Hindu–Arabic numeral system

"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…"