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

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?

• Maybe should exclude Wiki as source of information... and +1 interesting question. – draks ... Aug 1 '12 at 13:17
• @draks: Why would you exclude information just because it is found in some Wiki? – celtschk Aug 1 '12 at 14:27
• @celtschk Because it might not be reliable. – Inkbug Aug 1 '12 at 14:28
• Since when does the reliability of information depend on the medium it was written on? – celtschk Aug 1 '12 at 14:32
• 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 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. – Ben Crowell Aug 1 '12 at 15:11

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.

• 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
• 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

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)

• Here you'll find what Aryabhata says about Irrationality of powers of $\pi$ – draks ... Aug 1 '12 at 13:21
• @draks Where? I don't see anything there. – Inkbug Aug 1 '12 at 13:43
• @Inkbug Scroll to the only answer (picked the link to the question, sorry) and check the author... – draks ... Aug 1 '12 at 13:50
• @draks Funny! I didn't notice that. – Inkbug Aug 1 '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. math.ufl.edu/~rcrew/texts/pythagoras.html Therefore, as Xoff says, $\pi$ couldn't have been conjectured to be irrational before about 500 BC. – Ben Crowell Aug 1 '12 at 15:05

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).

• Do you know or has Wiki told you? – draks ... Aug 1 '12 at 13:16
• @draks: There is always someone who tells you. (And if no-one else tells you, you tell yourself.) How can you possibly know anything? – Dejan Govc Aug 1 '12 at 13:39
• @draks I obviously didn't remember the year and how to spell the whole name and thus had to look it up, but does this matter? – simmmons Aug 1 '12 at 13:48
• @DejanGovc hmm, true. and _at_simmmmons no, don't mind, don't worry, no offense. – draks ... Aug 1 '12 at 13:50

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 with a reference to Archimedes' work...

• Noli turbare circulos meos. – draks ... Aug 1 '12 at 13:47
• It's not clear what that has to do with the irrationality of $\pi$, however. – Thomas Andrews Aug 1 '12 at 14:38
• Unfortunately, I can't find, where I read, that Archimedes never believed that $\pi$ is a rational. I'll post it, if I ever find it. – draks ... Aug 1 '12 at 18:05