
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    Just because PI is irrational (and even transcendental), doesn't mean you cannot describe it accurately in a finite number of syllables. Just as an example, there are numerous infinite series that converge to Pi. E.g. https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80 – Nic Jul 30 '18 at 14:11
  • Right there is an algorithm to calculate it – kouty Jul 30 '18 at 17:44
  • I don't understand. I posted twice an answer, and they disappeared. – yO_ Aug 05 '18 at 07:58
  • The Vilna Gaon famously explains how a passuk in Melachim talking about circumference alludes to Pi (correct to five decimal points). https://judaism.stackexchange.com/questions/16883/is-the-passage-describing-an-approximation-of-pi-or-is-it-one-of-the-miracles-of – chortkov2 Dec 18 '18 at 21:16
  • https://s3.amazonaws.com/mywesternwallnet/wp-content/uploads/2014/04/24191748/story-of-pi-1.pdf – chortkov2 Jan 09 '20 at 15:46

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Pi can be derived from the Torah by rolling up a Torah scroll and measuring the end's diameter and circumference. The ratio between them will be pi.

Clint Eastwood
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