Aryabhatta, Brahmagupta: Ancient Indian Mathematics
Is it true that Aryabhatta discovered the Zero (0)? Were we the first to say that the solar system was heliocentric? There is need for a scientific approach to tackling these questions without either religious dogma or the opposite, of rejecting all things Indians discovered as myth-based jingoism
Many moons ago, I was delighted to read somewhere that an Indian named Aryabhatta had created the integer Zero, and if it had been patented, every Indian would have to be paid a princely sum of a few thousand dollars per year. Who was Aryabhatta I wondered then.
The naïve pride of childhood over one’s ancient culture was natural. But having arrived thus far in my own studies, I find that though Aryabhatta indeed was the first to invent the concept of Zero, there are examples of other civilisations trying to grapple with the puzzle of long numbers, especially those of their scientists dealing with astronomy.
The issue is two-fold, and hence, a senseless ‘hip-hip hurray… we created the zero’ demolishes the true history of science at the altar of jingoism.
In one of the most astounding ancient Indian treatises named Surya Siddhant (The Solar Principle), which comes down to us from 600 CE, we see the numbers 203, or 232, or 488 written “by just their digits spelt out one after the other, without taking recourse to the magnitude of the digit in the number”. (Roy).
One of the biggest tragedies of ancient Indian science is the clash of egos of mathematician Brahmagupta (598 to 670 CE) with his predecessor by more than a century, Aryabhatta (476 to 550 CE).
Place Marker and Integer
The issue with Zero is twofold. One is Zero as an integer, and the other is Zero as a place marker. The concept of Zero as a place marker was known to the Babylonians and also in contemporary Mayan civilization, says Dr Rahul Roy of the Indian Statistical Institute, in his paper Babylonian Pythagoras’ Theorem, the Early History of Zero and a Polemic on the Study of the History of Science (Source: “Resonance”. January 2003).
Read here
“Resonance”, by the way, is the internationally peer-reviewed journal of the Indian Academy of Sciences, so raising a finger at it shall not work.
Roy contends that though the concept of Zero as a place marker was known earlier, it is Aryabhatta who indeed discovered Zero as an integer, without which there could not have been any binary system used in computing.
Roy writes: “The importance of Zero as a place marker can be understood from the fact that if we did not have it, then there would be no way of distinguishing between the numbers 21 and 201. But the Babylonians did not have that for more than 1,000 years.”
Instead, they used 21 and other symbols such as 2/1 to make that kind of distinction, Roy says.
In one of the most astounding ancient Indian treatises named Surya Siddhant (The Solar Principle), which comes down to us from 600 CE, we see the numbers 203, or 232, or 488 written “by just their digits spelt out one after the other, without taking recourse to the magnitude of the digit in the number”. (Roy).
“The sum of zero and a negative number is negative, the sum of a positive number and zero is positive, the sum of zero and zero is zero. A negative number subtracted from zero is positive, a positive number subtracted from zero is negative, zero subtracted from a negative number is negative, zero subtracted from a positive number is positive, zero subtracted from zero is zero.” – Brahmagupta
Sattrajit Mukherjee, a young engineer, writes in the secular study group Pancham Vaidik: “A L Basham rightly said: “The unknown scholar who discovered zero is the second greatest son of India of all time, after Buddha.”
(NOTE: Basham is a noted historian, Indologist and author of a number of books. As a Professor at the School of Oriental and African Studies, London, 1950s and 1960s, he had said this in his book “The Wonder That Is India”)
“The decimal system was first mentioned in the Bakshali Manuscripts of 250 CE, discovered in 1881, in which the decimal system finds first mention in ancient India, and there is no record of the Zero being used before that.
“In fact, the name of the inventor of the decimal system is not known, though the system undoubtedly originated in India and after centuries, travelled to Europe via Arab scholars who had come to India,” writes Mukherjee.
Devnagri and Computing
Roy says that Aryabhatta devised an ingenious scheme of recording numbers based on the vowels and consonants of the Devnagri alphabet, arguably the most organised and scientific system any alphabet anywhere.
In fact, such is the precision of Sanskrit written in Devnagri that towards the end of the 1990s, western scientists were trying to see how Devnagri could be used to write advanced computer languages.
Roy writes: “It is from here that the Indian mathematicians take the profound step to consider zero as an integer and carry-on mathematical operations with it.
“While Varahamihira (in his Vrihata Samhita (575 AD) mentions the use of zero in mathematical operations, Brahmagupta (628 AD) elaborates on these operations in his remarkable treatise, Brahmasphutasiddhanta.
“Brahmagupta describes Zero as that quantity which is obtained when a number is subtracted from itself and he goes on to elaborate on the procedure of addition, subtraction, multiplication and division with zero,” Roy concludes.
Roy quotes Brahmagupta from his treatise thus:
“The sum of zero and a negative number is negative, the sum of a positive number and zero is positive, the sum of zero and zero is zero. A negative number subtracted from zero is positive, a positive number subtracted from zero is negative, zero subtracted from a negative number is negative, zero subtracted from a positive number is positive, zero subtracted from zero is zero.”
Could there be a more precise scientific definition of any single integer? That was ancient Indian mathematics.
We get a feel of the richness of Aryabhatta’s work in such research papers as S. Balachandra Rao (1998) [First published 1994]. Indian Mathematics and Astronomy: Some Landmarks. Bangalore; Jacobs, Harold R. (2003). Geometry: Seeing, Doing, Understanding (Third ed.),
Harold R Jacobs writes about Aryabhata’s work on the mathematical approximation for pi (π), “and may have come to the conclusion that π is irrational. In the second part of the Aryabhatiyam (gaṇitapāda 10).”:
He had said: “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, π = 62832/20000 = 3.1416, which is accurate to two parts in one million. On this measurement, S. Balachandra Rao writes: “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.
Considered in modern English units of time, Aryabhata calculated the sidereal rotation (the rotation of the earth referencing the fixed stars) as 23 hours, 56 minutes, and 4.1 seconds. The modern value is 23:56:4.091. Similarly, his value for the length of the sidereal year at 365 days, 6 hours, 12 minutes, and 30 seconds (365.25858 days) is an error of 3 minutes and 20 seconds over the length of a year (365.25636 days), writes Mohan Apte in his Marathi book Aryabhatia, Rajhans Publications, 2009.
So it is futile to say that Indian scientists did nothing, and just as futile to say that we had developed Indian sciences in a sort if bubble, segregated from the world, for there are lots of instances of Indian scientists and others working concurrently with scientists of other developed civilisations. And very often, the itinerant Arab travellers exchanged – or aided in exchanging ‑ information between Indian scientists with those from far-off climes.
It must be remembered that trade played a critical role in the development of basic sciences such as writing and mathematics. The writing scripts were developed as and when the volume of trade grew and from simple additions orally, traders needed to keep records of large volumes of business.
Similarly, when traders travelled to countries outside their own, they were faced with different methods of calculations. Thus it is that algebra was first largely developed in India, but Arba traders, again, took this system to their own scholars, and the term al zewar wal muqabla (reunion of btoken parts) came into usage and later took on the European term algebra.
(Coming up: Ancient Indian Geometry: Math Maze)
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