Sunday, August 25, 2013




N S Rajaram

An invention, unlike a discovery must be entirely a human creation. Fire was a discovery while mathematics was an invention. The greatest invention was the ‘modern’ number system based on the zero and the place value. Without it science and technology would be impossible.

N.S. Rajaram

Greatest invention

            What is humanity’s greatest invention? What makes us human is language, but language was an evolutionary development, the result of fortuitous genetic mutations and adaptation to the needs of the environment. So we didn’t invent it. Writing? Important but many civilizations invented writing but none came to be universally adopted. The only invention that went on to become universal is the modern number system based on values assigned to positions as well as symbols (numerals). The decimal system that we use is a special case of it, as is the binary system used in computers. Both were invented by Indians two thousand years ago, perhaps earlier. It was a lengthy painstaking process, not a bolt from the blue.

            In his monumental work Histoire Universelle des Chiffres 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 that surpassed and supplanted all others— one without which science, technology and everything else that we take for granted would be impossible. In his memorable words:

            “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.”

            His brilliant if somewhat flawed work, available in English as The Universal History of Numbers (Penguin, 2005) narrates the story of the dramatic journey that led to this, the greatest of all human inventions. The English version (in three-volumes) is adequate but lacks the literary flair and the dramatic flow of the French original. It tells the story of humanity’s 3000-year struggle to solve the most basic and yet the most important mathematical problem of all— counting. The first two volumes recount the tortuous history of the long search that culminated in the discovery in India of the ‘modern’ system and its westward diffusion through the Arabs. The third volume, on the evolution of modern computers, is not on the same level as the first two but still worth reading.

Origins: poetry to mathematics

            Ifrah places the achievement in its historical context by noting: "...real inventors of this fundamental discovery, which is no less important than such feats as the mastery of fire, the development of agriculture, or the invention of the wheel, writing or the steam engine, were the mathematicians and astronomers of the Indian civilization: scholars who, unlike the Greeks, were concerned with practical applications and who were motivated by a kind of passion for both numbers and numerical calculations."

            This is only partly true, for as described next, the idea of the positional or the place value system grew out of Chandas Shastra of Pingala (before 300 BCE)— his work on poetics! This Pingala according to tradition was a brother of the great grammarian Panini. The Chandaḥśāstra presents the first known description of a binary numeral system in connection with the systematic enumeration of meters with fixed patterns of short and long syllables as in anushtup, gayatri, etc.)

            We now use zero and one (0 and1) in representing binary numbers, but it is not known if the concept of zero was known to Pingala— as a number without value and as a positional location. Pingala's work also contains the Fibonacci number, called mātrāmeru, and now known as the Gopala–Hemachandra number. Pingala knew also the special case of the binomial theorem for the index 2, i.e. for (a + b)2, as did his Greek contemporary Euclid.

            Halayudha (10th century AD) who wrote a commentary on Pingala’s work understood and used zero in the modern sense but by then it was commonplace in India and had also begun to make its way to West Asia as well to countries like Indonesia, Cambodia and others in East and Southeast Asia. It took several centuries more before being accepted in Europe. It was Leonardo of Pisa, better known as Fibonacci who seems to have introduced it in Europe in the 13th century. (He learnt it from the Arabs but noted that it came from India. His successors were not so careful, and for centuries they were called Arabic numerals.)

            Halayudha was himself a mathematician of no mean order. His discussion of combinatorics of poetic meters led him to a general version of the binomial theorem centuries before Newton. (This was the integer version only and not the full general version with arbitrary exponent given by Newton.) This too traveled east and west with the Persian mathematician and poet Omar Khayyam using the results in the 13th century. Halāyudha's commentary includes a presentation of the Pascal's triangle for binomial coefficients (called meruprastāra).

March of numbers

            The modern number system is commonly known as the Arabic or the Hindu-Arabic. The term ‘Arabic numerals’ is a misnomer; the Arabs always called them ‘Hindi’ numerals. As noted earlier Fibonacci also knew of their Indian origin. 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. The Greek Miracle is a modern European fantasy.

            The discovery of the positional number system 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.

            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 or synthesize it with arithmetic operations like addition and subtraction, and more especially multiplication and division.

Golden Age of Indian mathematics

            As might be expected, the discovery of the place value system and the synthesis involving zero gave a tool of great power to mathematicians. This led to a burst of activity and a mathematical golden age in India. (A similar thing happened following the invention of the computer.) It is not known who invented the full fledged zero symbol with all its properties, but Aryabhata (476-550 AD) was the first mathematician known to have used it with the mastery needed to solve advanced problems even if he did not use the modern symbol for it. (We are not sure he didn’t, because we don’t have any manuscripts in his own hand!)

            Ifrah argues that knowledge of zero was implicit in Aryabhata's place-value system as a place holder for the powers of ten with null coefficients. The supposition is based on the following two facts: first, the invention of his alphabetical counting system would have been impossible without zero or the place-value system; secondly, he carries out calculations on square- and cube roots that are impossible if the numbers in question are not written according to the place-value system and zero.

            Like Ramanujan, Aryabhata too was a prodigy. But unlike Ramanujan who was essentially self-taught, Aryabhata seems to have received a thorough education. He is believed to have been born in Patliputra near modern Patna in Bihar. Many are of the view that he was born in the south of India possibly Kerala and lived in Magadha at the time of the Gupta rulers. Whatever the case, it is beyond doubt that he lived in Patliputra where he wrote his seminal treatise the Aryabhata-siddhanta. A compilation famous as the Aryabhatiya is the only work by him to have survived (in several versions including Persian, Arabic and Greek).

            The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry and spherical trigonometry. It also contains continued fractions, quadratic equations, sums of power series and a table of sines. This work is the first known that studies integer solutions to equations of the form by = ax + c and by = ax - c, where a, b, c are integers. These indeterminate equations known as Diophantine are still of interest. (Pell’s equation is a special case of it.)

            He provided an approximation to π (pi, the ratio of circumstance to diameter) that works out to 3.1416 or correct to the fourth decimal. Bhaskara I who wrote a commentary on the Aryabhatiya about 100 years later wrote of his genius: "Aryabhata is the master who, after reaching the furthest shores and plumbing the inmost depths of the sea of ultimate knowledge of mathematics, kinematics and spherics, handed over the three sciences to the learned world."

            Aryabhata knew that the earth spins on its axis, the earth moves round the sun and the moon orbits the earth— all a thousand years before Copernicus. He knew also that planets and the moon shine by reflecting sunlight. Aryabhata gives the radius of the planetary orbits in terms of the radius of the Earth/Sun orbit as essentially their periods of rotation around the Sun that are remarkably close to the modern value. He gave correct explanations for the causes of eclipses of the Sun and the Moon.

            (Unlike Galileo and Giordono Bruno, Indian scientists did not have to fight religious dogma or church authority. So they used without fuss whichever model suited them—heliocentric or geocentric—for the calculations at hand. Their approach was pragmatic, not dogmatic. So their work did not create the turmoil that Copernicus and Galileo did in Europe a thousand years later. So it is pointless to compare Indian and European scientists in this regard— the conditions were totally different.)

            This remarkable man was a genius who like Ramanujan continues to baffle mathematicians today. His works was then later adopted by the Greeks (of the Eastern Roman Empire), Persians and then the Arabs.

Capacity for abstract thought

            Ifrah’s admiration for the Indian achievement leads him to the statement: "The Indian mind has always had for calculations and the handling of numbers an extraordinary inclination, ease and power, such as no other civilization in history ever possessed to the same degree. So much so that Indian culture regarded the science of numbers as the noblest of its arts...A thousand years ahead of Europeans, Indian savants knew that the zero and infinity were mutually inverse notions."

            This is not entirely correct. While the Indian mind takes readily to numbers what made the synthesis as well as the discovery of zero-infinity duality possible was 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 this kind of abstraction in other fields also. Grammarian Panini describes the Indian alphabet in purely phonetic terms, without reference to symbols. It is the same in music. While the Western 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 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.

            In summary, Georges Ifrah’s Universal History of Numbers is an impressive achievement but not a definitive work. It has several drawbacks— errors of omission and commission that are perhaps unavoidable when one tries to cover a vast area spanning space, time and civilizations. The author’s discussion of palaeography sometimes goes awry due to his reliance on secondary sources, some of which go back to the nineteenth century. He accepts as proven conclusions that are contentious and even demonstrably false. (Like his acceptance of the non-existent Aramaeo-Brahmi as the source of the Brahmi alphabet.) These, however, do not seriously detract from a marvelous work. It is up to Indian scholars to carry forward his pioneering effort.

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