Aryabhata and the Discovery of Pi

Ancient India’s most famous mathematician and astronomer Aryabhata is widely recognized for contributing the concept of zero to the world. As we know zero has formed the basis for the evolution of modern mathematics.
A lesser known fact perhaps is his work on the discovery of the important mathematical constant pi (π). Pi has applications in mathematical calculations and various aspects of science and engineering.
Ancient India in Vedic Times
The Vedic period was a particularly golden period in Indian history, flush with discoveries and inventions in various areas of science, mathematics, arts and culture. Unfortunately, these discoveries (including Aryabhata’s discovery of pi) were buried in the sands of time. And when the western world discovered these concepts, they were hailed as breakthroughs, with no one suspecting that the knowledge was already present in India ages ago.
Aryabhata and the Discovery of Pi
Born circa 476AD, Aryabhata was present during the Vedic period of India’s history. A highly intelligent individual, he was a Sanskrit scholar with deep interest in astronomy and mathematics. His seminal work ‘Aryabhatiya’ is a compendium of mathematics and astronomy, which has survived till modern times. Studying the ‘Aryabhatiya’ shows beyond doubt that Aryabhata had indeed discovered and worked on concept of pi long before the Western world was even aware of its existence.
References to Pi in Aryabhatiya
The Aryabhatiya, written in Sanskrit consists of 108 verses divided into 4 padas or chapters. The second pada called the Ganitapada (Ganita = mathematics) bears a reference to the concept (and approximate value) of pi. In Ganitapada 10, Aryabhata says
“caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ”
(Please note the format of this verse. It is extremely similar to the Vedic Mathematics sutras popularized by Shri. Bharati Tirtha Maharaj and refers to a form of writing in the Vedic times that was intended to be easy to memorize and thus recall).
Translated into English, this verse means: “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.
In other words, what Aryabhata said was that circumference of a circle with a diameter of 20000 is (4+100) x8 +62000= 62832. And we know that the value of pi is the ratio of the circumference to the diameter, so in this case 62832/20000, which is incredibly 3.1416. This is the value of pi accurate to five figures.
So we can definitely infer the following from the above verse and subsequent calculations
a. Aryabhata knew that he was talking about a mathematical constant, because he uses the term rule, indicating the value remains the same, even when the numbers change. And we know the ratio of the circumference of any circle (whatever its size) to its diameter is always the same – pi.
b. He uses the term ‘approached’, indicating that the value is not exact, but rather an approximation. This could well have been very first reference to the irrational nature of pi. Modern world in fact discovered that pi was irrational when proved so by Lambert in Europe as late as 1761.
There is no explanation in the Aryabhatiya as to how Aryabhata arrived at these values. Perhaps, he found it so obvious that he did not feel the need to explain.
That we shall never know, but the fact remains that Aryabhata had indeed discovered pi.
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Following is an excerpt from a research article titled “Best k-digit rational approximation of irrational numbers: Pre-computer versus computer era” by S.K. Sena, and Ravi P. Agarwal
Department of Mathematical Sciences, Florida Institute of Technology

More than 4700 years ago, the famous Indian mathematician and astronomer Aryabhatta (b. 2765 BC) gave 62832/20000 = 31416/10000 = 3.1416 as an approximation of π [21]. He calculated π by measuring the diameter of the circle in a remainderless unit and then measuring the circumference in the same unit. He made the units increasingly smaller in such a way that the diameter would be an integer in the unit used and improved the accuracy [21]. It is interesting to note that the two parameters, viz., (i) the size of the circle and (ii) the unit employed will be vital to improve the accuracy of π. By increasing the size of the circle and keeping the unit of measurement fixed or by making the unit size smaller and keeping the circle fixed, one can improve accuracy significantly. Thus keeping the circle as large as possible within the limits of the concerned measuring device and then making the unit of measurement as small as possible, one can achieve the maximum possible accuracy. 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% [3]. This translates to four significant digits. Thus, even over 4700 years ago when measuring devices were believed to be less sophisticated, Aryabhatta obtained π(=3.1416) to an accuracy of almost four digits! Beyond the accuracy of four significant digits, it is not possible to compute π (by measuring the circumference and the diameter of a circle) even today when much more sophisticated measuring devices are available! 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 [21]. This fact clearly demonstrates that Aryabhatta and for that matter the then Indian mathematicians/astronomers knew that π is an irrational number, however, finitely small the unit of measurement be. Another Indian mathematician, Bhaskara (who was in between Aryabhatta (2675 BC) and another famous Indian astronomer Varahamihira (123 BC) and whose exact time is not known) was the earliest known commentator of Aryabhatta’s works [21]. He suggested several approximations for π − 3927/1250 (=3.1416) for accurate work, 22/7 (=3.142857142857143) for less accurate calculation, while (=3.162277660168380) for ordinary work [1].

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