# 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. Commented Aug 1, 2012 at 13:17
• @draks: Why would you exclude information just because it is found in some Wiki? Commented Aug 1, 2012 at 14:27
• @celtschk Because it might not be reliable. Commented Aug 1, 2012 at 14:28
• Since when does the reliability of information depend on the medium it was written on? Commented Aug 1, 2012 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.
– user13618
Commented Aug 1, 2012 at 15:11

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 $\pi$ Commented Aug 1, 2012 at 13:21
• @draks Where? I don't see anything there. Commented Aug 1, 2012 at 13:43
• @Inkbug Scroll to the only answer (picked the link to the question, sorry) and check the author... Commented Aug 1, 2012 at 13:50
• @draks Funny! I didn't notice that. Commented Aug 1, 2012 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.
– user13618
Commented Aug 1, 2012 at 15:05

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
Commented Mar 19, 2018 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
Commented Mar 19, 2018 at 22:48

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? Commented Aug 1, 2012 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? Commented Aug 1, 2012 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? Commented Aug 1, 2012 at 13:48
• @DejanGovc hmm, true. and _at_simmmmons no, don't mind, don't worry, no offense. Commented Aug 1, 2012 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: Pi, with a reference to Archimedes' work...

• Noli turbare circulos meos. Commented Aug 1, 2012 at 13:47
• It's not clear what that has to do with the irrationality of $\pi$, however. Commented Aug 1, 2012 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. Commented Aug 1, 2012 at 18:05