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It is well known that Aryabhata, the prominent Indian mathematician and astronomer of the 5th century CE, made significant contributions to mathematics, including approximations of $\pi$ (pi). In his work Aryabhatiya or Aryabhatiyam, he approximated $\pi$ as $\pi\approx3.1416$ and stated that it is an approximation (not exact). In the second part of the Aryabhatiyam (gaṇitapāda 10), he writes (as per Wikipedia):

caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.

"Add four to $100$, multiply by eight, and then add $62,000$. By this rule, the circumference of a circle with a diameter of $20,000$ can be approached."

This implies that for a circle whose diameter is $20000$, the circumference will be $62832$, i.e, $\pi=\frac{62832}{20000}=3.1416$, which is accurate to two parts in one million.

This shows that Aryabhata had a remarkably accurate understanding of $\pi$ for his time. However, it is also speculated that Aryabhata used the word āsanna (approaching in Sanskrit) to mean that this is an approximation and the value is incommensurable (irrational). Did he state that $\pi$ is irrational? Did he explicitly understand that $\pi$ is irrational? Is there any historical or direct evidence/record that he recognized its irrationality? I know some regional scholars write about such a topic by exaggerating it. What is the history? We all know that mathematics was not ready to prove $\pi$'s irrationality at that time. Lambert provided the proof in the 1760s for the first time.

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No, certainly not "explicitly". The question would have made sense already to ancient Greeks in the ratio form: is the ratio of the circumference to the diameter incommensurable? Did they suspect it? Maybe, but there is no evidence that the question was actually asked either by Archimedes or Ptolemy, who both estimated the ratio, let alone that they conjectured an answer. The claim that Aryabhata did is repeated on MacTutor, which usually knows better, but seems to hinge on overinterpreting a single word āsanno.

There is a somewhat better case for Aryabhata's commentator Bhaskara I, which is advocated in Chemla-Keller, The Sanskrit karanīs and the Chinese mian from the volume From China to Paris. But even his is quite weak. Chemla-Keller quote the following dialogical passage of Bhaskara's, commenting on Aryabhata's āsanno vṛttapariṇāhaḥ (taken from Shukla's Hindu mathematics in the seventh century as found in Bhāskara I’s commentary on the Āryabhaṭīya):

"Now why is the approximate circumference mentioned and not indeed the exact circumference?
They believe the following: there is no such method by which the exact circumference is computed.
Now, some think that the circumference of (a circle) having one for diameter when measured directly, is ten karanīs.
This is not so because karanīs do not have a statable size.
"

Karanī are square roots of imperfect squares or, more generally, quadratic surds, and ten karanīs is $\sqrt{10}$, a popular Indian approximation for $\pi$. Chemla-Keller make quite a lot out of this:

"This answer given by Bhaskara may very well be his way of expressing the irrationality of the ratio of the diameter of a circle to its circumference. Note that this ratio is approximated by what can be seen as a non-reduced fraction (62 832 /20 000). The plural form "they" is probably a reference to elder teachers, a tradition which Bhaskara claims to belong to. The dialogue continues. It develops as a series of objections to Bhaskara's answers, objections that Bhaskara in turn attempts to refute. The ten karanīs are thus introduced as an objection to the approximate result presented by Aryabhata. They are presented as the exact circumference of a circle whose diameter is one... This answer seems to evoke very precisely the irrationality of $\sqrt{10}$. We can note that its status as being un-measurable is not related to the impossibility of stating an exact circumference. It seems, as a matter of fact, that these two problems are distinguished."

However, there is a big distance from "there is no such method by which the exact circumference is computed" to "$\pi$ is irrational". Even if we take "karanīs do not have a statable size" for "surds are irrational", this is not said about $\pi$ in the text.

Bhaskara's commentary was discussed on MathSE, where users pointed out that Aryabhata's method was likely similar to Archimedes's, approximating the circumference by perimeters of inscribed regular polygons, and involved iterative computations with surds when implemented numerically, hence the appearance of karanīs. Archimedes's was, essentially, the only systematic method of approximating $\pi$ until the late middle ages, used by Indian and Chinese authors in particular. So "there is no such method by which the exact circumference is computed" may simply refer to perimeters of inscribed (or circumscribed) polygons not giving the exact circumference. Even if Bhaskara did assert the irrationality of surds, which he might have since incommensurability proofs from Euclid's Elements were likely transmitted to India by his time, this alone would not have given him a reason to transfer it to $\pi$. All the more so if "some thought" that ten karanīs was the "exact circumference" despite not having a "statable size".

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  • $\begingroup$ Thank you for sharing information. $\endgroup$ Commented Jun 2 at 14:39

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