The story of the search for the perfect number system
By N.S. Rajaram
The Universal History of Numbers, 3 volumes by Georges Ifrah (2005 PB). Penguin, India.
“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, the author of the monumental but somewhat flawed The Universal History of Numbers, 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 was the positional or the place value number system. It is without a doubt the greatest mathematical discovery ever made, and arguably India’s greatest contribution to civilization.
The three-volume Indian edition is the English version of the 1994 French edition. 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. Better accounts exist.
The term ‘Arabic numerals’ 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. 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.
In Ifrah’s words: “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 system is just a special case of it.
The 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 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 be on hand to cross check numbers for accuracy as well as interpretation.
The 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. The books may be read by anyone with an interest in mathematics.
In summary, Georges Ifrah has opened the gates for what promises to be a major new pathway for research. It is now for others to rise to the challenge.
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N.S. Rajaram is a mathematician who has written extensively on ancient history with an emphasis related to Indian history.