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When was $\pi$ first suggested to be irrational?

According to Wikipedia, this was proved in the 18th century.

Who first claimed / suggested (but not necessarily proved) that $\pi$ is irrational?

I found a passage in Maimonides's Mishna commentary (written circa 1168, Eiruvin 1:5) in which he seems to claim that $\pi$ is irrational. Is this the first mention?

Inkbug
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  • Maybe should exclude Wiki as source of information... and +1 interesting question. – draks ... Aug 01 '12 at 13:17
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    @draks: Why would you exclude information just because it is found in some Wiki? – celtschk Aug 01 '12 at 14:27
  • @celtschk Because it might not be reliable. – Inkbug Aug 01 '12 at 14:28
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    Since when does the reliability of information depend on the medium it was written on? – celtschk Aug 01 '12 at 14:32
  • @celtschk I also usually trust Wikipedia. Not everyone does. – Inkbug Aug 01 '12 at 14:43
  • Wiki $\neq$ Wikipedia. You can make an argument against trusting Wikipedia (just as you might e.g. make an argument against some specific journal of which you don't think they do peer review the way they should), however there's no reason to mistrust all Wikis (just as there is no reason to mistrust all journals). – celtschk Aug 01 '12 at 15:07
  • @celtschk Of course a wiki (lowercase) is not Wikipedia. However, I assumed draks was referring to Wikipedia. – Inkbug Aug 01 '12 at 15:09
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    It's not obvious to me that, e.g., Euclid and Archimedes would have considered $\pi$ to be a number in the same sense that they considered $\sqrt{2}$ to be a number. In Euclid, numbers are represented as comparisons of one segment to another, e.g., $\sqrt{2}$ would be the ratio of a square's diagonal to its edge. You can't construct segments in the ratio of $\pi$ using Euclid's postulates. Take a look at Proposition 1 here http://en.wikipedia.org/wiki/Measurement_of_a_Circle , and note how Archimedes states what we'd express as $A=\pi r^2$ without referring to $\pi$ as a number. –  Aug 01 '12 at 15:11
  • @celtschk I just thought, that the OP was already aware of the Wikipedia pages, that show up when I start to google it. I didn't write it like that, but it was my intention. Wiki is also one of my primary source of information, but if I had asked this question, I would something more than just Wikilinks... for example, the OP mentioned Maimonides's Mishna commentary, which would be an acceptable answer... – draks ... Aug 01 '12 at 17:43
  • To translate Maimonides: "You should know that the ratio of a circle's diameter to its circumference is unknown and can never be found. And this lack of knowing is not due to our mathematical shortcomings, as the group called Jahiliyyah [likely a reference to pre-Islamic mathematicians -Fred] thinks. Rather, it is an intrinsic property of this ratio that a precise value is unknowable, and its nature is that it cannot be known, although approximations are known." – Fred Mar 19 '18 at 23:02

4 Answers4

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There is a claim on the wikipedia article on irrational numbers that Aryabhata wrote that pi was incommensurable (5th century) but the question had to be asked as soon someone realized there was such numbers... (that was 5th century Before Christ)

Xoff
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    Here you'll find what Aryabhata says about Irrationality of powers of $\pi$ – draks ... Aug 01 '12 at 13:21
  • @draks Where? I don't see anything there. – Inkbug Aug 01 '12 at 13:43
  • @Inkbug Scroll to the only answer (picked the link to the question, sorry) and check the author... – draks ... Aug 01 '12 at 13:50
  • @draks Funny! I didn't notice that. – Inkbug Aug 01 '12 at 13:58
  • Amplifying on this, the notion that lengths could be incommensurable is thought to date back to the Pythagorean school, ca. 500 BC, and it was probably around this time that $\sqrt{2}$ was proved to be irrational. http://www.math.ufl.edu/~rcrew/texts/pythagoras.html Therefore, as Xoff says, $\pi$ couldn't have been conjectured to be irrational before about 500 BC. –  Aug 01 '12 at 15:05
  • This claim has no reference, and it hinges on the interpretation of one word ("approaches"), where the simplest interpretation is that rational value given is only an approximation to the true value. – Per Erik Manne Aug 03 '12 at 08:50
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Bhaskara I (the less famous of the two Bhaskara) wrote a comment on Aryabhata in 629, where he gives Aryabhata's approximation $\pi\approx {62832\over 20000}=3.1416$, and states that a nonapproximate value for this ratio is impossible. See Kim Plofker: Mathematics in India, p. 140.

On the surface this would seem like a clear statement equivalent to the irrationality of $\pi$, but perhaps it is not quite so easy. The "reason" Bhaskara gives is that "surds (square roots of nonsquare numbers) do not have a statable size". It is believed that Aryabhata's method of approximating $\pi$ is essentially the same as Archimedes', computing the circumference of an inscribed regular polygon in a circle with 384 sides. This requires the computation of many surds, so perhaps Bhaskara only meant that he could not get an exact value of the circumference of the inscribed polygon.

Per Erik Manne
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    The reason Bhaskara I attributes to Aryabhata giving an approximation rather than a precise value is, "They believe there is no such method by which the exact circumference is computed" [Shukla 1976: 72]. Then, in response to those who believe that a precise value can be given as π = √10, Bhaskara says, "This is not so because [surds] do not have a statable size." So while he might be saying surds are irrational, he isn't commenting on whether pi is irrational (only that there's no method for expressing a precise value for pi). – Fred Mar 19 '18 at 22:42
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    For more details, see the article "The Sanskrit karanis and the Chinese mian" by Karine Chemla and Agathe Keller (From China to Paris: 2000 Years Transmission of Mathematical Ideas, 2002, pp. 98 - 99). – Fred Mar 19 '18 at 22:48
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I don't know about the first suggestion but as far as I know the first proof was in 1761 by Johann Heinrich Lambert (Wikipedia link).

simmmons
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From a non-wiki source:

Archimedes [1], in the third century B.C. used regular polygons inscribed and circumscribed to a circle to approximate : the more sides a polygon has, the closer to the circle it becomes and therefore the ratio between the polygon's area between the square of the radius yields approximations to $\pi$. Using this method he showed that $223/71<\pi<22/7$.

Found on PlanetMath: Pi, with a reference to Archimedes' work...

hardmath
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draks ...
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