Recently, there were many topics in sci.math discussed by so many (mathematicians, logicians, physicians, cranks and anti-cranks,..etc) the old definition of $\pi$ that is still considered valid up to our current date despite so much alleged progress in understanding what is actually the real number? where they finally create so much doubt about the well-known concept of $\pi$ being truly a real number especially that they claim it exists only in the perfect circle but so, unfortunately, the perfect circle itself doesn't exist in any physical reality

So, I thought that asking if really was there an old rigorous and a historical proof for the fact that $\pi$ must be really a real number, or was it only an innocent and so naive conclusion based on direct observation of a layperson, wonder!

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    $\begingroup$ You have asked similar questions in the past, where the issue ends up being more a matter of definitions than history. Please clarify what you mean by "a real number". Does Archimedes's work qualify? $\endgroup$ Aug 2 '17 at 16:58
  • $\begingroup$ Any work based on sensible logic confine to our physical reality would certainly qualify and asking a similar question might indicate truly not very well convincing answers, I may conclude that there isn't a historical proof from your comment, but only an old definition which is still working smoothly, fine, about a real number definition, of course, that was defined in mathematics but still evolving up to our dates, I personally find only those only positive constructible numbers are the truly real numbers since they obey exactly the existing principle of physical reality where no else $\endgroup$ Aug 2 '17 at 17:16
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    $\begingroup$ You may conclude what you want, but it would be incorrect. The issue as usual is that your definitions are not quite formal and very idiosyncratic. My comment actually answers your question in one of the standard ways in which one would interpret it. Again, once you provide a formal statement of what you actually accept as a real number, then we could proceed. ("Sensible logic confine[d] to our physical reality" is too vague.) Of course, restricting the problem to (ruler-and-straight-edge) constructible reals also gives an easy answer, but I doubt that's what you want. $\endgroup$ Aug 2 '17 at 17:23
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    $\begingroup$ (What I mean is this: There is a geometric definition of $\pi$, and classical proofs using Greek geometry that there is indeed such a constant. Of course classical Greek geometry does not use the concept of real number, so the question is perhaps one of when we identified geometric ratios and the like with "numbers". But this has nothing to do with $\pi$.) Meanwhile, you may find this interesting, even if perhaps not strictly what you intend: mathoverflow.net/q/72792/6085. $\endgroup$ Aug 2 '17 at 17:26
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    $\begingroup$ Do you notice how far your comments landed from what I asked? It suggests you are not asking in good faith. Perhaps that is not what you intended, but that is the message they convey. Anyway, using your notion, $\pi$ is not a "real number" and you surely know that already. The way your question is phrased suggests that this is not the notion you expect people to use. $\endgroup$ Aug 2 '17 at 17:37

Archimedes of Syracuse (287–212 BCE) is thought to have been the first (in Measurement of the Circle) to show that the “two possible Pi’s” are the same. For a circle of radius $r$ and diameter $d$, Area= $π_1 r^2$ while Perimeter = $π_2 d$, but that $π_1 = π_2$ is not obvious, and is often overlooked.

To be precise, $π$ is a transcendental number. The mathematical ontology runs as follows:

   +- rational               (5/6, -1, etc..)
   +- irrational
      +- algebraic           (sqrt(2), 2^(1/3), etc..)
      +- transcendental      (pi, e, etc..)

The irrationality of $π$ was first shown by Lambert in 1761 using continued fractions. Legendre conjectured that $π$ is non algebraic, that is, that $π$ is transcendental. Legendre was validated when in 1882 Lindemann proved $π$ transcendental.

See also:
I Prefer Pi: A Brief History and Anthology of Articles
in the American Mathematical Monthly,
Jonathan M. Borwein and Scott T. Chapman
American Mathematical Monthly 121:1 February 10, 2015

  • $\begingroup$ If $\pi$ is a transcendental number then why does its representation is always and forever in constructible numbers only and generally in rational form?, I do understand that (in mind) is so different, but that remains forever in mind only and never in any physical reality, so the point that the human mind can contain many things but if failed to express exactly then guess what is it really?, for instance I tell you simply that the irrational number say $\sqrt{5}$ is a diagonal of a rectangle with sides (1, 2) units (FINISHED), both in mind and also in our physical and checkable reality too ! $\endgroup$ Aug 2 '17 at 18:44
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    $\begingroup$ I dont understand a single word of your comment, is this supposed to be english? $\endgroup$ Aug 2 '17 at 18:47
  • $\begingroup$ Of course, since we have tonnes of discussions about it else where, wonder! $\endgroup$ Aug 2 '17 at 19:36
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    $\begingroup$ You are free to answer your own question. There is even a guideline somewhere that says you should do so quickly, when you have posted a question, and it occurs to you that you have an answer. So you say, some things were discussed often, so then you have an answer and you should post it to your own question. meta.stackexchange.com/a/17847/165536 $\endgroup$ Aug 2 '17 at 21:22

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