India’s contribution to the world of Numbers: Dr. Navaratna S. Rajaram

Posted on August 8, 2010

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Center for Indic Studies
University of Massachusetts- Dartmouth
U.S.A.

First I want to thank Prasara Bharati and the All India Radio for inviting me to speak about numbers, a subject that is close to my heart. I am a mathematician by training and have taught, conducted research and written about mathematics for more than thirty years. Based on this experience I would not be exaggerating when I say that the discovery of numbers and how to use them is the greatest invention ever made by man.

Without an efficient number system there can be no mathematics and without mathematics there can be no science and no technology. Humanity struggled with the idea of numbers and counting for thousands of years until Indian mathematicians and philosophers solved the problem by inventing the positional or the place value number system more than two thousand years ago.

The key idea was identifying positions by numbers that we call a base. In the decimal system that is now universal, the base is ten; in the binary system used in computers, the base is two. With the invention of the positional or the place value system, multiplication, which used to be an incredibly difficult problem, became so easy that we now teach it to children. All you need to multiply by ten is to shift one position. In the binary system shifting doubles the number.

In binary form it is perfectly suited to computers also because numbers can be represented by rows of switches or some other device. You need only zero and one to represent any number. So, in the binary notation, a number becomes a row of zeroes and ones. This can be represented by a row of switches or lights with on for one and off for zero. This was how it was done in early computers though later, transistors and microprocessors greatly simplified the engineering. But the basic mathematics is still based on the positional number system invented more than two thousand years ago.

Today we use both the decimal system and the binary system. Both were invented in ancient India. The key breakthrough, as I just said is the idea of identifying the position with a base- ten for the decimal and two for the binary. Other bases are also used especially in computer design. It is a mistake therefore to say that the Indians invented the decimal system and the zero: they invented the whole concept of the place value arithmetic of which the decimal is a special case. The real breakthrough was achieving a synthesis of diverse ideas.

What makes the positional system perfect is the synthesis of three simple yet profound ideas: zero as a numerical symbol; zero having ‘nothing’ as its value; and the zero as a position in a number string. Other civilizations, including the Babylonian and the Maya, discovered one or other feature but failed to achieve the grand synthesis that gave us the modern system. Of the world’s civilizations, the Mayas came closest. They, like the Babylonians, had an idea of the zero, but never learnt how to operate with it. The rules were laid out most clearly by the Indian mathematician Brahmagupta 1500 years ago. It is still the method we use.

This synthesis was possible due to the Indians’ capacity for abstract thought: they saw numbers not as visual aids to counting, but as abstract symbols. While other number systems, like the Roman numerals for example, expressed numbers visually, Indians early broke free of this shackle and saw numbers as pure symbols with values. We see it in other fields also. The great grammarian Panini describes the Indian alphabet in purely phonetic terms, without reference to symbols. It is the same in music. While the Western musical notation depends on both the form and the location of notes written across staves, the Indian notation can use any seven symbols.

The economy and precision of the positional system has made all others obsolete. Some systems could be marvels of ingenuity, but led to incredible complexities. The Egyptian hieroglyphic system needed 27 symbols to write a simple four-digit number like 7659. Another indispensable feature of the Indian system is its uniqueness. Once written, it has a single value no matter who reads it. This was not always the case with other systems. In one Maya example, the same signs can be read as either 4399 or 4879. It was even worse in the Babylonian system, where a particular number string can have a value ranging from 1538 to a fraction less than one! So a team of scribes had to be on hand to cross check numbers for accuracy as well as interpretation.

I am prepared to argue that it is mankind’s greatest invention. It is without a doubt the greatest mathematical discovery ever made, and arguably India’s greatest secular contribution to civilization. And this is not just my opinion. Georges Ifrah in his monumental three-volume Universal History of Numbers writes:

“Finally it all came to pass as though across the ages and the civilizations, the human mind had tried all the possible solutions to the problem of writing numbers, before universally adopting the one which seemed the most abstract, the most perfected and the most effective of all.”

In these memorable words, the French-Moroccan scholar Georges Ifrah sums up the many false starts by many civilizations until the Indians hit upon a method of doing arithmetic which surpassed and supplanted all others— one without which science, technology and everything else that we take for granted would be impossible. This of course was the positional or the place value number system.

Ifrah appreciates the true magnitude of the achievement when he writes: “The measure of genius of the Indian civilization, to which we owe our modern system, is all the greater in that it was the only one in all history to have achieved this triumph.” Modern civilization rests on the modern number system. The decimal and binary systems are special cases of it as I just noted.

The Indian system is sometimes called ‘Arabic numerals’ but that is a misnomer; the Arabs always called them ‘Hindi’ numerals. What is remarkable is the relatively unimportant role played by the Greeks. They were poor at arithmetic and came nowhere near matching the Indians. Babylonians a thousand years before them were more creative, and the Maya of pre-Colombian America far surpassed them in both computation and astronomy. Some European scholars tend to exaggerate Greek contribution and attribute everything to the Greeks. This so-called Greek Miracle is a modern European fantasy.

The discovery of the positional number system was not just a technical advance; it is a defining event in history, like man’s discovery of fire. It changed the terms of human existence. While the invention of writing by several civilizations was also of momentous consequence, no writing system ever attained the universality and the perfection of the positional number system. Today, in the age of computers and the information revolution, computer code has all but replaced writing and even pictures. This would be impossible without the Indian number system, which is now virtually the universal alphabet as well.

The next question is when and how did this number system evolve? This is not easy to answer, but I will give it a try. First we need to clear the confusion created by the claims of some recent books going by the name of Vedic Mathematics. These are interesting books but modern in composition and style. What they call Vedic mathematics is not found in the Vedic literature and the claims they make are highly exaggerated. So we can put this out of our mind and look at authentic ancient sources.

We also need to understand a couple of things. The important thing is the use of the place value method, not the use of ten as the base. The decimal system is the most widely used, but this must be separated from the concept of assigning values to places. In the decimal system, the first position or the rightmost indicates multiplication by one, the second multiplication by ten, the third by hundred and so on. In the binary system, two takes the place of ten. The first position shows multiplication by one, the second by two, the third by four and so forth. It is this positional method more than the base of ten or two that makes the Indian system so powerful.

The use of ten as base is ancient and can be found in the Vedas, but we don’t know how they represented them in writing. Incidentally, there was writing in Vedic times though the texts were memorized. The oldest mathematical texts that we have are what are known as the Shulba-sutras. These are parts of Vedic texts that are more or less contemporary with the Harappan archaeology. They too give no indication of the use of the decimal place value system.

Curiously, the first indication of the idea of a place value system is found not in any mathematical text but in a work on poetics called
Chandah-sutra by Pingala. He describes a binary system and solves several interesting problems involving multiplication and division of binary numbers. So we can definitely say that the idea of both the decimal or base ten and the use of position or place value, at least in the binary case were known in India several centuries before the beginning of the Christian era.

From this we may reasonably conclude that the modern decimal place value system arose by adopting to base ten the idea found in Pingala’s binary system. But this is only guesswork at this time. It is quite possible that further research will reveal that the decimal place value system was an independent development.

I would like to sound an important cautionary note before I proceed further: we have to be very careful in accepting the claims of Western scholars especially of the colonial period. Most of them try to trace all major Indian achievements to non-Indian sources like the Greeks, Egyptians and even the Chinese as a recent book claims. Indian researchers must go to the original sources and not accept these ‘experts’ and ‘scholars’ so-called. It is best assume that their interpretations are wrong and study the original sources without prejudice.

The next question is when do we first find the use of this system? The earliest mathematical text that uses a primitive form of the decimal system is what is known as the Bakhshali Manuscript. It is about two thousand years old. It was discovered in 1881 in a village called Bakhshali near Taxila in present day Pakistan. It is a very valuable source for gaining some idea of the state of mathematical knowledge two thousand years ago.

Although the use of zero and the decimal system were known more than two thousand years ago, it took several centuries before methods using it were invented and mastered. Aryabhata in the fifth century seems to have been the first great mathematician to have used it with great mastery. At least that is the record we have. Following him, Brahmagupta gave the clearest rules for operations with zero and also the use of negative numbers. His rules are essentially what we use today also.

There was a great spurt in mathematical activity in the period surrounding Aryabhata, and his successors. This may be attributed to the new methods made possible by the widespread use of zero and the decimal system. The same thing happened after computers were invented and their use became widespread some sixty years ago. Problems which were previously impossible could now be solved thanks to computers, and new methods were invented. The same thing happened in India with the zero and the decimal system.

A great discovery like this doesn’t go unnoticed. Indian methods were soon borrowed by the Arabs and the Persians and found their way to Europe. Leonardo of Pisa better known as Fibonacci was one of the first Europeans to learn and use it. Fibonacci lived in the thirteenth century, which means it took more than five centuries before the Indian system was widely used in Europe. Even then it met with resistance.

Because Europeans learned the Indian number system from the Arabs, they called it Arabic numerals. But the Arabs always stated that they got it from the Indians. Fibonacci also stated in his book that the new number system was of Indian origin. Arab records also state that they translated Indian works going as far back as 425 BC. This should be a valuable source of research for Indian scholars. They may discover ancient Indian works that are no longer available in India.

This in brief is the story of how the modern Indian number system evolved. It took not centuries, but more than two thousand years.

Broadcasted at 9:10 PM (IST) on July 21, 2010 in AIR.

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Posted in: History of India