More than 4700 years ago, the famous Indian mathematician and astronomer Aryabhatta (b. 2765 BC) calculated 62832/20000 = 31416/10000 = 3.1416 as an approximation of π.
Aryabhata worked on the approximation for pi (π), and came to the conclusion that π is irrational. In the second part of the Aryabhatiyam (gaṇitapāda 10), he wrote: “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, = 62832/20000 =3.1416 , which is accurate to three decimal places. It is speculated that Aryabhata used the word āsanna (approaching), to mean that not only is this an approximation but that the value is incommensurable (or irrational).
If this is correct, it is quite a sophisticated insight, because the irrationality of pi (π) was proved in Europe only in 1761 by Lambert. After his works were translated into Arabic (c. 820 CE) this approximation was mentioned in Al-Khwarizmi's book on algebra.
It is interesting to note that there is, in general, no measuring device – optical or electronic or any other based on any other technology – that can measure a quantity with an accuracy more than 0.005% . This translates to four significant digits. Thus, even over 4700 years ago when measuring devices were believed to be less sophisticated, Aryabhatta derived π(=3.1416) to an accuracy of almost four digits.
This is definitely a remarkable achievement by Aryabhatta in mathematics in ancient India. Aryabhatta also discovered the non-remainderlessness of the circumference of a circle when the diameter is measured in a unit which provides an exact integer (within the limits of device error) for the value of the diameter.