numeral-systems

"The ancient Jews used Hebrew as their numerical system. May I? Each letter's a number. Like, the Hebrew A, Aleph [], is 1. B, Bet [], is 2. You understand?"

"1. All things proclaim an active, thinking cause. 2. The living unity heeds number's laws. 3. That which contains all is by nothing bound. 4. Before all else, He everywhere is found. 5. He, the sole Master—praise to Him alone! 6. To pure hearts His true doctrine He makes known. 7. Faith's works a single guide have, under heaven. 8. So one sole altar and one law are given. 9. What the Eternal founds, forever stays 67"

"In case of multiples from the units place, the value of each place (sthana) is ten times the value of the preceding place."

"The sum of two positive quantities is positive; of two negative is negative; of a positive and a negative is their difference; or, if they are equal, zero. The sum of zero and negative is negative; of positive and zero is positive; of two zeros is zero (31)."

"In subtraction, the less is to be taken from the greater, positive from positive; negative from negative. When the greater, however, is subtracted from the less, the difference is reversed. Negative taken from zero becomes positive; and positive [taken from zero] becomes negative. Zero subtracted from negative is negative; from positive, is positive; from zero, is zero. When positive is to be subtracted from negative, and negative from positive, they must be thrown together (32-33)."

"The product of a negative quantity and a positive is negative; of two negatives, is positive; of two positives, is positive. The product of zero and negative, or of zero and positive, is zero; [the product] of two zeros, is zero. (34)."

"Positive, divided by positive, or negative by negative, is positive. Zero, divided by zero, is zero. Positive, divided by negative, is negative. Negative, divided by positive, is negative. Positive, or negative, divided by zero, is a fraction with that for denominator: or zero divided by negative or positive. (35-36)."

"The square of negative or positive is positive; of zero, is zero. The root of a square is such as was that from which it was raised [i.e. either positive or negative]. (37)."

"The grandest achievement of the Hindus and the one which, of all mathematical inventions, has contributed most to the general progress of intelligence, is the invention of the principle of position in writing numbers. Generally we speak of our notation as the “Arabic” notation, but it should be called the “Hindu” notation, for the Arabs borrowed it from the Hindus. That the invention of this notation was not so easy as we might suppose at first thought, may be inferred from the fact that, of other nations, not even the keen-minded Greeks possessed one like it."

"‘…the transition [to the Hindu number system], far from being immediate, extended over long centuries. The struggle between the Abacists, who defended the old traditions, and the Algorists, who advocated the reform, lasted from the eleventh to the fifteenth century and went through all the usual stages of obscurantism and reaction. In some places, Arabic numerals [more precisely, Hindu numerals] were banned from official documents; in others, the art was prohibited altogether. And, as usual, prohibition did not succeed in abolishing, but merely served to spread bootlegging, ample evidence of which is found in the thirteenth century archives of Italy, where, it appears, merchants were using the Arabic numerals as a sort of secret code.’"

"It is India that gave us the ingenious method of expressing all numbers by ten symbols, each receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit."

"As my father was a public official away from our homeland in the Bugia customs house established for the Pisan merchants who frequently gathered there, he had me in my youth brought to him, looking to find for me a useful and comfortable future; there he wanted me to be in the study of mathematics and to be taught for some days. There from a marvelous instruction in the art of the nine Indian figures, the introduction and knowledge of the art pleased me so much above all else, and I learnt from them, whoever was learned in it, from nearby Egypt, Syria, Greece,Sicily and Provence, and their various methods, to which locations of business I travelled considerably afterwards for much study, and I learnt from the assembled disputations. But this, on the whole, the algorithm and even the Pythagorean arcs, I still reckoned almost an error compared to the Indian method.’"

"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.’"

"The difficulty of understanding why it should have been the Hindus who took this step, why it was not taken by the mathematicians of antiquity, why it should first have been taken by practical man, is only insuperable if we seek for the explanation of intellectual progress in the genius of a few gifted individuals, instead of in the whole social framework of custom thought which circumscribes the greatest individual genius. What happened in India about AD 100 had happened before. May be it is happening now in Soviet Russia…. To accept it (this truth) is to recognise that every culture contains within itself its own doom, unless it pays as much attention to the education of the mass of mankind as to the education of the exceptionally gifted people.’"

"All the algorithms for fractions now used were invented by the Hindus. The Greek treatment of fractions never advanced beyond the level of the Egyptian Rhind papyrus. […] This inability to treat a fraction as a number on its own merits is the explanation of a practice [which] was as useless as it was ambiguous. […] When we remember that the Greeks and Alexandrians continued this extraordinary performance, there is nothing remarkable about the small progress which they achieved in their arithmetic. What is remarkable is that a few of them like Archimedes should have discovered anything at all about series of numbers involving fractional quantities."

"‘The change did not come about without obstruction from the representatives of custom thought. An edict of A.D. 1259 forbade the bankers of Florence to use the infidel symbols, and the ecclesiastical authorities of the University of Padua in A.D. 1348 ordered that the price list of books should be prepared not in “ciphers”, but in plain letters.’"

"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."

"Just as, although the stroke [line] is the same, yet by a change of place it acquires the values, one, ten, hundred, thousand, etc…"

"Probably the most striking testimony from Arabic sources is that given by the Arabic traveler and scholar Mohammed ibn Ahmed, Abū ’l-Rīahān al-Bīrūnī (973-1048), who spent many years in Hindustan. He wrote... the “Book of the Ciphers,” unfortunately lost, which treated... of the Hindu art of calculating... being versed in Arabic, Persian, Sanskrit, Hebrew, and Syriac, as well as in astronomy, chronology, and mathematics. In his work on India he... states explicitly that the Hindus of his time did not use the letters of their alphabet for numerical notation, as the Arabs did. He also states that the numeral signs called aṅka had different shapes in various parts of India, as was the case with the letters. In his Chronology of Ancient Nations he gives the sum of a geometric progression... in three different systems..."

"Preceding Al-Bīrūnī... another Arabic writer of the tenth century, Motahhar ibn Tāhir, author of the Book of the Creation and of History... gave... in Indian (Nāgarī symbols), a large number asserted by the people of India to represent the duration of the world."

"Al-Mas'ūdī (885?-956), whose journeys carried him from Bagdad to Persia, India, Ceylon, and... across the China sea, and at other times to Madagascar, Syria, and Palestine... neglected no accessible sources of information, examining also the history of the Persians, the Hindus, and the Romans. ...[H]is ...Meadows of Gold ...states that the wise men of India, assembled by the king, composed the Sindhind...that by order of Al-Mansur many works of science and astrology were translated into Arabic, notably the Sindhind (Siddhanta). Concerning the meaning and spelling of this name... Colebrooke ascribes... the meaning "the revolving ages." Similar designations are collected by Sedillot... Casiri... refers to the work as the Sindum-Indum... meaning "perpetuum aeternumque [eternal perpetuity].""

"This Sindhind is the book, says Mas'ūdī, which gives all that the Hindus know of the spheres, the stars, arithmetic, and the other branches of science. He mentions... Al-Khowārazmī and Habash as translators of the tables of the Sindhind."

"The oldest work... complete, on the history of Arabic literature and history is the Kitah al-Fihrist, written in the year 987 a.d., by Ibn Abī Ya'qūb al-Nadīm. ...Of the ten chief divisions of the work, the [second subdivision of the] seventh... treats of mathematicians and astronomers. The first of the Arabic writers mentioned is Al-Kindī (800-870 A.D.), who wrote five books on arithmetic and four books on the use of the Indian method of reckoning. Sened ibn 'Alī... is also given as the author of a work on the Hindu method of reckoning. ...[T]here is a possibility that some of the works ascribed to Sened ibn 'Alī are really works of Al-Khowārazmī ...However, ...Casiri also mentions the same writer as the author of a most celebrated work on arithmetic. To Al-Sūfī... is also credited a large work... and similar treatises by other writers..."

"[T]herefore... the Arabs from the early ninth century on fully recognized the Hindu origin of the new numerals."

"Leonard of Pisa... wrote his Liber Abbaci in 1202. ...[H]e refers frequently to the nine Indian figures, thus showing again the general consensus of opinion in the Middle Ages that the numerals were of Hindu origin."

"One of the earliest treatises on algorism is... the Carmen de Algorismo [Poem about Arithmetic], written by Alexander de Villa Dei (Alexandre de Ville-Dieu), a Minorite monk of about 1240 a.d. The text of the first few lines is as follows: "Hee algorism’ ars p’sens dicit’ in qua Talib; indor\mathit{4} fruim bis quinq; figuris." "This boke is called the boke of algorim or augrym after lewder use. And this boke tretys of the Craft of Nombryng [arithmetic], the quych crafte is called also Algorym. Ther was a kyng of Inde the quich heyth Algor & he made this craft. . . . Algorisms, in the quych we use teen figurys of Inde.""

"While it is generally conceded that the scientific development of astronomy among the Hindus towards the beginning of the Christian era rested upon Greek or Chinese sources, yet their ancient literature testifies to a high state of civilization, and to a considerable advance in sciences, in philosophy, and along literary lines, long before the golden age of Greece."

"From the earliest times... to... present day the Hindu has been wont to put his thought into rhythmic form. The first of this poetry... being also worthy from a metaphysical point of view... consists of the Vedas, hymns of praise and poems of worship, collected during the ... from approximately 2000 B.C. to 1400 B.C. Following this work, or possibly contemporary... is the Brahmanic literature, which is partly ritualistic (the s), and partly philosophical (the Upanishads). Our... interest is in the s... which contain... geometric material used in connection with altar construction, and also numerous examples of... "Pythagorean numbers," although this was long before Pythagoras lived."

"Whitney places the whole of the Veda literature, including the Vedas, the Brahmanas, and the Sutras, between 1500 B.C. and 800 B.C., thus agreeing with [Albert] Bürk who holds that the knowledge of the Pythagorean theorem revealed in the Sütras goes back to the eighth century B.C."

"The importance of the Sutras as showing an independent origin of Hindu geometry, contrary to the opinion long held by Cantor of a Greek origin, has been repeatedly emphasized in recent literature, especially since... Von Schroeder."

"Further fundamental mathematical notions such as the conception of irrationals and the use of s, as well as the philosophical doctrine of the transmigration of souls, —all of these having long been attributed to the Greeks, —are shown in these works to be native to India."

"[W]e are not at all sure that the most ancient forms of the numerals commonly known as Arabic had their origin in India. ...[T]heir forms may have been suggested by those used in Egypt, or in Eastern Persia, or in China, or on the plains of Mesopotamia. We are... in the dark as to these early steps; but as to their development in India, the approximate period of the rise of their essential feature of place value, their introduction into the Arab civilization, and their spread to the West, we have more or less definite information."

"When... we consider the rise of the numerals in the land of the Sindhu... only the large movement... is meant, and that there must... be... numerous possible sources outside of India... and long anterior to the first prominent appearance of the number symbols."

"[I]n the history of ancient India... primary schools... existed in earliest times, and of the seventy-two recognized sciences writing and arithmetic were the most prized. In the Vedic period [~]2000 to 1400 B.C., there was the same attention to astronomy that was found in the earlier civilizations of Babylon, China, and Egypt... Such advance... presupposes a fair knowledge of calculation, but of the manner of calculating we are quite ignorant..."

"[T]he Lalitavistara, relates that when the Bödhisattva was of age to marry, the father of Gopa, his intended bride, demanded an examination of the five hundred suitors, the subjects including arithmetic, writing, the lute, and archery. Having vanquished his rivals in all else, he is matched against Arjuna the great arithmetician and is asked to express numbers greater than 100 kotis. In reply he gave a scheme of number names as high as 1053, adding that he could proceed as far as 10421... which suggests the system of Archimedes and the unsettled question of the indebtedness of the West to the East in the realm of ancient mathematics."

"Sir Edwin Arnold, in The Light of Asia... speaks of Buddha’s training at the hands of the learned : "And Viswamitra said, 'It is enough, Let us to numbers. After me repeat Your numeration till we reach the lakh, One, two, three, four, to ten, and then by tens To hundreds, thousands.’ After him the child Named digits, decads, centuries, nor paused, The round lakh reached, but softly murmured on, Then comes the koti, nahut, ninnahut, Khamba, viskhamba, abab, attata, To kumuds, gundhikas, and utpalas, By pundarikas into padumas, Which last is how you count the utmost grains Of Hastagiri ground to finest dust; But beyond that a numeration is, The Kātha, used to count the stars of night, The Kōti-Kātha, for the ocean drops; Ingga, the calculus of circulars; Sarvanikchepa, by the which you deal With all the sands of Gunga, till we come To Antah-Kalpas, where the unit is The sands of the ten crore Gungas. If one seeks More comprehensive scale, th’ arithmic mounts By the Asankya, which is the tale Of all the drops that in ten thousand years Would fall on all the worlds by daily rain; Thence unto Maha Kalpas, by thé which the gods compute their future and their past.'""

"Thereupon Ācārya expresses his approval... and asks to hear the "measure of the line" as far as yōjana, the longest measure bearing name. This given, Buddha adds: ..." 'And master! If it please, I shall recite how many sun-motes lie From end to end within a yōjana.’ Thereat, with instant skill, the little prince Pronounced the total of the atoms true. But Viswamitra heard it on his face Prostrate before the boy; 'For thou,' he cried, 'Art Teacher of thy teachers—thou, not I, Art Guru.' ""

"[T]his is far from being history. And yet it puts in charming rhythm only what the ancient Lalitavistara relates of the number-series of the Buddha’s time. ...[I]t reveals a condition that would have been impossible unless arithmetic had attained a considerable degree of advancement."

"[W]e are uncertain as to the time and place of [the] introduction [of these numeral forms] into Europe. There are two general theories... The first is that they were carried by the Moors to Spain in the eighth or ninth century, and thence were transmitted to Christian Europe... The second, advanced by Woepcke, is that they... were already in Spain when the Arabs arrived there, having reached the West through the Neo-Pythagoreans. There are two facts to support this second theory: (1) the forms of these numerals are characteristic, differing materially from those which were brought by Leonardo of Pisa from Northern Africa early in the thirteenth century (before 1202 a.d.); (2) they are essentially those which tradition has so persistently assigned to Boethius (c. 500 A.D.), and which he would naturally have received, if at all, from these same Neo-Pythagoreans or from the sources from which they derived them."

"Woepcke points out that the Arabs on entering Spain (711 A.D.) would naturally have followed their custom of adopting for the computation of taxes the numerical systems of the countries they conquered... The theory is that the Hindu system, without the zero, early reached Alexandria (say 450 a.d.), and that the Neo-Pythagorean love for the mysterious and... the Oriental led to its use... that it was then passed along the Mediterranean, reaching Boethius in Athens or in Rome, and to the schools of Spain, being discovered in Africa and Spain by the Arabs even before they themselves knew the improved system with the place value."

"Bubnov holds that the forms first found in Europe are derived from ancient symbols used on the abacus, but that the zero is of Hindu origin. This theory does not seem tenable..."

"The Spanish forms of the numerals were called the hurūf al-ģobār, the ģobār or dust numerals, as distinguished from the hurūf aljumal or alphabetic numerals. Probably the latter... were also used by the Arabs. ...[D]oubtless ...these numerals were written on the dust abacus, this plan being distinct from the counter method ...Al-Bīrūnī states that the Hindus often performed numerical computations in the sand. ...The system has nine characters, but no zero. A dot above a character indicates tens, two dots hundreds, and so on, \dot{5} meaning 50, and [\ddot{5} meaning 500]."

"When we consider... that the dot is found for zero in the Bakhsālī manuscript, and that it was used in subscript form in the Kitāb al-Fihrist in the tenth century... we are forced to believe that this form may also have been of Hindu origin."

"The Indian use of subscript dots to indicate the tens, hundreds, thousands, etc., is established by a passage in the Kitāb al-Fihrist (987 A.D.)... The numeral forms given are those which have usually been called Indian, in opposition to ģobār. In this document the dots are placed below the characters, instead of being superposed... The significance was the same."

"Anicius Manlius Severinus Boethius was born at Rome c. 475. Not many generations after his death, the period being one in which historical criticism was at its lowest ebb, the church found it profitable to look upon his execution as a martyrdom. He was accordingly looked upon as a saint, his bones were enshrined, and as a natural consequence his books were among the classics in the church schools for a thousand years. It is pathetic, however, to think of the medieval student trying to extract mental nourishment from a work so abstract, so meaningless, so unnecessarily complicated, as the arithmetic of Boethius."

"The numerals had existed, without the zero, for several centuries; they had been well known in India; there had been a continued interchange of thought between the East and West; and warriors, ambassadors, scholars, and the restless trader, all had gone back and forth, by land or more frequently by sea, between the Mediterranean lands and the centers of Indian commerce and culture. Boethius could very well have learned one or more forms of Hindu numerals from some traveler or merchant."

"[I]t is one of the mistakes of scholars to believe that they are the sole transmitters of knowledge. ...[T]he characters, the methods of calculating, the improvements that took place from time to time, the zero when it appeared, and the customs as to solving business problems, would all have been made known from generation to generation along... trade routes from the Orient to the Occident. [I]t was to the tradesman and the wandering scholar that the spread of such learning was due, rather than to the school man."

"Avicenna (980-1037 a.d.)... relates that when his people were living at Bokhara his father sent him to the house of a grocer to learn the Hindu art of reckoning, in which this grocer (oil dealer, possibly) was expert. Leonardo of Pisa, too, had a similar training."

"It could not have been at all unusual for the ancient Greeks to go to India, for Strabo lays down the route, saying that all who make the journey start from and traverse and before taking the direct road. The products of the East were always finding their way to the West, the Greeks getting their ginger from Malabar, as the Phoenicians had long before brought gold from Malacca."

"The Chinese historians tell us that about 200 B.C. their arms were successful in the far west, and that in 180 B.C. an ambassador went to , then a Greek city, and reported that Chinese products were on sale in the markets there. There is also a noteworthy resemblance between certain Greek and Chinese words, showing that in remote times there must have been more or less interchange of thought."