The History of Numbers
by Swami B B Visnu
[Continued from History of Numbers - Part 1]
Glorification of the Decimal Number System
The Indian numerals and the positional number system were introduced to the Islamic civilization by Al-Khwarizmi, the founder of several branches and basic concepts of mathematics. Al-Khwarizmi's book on arithmetic synthesized Greek and Indian knowledge and also contained his own fundamental contribution to mathematics and science including an explanation of the use of zero. It was only centuries later, in the 12th century, that the Indian numeral system was introduced to the Western world through Latin translations of his Arithmetic.
Michel de Montaigne, Mayor of Bordeaux (France) and one of the most learned men of his day, confessed in 1588 (prior to the widespread adoption of decimal arithmetic in Europe) that in spite of his great education and erudition, “I cannot yet cast account either with penne or counters.” That is, he could not do basic arithmetic.6
Dantzig notes in regards to the discovery of the positional decimal arithmetic, "… it assumes the proportions of a world-event… without it no progress in arithmetic was possible."7
Pierre-Simon Laplace, the famous 19th century mathematician, explained: "The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged from India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and places arithmetic foremost amongst useful inventions. The importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius." 8
Ifrah describes the significance of this discovery in these terms: "Now that we can stand back from the story, the birth of our modern number-system seems a colossal event in the history of humanity, as momentous as the mastery of fire, the development of agriculture, or the invention of writing, of the wheel, or of the steam engine." 9
Indian mathematicians used their revolutionary number system to advance human knowledge at great speed. The Sthananga Sutra, an Indian religious work from the second century AD, contains detailed operations that involve logarithms to the base 2. Modern texts credit the discovery of logarithms to the Scottish mathematician John Napier, who published his discovery in 1614. Indian knowledge of logarithms thus precedes Napier's discovery by more than 1,000 years.
Panini's Systematization of Sanskrit & The Binary Number System
Panini's precise systematization of the Sanskrit language in the 4th or 7th century BCE is widely considered as a forerunner of the Backus Normal Form (discovered by John Backus in 1959), which forms the basis of the current computer language. Panini is recognized as one of the foremost geniuses of ancient India and is credited with the systematization of Sanskrit as a language. Panini's work was so thorough that no one in the past 2,000 years has been able to improve on it. He codified every aspect of spoken communication, including pronunciation, tones and gestures. NASA scientist Rick Briggs, as part of his NASA research, showed that Sanskrit is the most perfectly suited, unambiguous, language for programming Artificial Intelligence.10
[F1-GRAPHIC of Sanskrit - Om bhur bhuvah...]
Jaina mathematicians (6th-7th century BCE) have the distinction of being a bridge between the Vedic Period in mathematics to the so-called Classical Period. They are also credited with extricating mathematics from religious rituals. The Jains’ fascination with large numbers directly led them to defining infinity into several types.
Pingala (300 to 200 BCE), a well-recognized Jaina mathematician, although not strictly a mathematician but a musical theorist, is credited with first using the Binary numeral system in the form of short and long syllables, making it similar to Morse code. He and his contemporary Indian scholars used the Sanskrit word śūnya to refer to zero or void. He is also credited with discovering the “Pascal triangle” and the Binominal coefficient. Basic concepts of the Fibonacci numbers have also been described by Pingala.
[F2-GRAPHIC of Fibonacci Numbers]
The Binary Number System Discovered in Europe 2,000 Years Later
Two thousand years later in 1679, the prominent mathematician, Gottfried Wilhelm Leibnitz, prompted by such huge mistakes as Columbus finding the West Indies in the Americas when in fact he thought he was in Japan, decided to stop human error with a better numerical system. In the process he invented the binary number system which allowed the representation of all numbers with only ones and zeros.
A simple diagram illustrates this easily. Our habit is to think in tens, hundreds, thousands, etc. However, the number nine written in binary is 1001. The first column (from the right) counts how many ones, the second, how many twos, then how many fours, eights etc. Thus nine in binary is one eight, no fours, no twos and one one .
[F3-GRAPHIC of Binary number example - eggs in cups]
This system provides the most efficient way of adding and subtracting numbers and is ideally suited for the computer, although Leibnitz never built the binary machine that he designed at that time. It wasn't until 1944, in the midst of World War II that the world's first binary computer, Colossus, was developed in Blecksly Park, England, utilizing the simple system of electrical currents being either off or on as representing zero or one. In this binary format millions of rapid calculations were made, allowing the Allies to crack the German coded messages with such skill that they often knew the contents of these messages even before Hitler did.
The Decimal Number System Spreads to Muslim Countries
Usage of the decimal number system spread to muslim countries where scholars were amazed by its usage and simplicity. By 776 AD the Arab empire was beginning to take shape. The Arabic world, in comparison to Europe, was much more accepting of the Indian system — in fact, the West owes its knowledge of the scheme to Arab scholars. Arabian scholars were always prepared to give Indian scientists credit for their number system. An early Arabian work states that,
"We also inherited a treatise on calculation with numbers from the sciences of India, which Abu Djafar Mohammed Ibn Musa al-Charismi developed further. It is the most comprehensive, most practical, and requires the least effort to learn; it testifies for the thorough intellect of the Indians, their creative talent, their superior ability to discriminate and their inventiveness." 11
On the other hand, the Europeans response to the extraordinary cultural and scientific achievements of India during the British occupation of India, was to postulate the Aryan Invasion Theory — that India's wondrous heritage came from Europe. Although this theory remains a controversial issue, more recent archaeological, linguistic, genetic and other evidence has effectively shown that there is no substantiation for this Aryan Invasion Theory. The earliest known use of the Indian decimal number system in Europe is in a Sicilian coin of 1134; in Britain the first use is in 1490.
Around the middle of the tenth century al-Uqlidisi wrote Kitab al-fusul fi al-hisab al-Hindi which is the earliest surviving book that presents the Indian system. In it al-Uqlidisi argues that this system is of practical value: "Most arithmeticians are obliged to use it in their work: since it is easy and immediate, requires little memorization, provides quick answers, and demands little thought ... " 12
In the fourth part of this book al-Uqlidisi showed how to modify the methods of calculating with Indian symbols, which had required a dust board, to methods which could be carried out with pen and paper. This requirement of a dust board had been an obstacle to the Indian system's acceptance. For example As-Suli, after praising the Indian system for its great simplicity, wrote in the first half of the tenth century: "Official scribes nevertheless avoid using [the Indian system] because it requires equipment [like a dust board] and they consider that a system that requires nothing but the members of the body is more secure and more fitting to the dignity of a leader." al-Uqlidisi's work is therefore important in attempting to remove one of the obstacles to acceptance of the Indian nine symbols. It is also historically important as it is the earliest known text offering a direct treatment of decimal fractions.
Another reference to the transmission of Indian numerals is found in the work of al-Qifti's Chronology of the scholars written around the end of the 12th century. This publication quotes much earlier sources.13
It was not simply that the Arabs took over the Indian number system. Rather different number systems were used simultaneously in the Arabic world over a long period of time. For example there were at least three different types of arithmetic used in Arab countries in the eleventh century:
1 - A system derived from counting on the fingers with the numerals written entirely in words—this finger-reckoning arithmetic was the system used by the business community
2 - The sexagesimal system with numerals denoted by letters of the Arabic alphabet
3 - The arithmetic of the Indian numerals and fractions with the decimal place-value system.
[F4-GRAPHIC of Decimal Numbers]
Persian author Mohammed ibn Musa al-Khwarizmi wrote a book, often claimed to be the first Arabic text written including the rules of arithmetic for the decimal number system, called Kitab al jabr wa‘l-muqabala (Rules of restoring and equating) dating from about 825 AD.14
Although the original Arabic text is lost, a twelfth century Latin translation, Algoritmi de numero Indorum (in English Al-Khwarizmi on the Hindu Art of Reckoning), gave rise to the word 'algorithm' deriving from his name in the title. Furthermore, from the Arabic title of the original book, Kitab al jabr w'al-muqabala, we derive our modern word 'algebra.' 15
"The imam and emir of the believers, al-Ma'mun, encouraged me to write a concise work on the calculations al-jabr and al-muqabala, confined to a pleasant and interesting art of calculation, which people constantly have need of for their inheritances, their wills, their judgements and their transactions, and in all the things they have to do together, notably, the measurement of land, the digging of canals, geometry and other things of that kind." 16
Al-Khwarizmi developed this numerical system further with quadratic equations, algebra, etc — enabling science, mathematics and astronomy in Islamic countries to dramatically develop. However, on the other side of the Mediterranean, Christian Europe doggedly continued with the awkward Roman numerals for centuries.
The Pope & Fibonacci Try to Introduce the Indian Decimal Number System into Europe
It is astonishing how many years passed before the Indian numeral system finally gained full acceptance in the rest of the world. There are indications that it reached southern Europe perhaps as early as 500 CE, but with Europe mired in the Dark Ages, few paid any attention. The first surviving example of the Indian numerals in a document in Europe was, however, long before the time of al-Banna in the fourteenth century. The Indian numerals appear in the Codex Vigilanus copied by a monk in Spain in 976.17
Significantly, the main part of Europe was not ready at that time to accept new ideas of any kind. Acceptance was slow, even as late as the fifteenth century when European mathematics began its rapid development, which continues today.
During this time counting tables were used by "bankers" in medieval Italian cities for exchanging currencies. If they cheated their table would be broken and this banker was then know as rukta or broken (banka-rukta), an early version of the modern word 'bankrupt.'
That the European monks depicted Indian numerals in a variety of orientations is clear evidence that they did not understand the usefulness of place-value number systems. Calculations in Europe were still made on calculation boards. Among the first uses of the Indian system in Europe was the introduction of Indian numerals for checker board calculations by Gerbert of Aurillac, who became Pope Sylvester II in 999. When he encountered Indian numerals in Arabic manuscripts held in a Spanish monastery he introduced round tokens with Indian numerals to his calculation board.
[G-GRAPHIC - Pope Sylvester]
However, this system encountered stiff resistance, in part from accountants who did not want their craft rendered obsolete, to clerics who were aghast to hear that the Pope had traveled to Islamic lands to study this foreign method. Because of this Islamic connection it was widely rumored that he was a sorcerer, and that he had sold his soul to Lucifer during his travels. This accusation persisted until 1648, when papal authorities reopened Sylvester’s tomb to make sure that his body had not been infested by Satanic forces.18
The early Christian world view was largely a product of Aristotelian conceptions, where the Earth was the center of the universe, set in motion by an "unmoved mover," or God. Because there was no place for a void in this cosmology it followed that the concept of zero and everything associated with it was a godless concept. Eastern philosophies however, rooted in ideas of eternal cycles of creation and destruction, had no such qualms.
Leonardo of Pisa, also known as Fibonacci, the young son of an Italian diplomat, who is now regarded as one of the greatest mathematicians of all time, discovered the "Arabic numerals" in the port of Bijaya, Algeria. The Indo-Arabic system was re-introduced to Europe by Fibonacci, in his 1202 CE book, Liber Abaci (Book of the Abacus or Book of Calculating), which was a showcase for the Indian numerals, with emphasis on its usage by merchants.19
Although this work persuaded many European mathematicians of the day to use this "new" system, usage of the ten digit positional system remained limited for many years, in part because the scheme continued to be considered “diabolical,” due to the mistaken impression that it originated in the Arab world (in spite of Fibonacci’s clear descriptions of the “nine Indian figures” plus zero).20
[H-GRAPHIC - Fibonacci]
Decimal arithmetic began to be widely used by scientists beginning in the 1400s, and was employed, for instance, by Copernicus, Galileo, Kepler and Newton, but it was not universally used in European commerce until after the French Revolution in 1793.21
Nicolas Copernicus, said to be the founder of modern astronomy, in his great work De Revolutionibus, published not long before his death in 1543, presented his (at the time) heretical idea that the earth rotated on its axis and traveled around the sun once yearly. This went against the philosophical and religious beliefs that the Catholic Church and all of Europe had held during medieval times.22
[I-GRAPHIC - Copernicus]
Copernicus never knew the great stir his work caused, but two other renowned Italian scientists, Galileo Galilei and Giordano Bruno, wholeheartedly supported Copernicus's system and suffered greatly at the hands of the church's inquisitors for daring to oppose the Church's views and stultifying authority.
Both were tortured extensively, Bruno for daring to go even beyond Copernicus to claim that space was boundless and that the sun was and its planets were but one of any number of similar systems. Bruno, after eight years in chains, was burned at the stake—his life a testimony to the drive for knowledge and truth that marked the incredible period of the Renaissance.
The old and frail Galileo was put in prison for the duration of his life. Nearly four hundred years later the Catholic Church grudgingly admitted that Galileo was right.
Nor was the usage of this streamlined decimal number system of counting easily accepted in Christian dominated Europe. Florence, Italy, banned the usage of this new number system in 1299 CE. Such attitudes forced the continuing usage of the awkward and difficult Roman numerals.
However, use of the calculation board and of the abacus coexisted with the Indian number system for centuries. Because most people in medieval Europe were illiterate (in addition to superstitious) and the Indian calculation method required the writing down of numbers, the abacus remained the preferred tool in commerce and administration. Science, on the other hand, adopted the Indian place-value number system early.
[J-GRAPHIC - Abacus]
Despite many scholars finding calculating with Indian symbols helpful in their work, the business community continued to use their finger arithmetic throughout the tenth century. Abu'l-Wafa, who was himself an expert in the use of Indian numerals, nevertheless wrote a text on how to use finger-reckoning arithmetic since this was the system used by the business community and teaching material aimed at these people had to be written using the appropriate system.23
The parallel use of competing systems for calculation and measurement is not an unusual occurrence. The use of the Fahrenheit temperature scale by the public of the USA and the Celsius temperature scale by the scientists of the USA is another current example. Scientists like Copernicus, Brahe and Kepler understood the superiority of the Indian number system over the Roman numbers and used it for their detailed observations and calculations. Medieval publications demonstrate the use of the Indian method parallel to the use of the abacus and calculation boards during their time.
When James Cook in 1776 planned the voyage that brought him to Australia, the financial commitment was comparable to the commitment made by the USA and the USSR to get a man to the moon. Yet the Colonial Office prepared his budget with tokens on a checker board.
The use of the abacus or calculation board for administrative purposes continued in Europe until 1791, when the French National Assembly, which was set up through the French Revolution two years earlier, adopted the Indian calculation method for France and banned the use of the abacus from schools and government offices. Government offices in England continued to calculate taxes on calculation boards for another decade.
The Catholic Church had always regarded charging interest on loans as sinful but with the Reformation in the late Middle Ages, the church became business friendly, dropping its rejection of capitalism. With this new interest in capitalism and the necessity of calculating interest and compound interest, the old Roman numeral system failed badly and the new system was finally accepted. This also allowed European ships to sail afield once they were able to calculate their position consistently and easily.
Finally, the Copernican revolution shook European mathematics free from the shackles of Aristotelian cosmology. René Descartes in the 17th century invented his cartesian coordinate system of positive and negative numbers with zero at it's center. This combined algebra and geometry and led the way to calculus and a complete acceptance of the decimal number system in the western world. 24
Other Scientific Contributions of India
Subsequent phases of developments in mathematics are found in Vedic texts, along with ritual practices, as well as in the Puranas. Calculations for the precise building of ritual altars were important, for obvious reasons. Arithmetical principles such as addition, subtraction, multiplication, fractions, cubes, squares and roots were developed during these periods: Narad Vishnu Purana (~1000 BCE). Geometric principles are found in the Sulva Sutras, authored by Baudhayana (800 BCE) and Apasthamba (600 BCE).
In 510 CE, the Indian mathematician Aryabhata explicitly described schemes for various arithmetic operations, even including square roots and cube roots — schemes likely known in India earlier than this date. Aryabhata’s actual algorithm for computing square roots is described in greater detail in a 628 CE manuscript by a faithful disciple named Bhaskara I. Additionally, Aryabhata gave a decimal value of pi = 3.1416. Ifrah further confirms that Aryabhata's works would have been impossible without the usage of zero and the place-value system.25
[K-GRAPHIC - Aryabhata I]
One of India’s greatest gifts to the world is in the field of mathematics. The adoption of zero and the decimal place-value system in India unbarred the gates of the mind to rapid progress in arithmetic and algebra.
India pioneered almost every field of mathematics, from the numeral system and arithmetical principles of addition, subtraction, multiplication and division, to the invention of zero and the notion of infinity, to the power and place value and decimal systems, geometry and many of the theorems traditionally attributed and named after the Greeks or other Europeans.
[L-GRAPHIC - Infinity]
Algebra, trigonometry and even significant parts of calculus were all developed by Indians to a significant degree of finesse, all long before any country or individual that the Europeans have given such credits.
This author has taken the liberty of directly quoting from some of the references given here, such as the quotations by some scholars or historians and, when necessary actual descriptions of the mathematical notations (rather than paraphrasing them). I am indebted to the original authors for their scholarly writings, without which justice could not have been done in narrating the contributions of Indians in Mathematics through history.
References for PART II
6-- Ifrah Ibid, p. 577.
7-- Tobias Dantzig and Joseph Mazur, Number: The Language of Science, Plume, New York, 2007. This is a reprint, with Preface by Mazur, of Dantzig’s book as originally published by MacMillan in 1930.
8-- Hogben, Mathematics for the Million, London, 1942.
9-- Ifrah Ibid, p. 346-347.
10-- Rick Briggs, Roacs, NASA Ames Research Center, Moffet Field, California, Knowledge Representation in Sanskrit and Artificial Intelligence, Artificial Intelligence Magazine, Volume 39, Spring 1985.
11-- Woepcke, F., Menoire sur la propagation des chiffres indiens, Journal Asiatique, 6e serie, 1863.
12-- A S Saidan (trs.), The arithmetic of al-Uqlidisi. The story of Hindu-Arabic arithmetic as told in 'Kitab al-fusul fial-hisab al-Hindi' Damascus, A.D. 952/3) (Dordrecht-Boston, Mass., 1978).
13-- al-Qifti, Chronology of the Stars, 12th century: http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Arabic_numerals.html
14-- Mohammed ibn Musa al-Khwarizmi, Kitab al jabr wa‘l-muqabala (Rules of restoring and equating), 825 AD (original Arabic text is lost).
15-- Algoritmi de numero Indorum (in English Al-Khwarizmi on the Hindu Art of Reckoning), a twelfth century Latin translation of Kitab al jabr.
16-- Benoît, P. and F. Micheau (1995), The Arab Intermediary. In: M. Serres (editor): A History of Scientific Thought, Elements of a History of Science. Blackwell, Oxford, 191 - 221. (Translation of Éléments d'Histoire des Sciences, Bordas, Paris, 1989)
17-- Codex Vigilanus, Completed by three monks in 976, the Codex Vigilanus is an illuminated compilation of historical documents from the Visigothic period in Spain.
18-- Ifrah Ibid.
19--Liber Abaci, Fibonacci: http://www.amazon.com/exec/obidos/ASIN/0387954198/fibonacnumbersan, translated by L E Sigler, Springer Verlag (2002), 672 pages, available for the first time in English in 2002 celebrating it's 800th anniversary, as a translation with notes of Fibonacci's Liber Abaci (The Book of Calculating) from 1202 but revised in 1228.
20-- David E. Smith, History of Mathematics, Volume 1, (Dover, 1958 reprint of 1923): http://www.amazon.com/exec/obidos/ASIN/0486204294/fibonacnumbersan
21-- Ifrah Ibid, pg. 590.
22-- Nicolaus Copernicus, De Revolutionibus orbium coelestium, 1530.
23-- Abu'l-Wafa, Kitab fi ma yahtaj ilayh al-kuttab wa'l-ummal min 'ilm al-hisab (Book on what Is necessary from the science of arithmetic for scribes and businessmen), between 961 and 976.
24-- Richard Webb, New Scientist, Nothingness: Zero, the number they tried to ban, November 22, 2011: http://www.newscientist.com/article/mg21228390.500-nothingness-zero-the- number-they-tried-to-ban.html?page=1 .
25-- Ifrah Ibid, pg. 497-498.