Numeral Systems

"The Romans also exchanged products with the East. Horace says, "A busy trader, you hasten to the farthest Indies, flying from poverty over sea, over crags, over fires." The products of the Orient, spices and jewels from India, from Persia, and silks from China, being more in demand than the exports from the Mediterranean lands, the balance of trade was against the West, and thus Roman coin found its way eastward."

"Augustus speaks of envoys received by him from India... and it is not improbable that he also received an embassy from China. ...In Pliny's time the trade of the Roman Empire with Asia amounted to a million and a quarter dollars a year, a sum far greater relatively then than now, while by the time of Constantine Europe was in direct communication with the Far East."

"In the fifth century the Persian medical school at Jondi-Sapur admitted both the Hindu and the Greek doctrines..."

"[N]ot far from the time of Boethius, in the sixth century, the Egyptian monk Cosmas, in his earlier years as a trader, made journeys to Abyssinia and even to India and Ceylon, receiving the name Indicopleustes (the Indian traveler). His map (547 a.d.) shows some knowledge of the earth from the Atlantic to India."

"Mohammedanism was to the world from the eighth to the thirteenth century what Rome and Athens and the Italo-Hellenic influence generally had been to the ancient civilization. ...The Arab empire was an ellipse of learning with its foci at Bagdad and Cordova, and its rulers not infrequently took pride in demanding intellectual rather than commercial treasure as the result of conquest."

"[T]he Hindu numerals found their way to the North... in the eighth century they were taken to Bagdad. It was early in that century that the Mohammedans obtained their first foothold in northern India, thus foreshadowing an epoch of supremacy that endured with varied fortunes until after the golden age of the Great (1542-1605) and Shah Jehan. They also conquered Khorassan and Afghanistan, so that the learning and the commercial customs of India at once found easy access to the newly-established schools and the bazaars of Mesopotamia and western Asia."

"It was just after the Sindhind was brought to Bagdad that Muhammad ibn Mūsā al-Khwārizmī... was called to that city. ...Appreciating at once the value of the position system so recently brought from India, he wrote an arithmetic based upon these numerals, and this was translated into Latin..."

"Contemporary with Al-Khowarazmi... Abū 'l-Teiyib, Sened ibn Allī... also wrote a work on Hindu arithmetic...[T]he struggle to have the Hindu numerals replace the Arabic did not cease for a long time thereafter."

"We thus have the numerals in Arabia, in two forms: one the form now used there, and the other the one used by Al-Khowarazmi. The question then remains, how did this second form find its way into Europe?"

"[T]he probability [is] that it was the trader rather than the scholar... [who] carried these numerals from their original habitat to various commercial centers... we shall never know when they first made their inconspicuous entrance into Europe."

"The power of the Goths, who had held Spain for three centuries, was shattered at the battle of Jerez de la Frontera in 711, and almost immediately the Moors became masters of Spain and so remained for five hundred years, and masters of Granada for a much longer period. Until 850 the Christians were... free as to religion and... holding political office, so that priests and monks were not infrequently skilled... in Latin and Arabic, acting as official translators... [W]hile it lasted the learning and the customs of the East must have be come more or less the property of Christian Spain. At thie time the ġobār numerals were probably in that country, and these may well have made their way into Europe from the schools of Cordova, Granada, and Toledo."

"This book gives in compact form a readable and carefully prepared account of the numerous researches... made in the endeavor to trace the origin and development of the Hindu-Arabic numerals. Teachers of mathematics will welcome it, while students specializing in the history of mathematics will derive great help... Like the arithmetician Tonstall the authors read everything in every language and spent much time in licking what they found into shape ad ursi exemplum, as the bear does her cubs."

"Greece must also have had early relations with China, for there is a notable similarity between the Greek and Chinese life, as is shown in their houses, their domestic customs, their marriage ceremonies, the public story tellers, the puppet shows which Herodotus says were introduced from Egypt, the street jugglers, the games of dice, the game of finger-guessing, the , the music system, the use of the , the calendars, and in many other ways."

"So familiar are we with the numerals that bear the misleading name of Arabic, and so extensive is their use... that it is difficult... to realize that their general acceptance... is a matter of only the last four centuries."

"It seems strange that such a labor-saving device should have struggled for nearly a thousand years after its system of place value was perfected before it replaced such crude notations as the one that the Roman conqueror made substantially universal in Europe."

"This story has often been told in part, but it is a long time since any effort has been made to bring together the fragmentary narrations and to set forth the general problem of the origin and development of these numerals."

"In this little work we have attempted to state the history of these forms in small compass, to place before the student materials for the investigation of the problems involved, and to express as clearly as possible the results, of the labors of scholars who have studied the subject..."

"We have had no theory to exploit, for the history of mathematics has seen too much of this tendency... we have weighed the testimony and have set forth what seem to be the reasonable conclusions from the evidence..."

"If this work shall show more clearly the value of our number system, and shall make the study of mathematics seem more real to the teacher and student, and shall offer material for interesting some pupil more fully in his work... the considerable labor involved in its preparation has not been in vain."

"We... acknowledge our especial indebtedness to Professor Alexander Ziwet for reading all the proof, [and] for the digest of a Russian work, to Professor Clarence L. Meader for Sanskrit transliterations, and to Mr. for Arabic transliterations... and... to other scholars in Oriental learning..."

"It has long been recognized that the common numerals used in daily life are of comparatively recent origin."

"The number of systems of notation employed before the Christian era was about the same as the number of written languages, and in some cases a single language had several systems."

"The Egyptians... had three systems of writing, with a numerical notation for each; the Greeks had two... sets of numerals, and the Roman symbols... changed... from century to century."

"It will be well... to think of the numerals... we... call Arabic, as only one of many systems in use just before the Christian era. As it then existed the system was no better than many others, it was of late origin, it contained no zero, it was cumbersome and little used, and it had no particular promise."

"In Europe the invention of notation was generally assigned to the eastern shores of the Mediterranean until the critical period of about a century ago,—sometimes to the Hebrews, sometimes to the Egyptians, but more often to the early trading ns."

"The idea that our common numerals are Arabic in origin is not an old one. The mediaeval and Renaissance writers generally recognized them as Indian, and many of them expressly stated that they were of Hindu origin."

"Others argued that they were probably invented by the Chaldeans or the Jews because they increased in value from right to left, an argument... [also made by] England’s earliest arithmetical textbook-maker, Robert Recorde (c. 1542): "In that thinge all men do agree, that the Chaldays, whiche fyrste inuented thys arte, did set these figures as thei set all their letters, for they wryte backwarde as you tearme it, and so doo they reade. And that may appeare in all Hebrewe, Chaldaye and Arabike bookes .. . where as the Greekes, Latines, and all nations of Europe, do wryte and reade from the lefte hand towarde the ryghte.""

"Tartaglia in Italy and Köbel in Germany, asserted the Arabic origin of the numerals, while still others left the matter undecided or simply dismissed them as "barbaric.""

"[T]he Arabs... never laid claim to the invention, always recognizing their indebtedness to the Hindus both for the numeral forms and for the distinguishing feature of place value."

"Foremost among these writers was the great master of the golden age of Bagdad, one of the first of the Arab writers to collect the mathematical classics of both the East and the West, preserving them and finally passing them on to awakening Europe. This man was Mohammed the Son of Moses, from Khowarezm, or, more after the manner of the Arab, Mohammed ibn Mūsā al-Khowārazmī, a man of great learning and one to whom the world is much indebted for its present knowledge of algebra and of arithmetic. ...[I]n the arithmetic which he wrote, and of which Adelhard of Bath (c. 1130) may have made the translation or paraphrase, he stated distinctly that the numerals were due to the Hindus. This is as plainly asserted by later Arab writers, even to the present day. Indeed the phrase ilm hindī, "Indian science," is used by them for arithmetic, as [is] also the adjective hindī alone."

"The importance of the creation of the zero mark can never be exaggerated. This giving to airy nothing, not merely a local habitation and a name, a picture, a symbol but helpful power, is the characteristic of the Hindu race from whence it sprang. It is like coining the Nirvana into dynamos. No single mathematical creation has been more potent for the general on-go of intelligence and power.’"

"My confidence in our shared future is grounded in my respect for India’s treasured past—a civilization that has been shaping the world for thousands of years. Indians unlocked the intricacies of the human body and the vastness of our universe. And it is no exaggeration to say that our information age is rooted in Indian innovations—including the number zero."

"He sometimes spoke of "zero" as the symbol of the absolute (Nirguna Brahman) of the extreme monistic school of Hindu philosophy, that is, the reality to which no qualities can be attributed, which cannot be defined or described by words and which is completely beyond the reach of the human mind. According to Ramanujan the appropriate symbol was the number "zero" which is the absolute negation of all attributes."