It was once mistakenly thought that the Arabs invented ‘Arabic’ numerals, including the number ‘0’ (zero), a great improvement over Roman numerals for algebra, another supposed Arabic invention. That’s complete nonsense. For example, Cajori’s A History of Mathematics (Fifth edition, 1991) informs us:
The grandest achievement of the Hindus, and the one which, of all mathematical inventions, has contributed most to the progress of intelligence, is the perfecting of the so-called “Arabic Notation.” That this notation did not originate with the Arabs is now admitted by everyone.
What we call ‘Arabic’ numerals are not the characters that appear in Arabic script. Even the Arabs themselves don’t call these signs Arabic numerals but Western numerals, or Hindu numerals, which was their actual source around 200 BC. By the fifth century AD the Indians were using decimal notation and had developed the use of zero from an earlier concept of it in Babylonian mathematics.
Through the writings of Gerbert D’Aurillac (c.946-1003) these numerals became known in Western ‘Latin’ Europe. In 1202, the mathematician Fibonacci in his Liber Abaci [The Book of the Abacus] describes the notation and calculation method that he learned from his youth thus:
…following my introduction…to the nine digits of the Hindus, the knowledge of the art very much appealed to me before all others…Therefore, embracing more stringently that method of the Hindus [Modus Indorum], and taking stricter pains in its study…I have striven to compose this book. The nine Indian figures are: 9 8 7 6 5 4 3 2 1. With these nine figures, and with the sign 0… any number may be written…
The nine digits were known in Christendom before Islam came onto the scene. In AD 662 Severus Sebokht, a Nestorian bishop who lived in Keneshra on the Euphrates before the Arabs arrived there, wrote
I will omit all discussion of the science of the Indians…and of their valuable methods of calculation which surpass description. I wish only to say that this computation is done by means of nine signs.
How the Indian numerals and calculation methods came to be transmitted to the Arabs is recorded in the Arabic Chronology of the scholars by al-Qifti
…a person from India presented himself before the Caliph al-Mansur in the year [AD 776] who was well versed in the siddhanta method of calculation…Al-Mansur ordered this book to be translated into Arabic, and a work to be written, based on the translation, to give the Arabs a solid base for calculating the movements of the planets…
The Siddhanta method introduced arithmetic, trigonometry, algebra and geometry, which had definitely been borrowed and improved from the Greeks. In 628, Brahmagupta had written his Brahma-sphita-siddhanta, which included two chapters on mathematics, and represents the high water mark of Indian mathematics. Even then, Cajori reminds us,
Brahmagupta…introduces a proof of Ptolemy’s theorem and in doing so follows Diophantus…and uses the actual examples given by Diophantus…Parallelisms of this sort show unmistakably that the Hindus drew from Greek sources.
Regarding algebra, Cajori remarks
In the Hindu solutions of determinate equations…[are] traces of Diophantine methods. Some technical terms betray their Greek origin.
Diophantus of Alexandria, a Greek, lived in the third century AD, and is regarded as the ‘father of algebra’. He wrote thirteen books of Arithmetica, of which six survive. This is a collection of problems giving numerical solutions of both determinate and indeterminate polynomial equations. It was his polynomial equations that famously inspired Pierre de Fermat to propose what is known as Fermat’s Last Theorem, which he wrote in the margin of his copy of Arithmetica.
Arithmetica became known to the Arabs hundreds of years after Diophantus when Abu’l-Wefa translated it into Arabic. In Arabic works on algebra the worked examples used by Diophantus keep turning up. The word algebra itself comes from Arabic al-jabr, ‘completion’ an operation for solving quadratic equations found in the book known in Latin as Liber algebrae et almucabala, or The Book on Calculation by Completion and Balancing by the Persian mathematician al-Khwarizmi (“Algoritmi”, from which the word algorithm is derived) in the ninth century. Al-Khwarizmi improved algebra, and also wrote The Book of Addition and Subtraction according to the Hindu Calculation and On Calculation with Hindu Numerals, known in Latin as Algoritmi de numero Indorum, which rather obviously betray his immediate sources for these.