History of the Hindu–Arabic numeral system
|Positional systems by base|
|Non-standard positional numeral systems|
|List of numeral systems|
Its glyphs are descended from the Indian Brahmi numerals. The full system emerged by the 8th to 9th centuries, and is first described in Al-Khwarizmi's On the Calculation with Hindu Numerals (ca. 825), and Al-Kindi's four volume work On the Use of the Indian Numerals (ca. 830). Today the name Hindu–Arabic numerals is usually used.
Evidence of early use of a zero glyph may be present in Bakhshali manuscript, a text of uncertain date, possibly a copy of a text composed as early as the 2nd century BC.
Historians trace modern numerals in most languages to the Brahmi numerals, which were in use around the middle of the 3rd century BC. The place value system, however, developed later. The Brahmi numerals have been found in inscriptions in caves and on coins in regions near Pune, Mumbai, and Uttar Pradesh. These numerals (with slight variations) were in use over quite a long time span up to the 4th century.
During the Gupta period (early 4th century to the late 6th century), the Gupta numerals developed from the Brahmi numerals and were spread over large areas by the Gupta empire as they conquered territory. Beginning around 7th century, the Gupta numerals developed into the Nagari numerals.
Similarity with Chinese numeral system
Singaporean historian of mathematics Lam Lay Yong (National University of Singapore) claims that the computation in Kitab al-Fusul fi al-Hisab al Hindi (925) by al-Uqlidisi, and another Latin translation of the Arab manuscript written by the Persian mathematician Khwarizmi (825), are completely identical to algorithms for square root extraction, addition, subtraction, multiplication and division in the rod calculus described in Mathematical Classic of Sun Zi, which was written five centuries earlier. Yong claims that these methods are too identical to be explained by independent development, and promotes a theory of Chinese origin of the Hindu-Arabic numerals.
There is indirect evidence that the Indians developed a positional number system as early as the 1st century CE. The Bakhshali manuscript (c. 3rd century BCE) uses a place value system with a dot to denote the zero, which is called shunya-sthAna, "empty-place", and the same symbol is also used in algebraic expressions for the unknown (as in the canonical x in modern algebra). However, the date of the Bakhshali manuscript is hard to establish, and has been the subject of considerable debate. The oldest dated Indian document showing use of the modern place value form is a legal document dated 346 in the Chhedi calendar, which translates to 594 CE. While some historians have claimed that the date on this document was a later forgery, it is not clear what might have motivated it, and it is generally accepted that enumeration using the place-value system was in common use in India by the end of the 6th century. Indian books dated to this period are able to denote numbers in the hundred thousands using a place value system. Many other inscriptions have been found which are dated and make use of the place-value system for either the date or some other numbers within the text, although some historians claim these to also be forgeries.
In his seminal text of 499, Aryabhata devised a positional number system without a zero digit. He used the word "kha" for the zero position. Evidence suggests that a dot had been used in earlier Indian manuscripts to denote an empty place in positional notation. . The same documents sometimes also used a dot to denote an unknown where we might use x. Later Indian mathematicians had names for zero in positional numbers yet had no symbol for it.
The use of zero in these positional systems is the final step to the system of numerals we are familiar with today. The first inscription showing the use of zero which is dated and is not disputed by any historian is the inscription at Gwalior dated 933 in the Vikrama calendar (876 CE.). Documents on copper plates, with the same small o in them, dated back as far as the 6th century AD, abound.
The oldest known text to use zero is the Jain text from India entitled the Lokavibhaga, dated 458 AD. Ifrah wrote that a sentence in Lkavibhaga "panchabhyah khalu shunyebhyah param dve sapta chambaram ekam trini cha rupam cha" meant "five voids, then two and seven, the sky, one and three and the form" was the expression of the number 13107200000, was the earliest place value decimal number with the concept of zero.
Adoption by the Arabs
- "I will omit all discussion of the science of the Indians, ... , of their subtle discoveries in astronomy, discoveries that are more ingenious than those of the Greeks and the Babylonians, 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. If those who believe, because they speak Greek, that they have arrived at the limits of science, would read the Indian texts, they would be convinced, even if a little late in the day, that there are others who know something of value."
According to al-Qifti's chronology of the scholars :
- "... a person from India presented himself before the Caliph al-Mansur in the year [776 AD] who was well versed in the siddhanta method of calculation related to the movement of the heavenly bodies, and having ways of calculating equations based on the half-chord [essentially the sine] calculated in half-degrees ... This is all contained in a work ... from which he claimed to have taken the half-chord calculated for one minute. 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 work was most likely to have been Brahmagupta's Brahmasphutasiddhanta (Ifrah)  (The Opening of the Universe) which was written in 628 . Irrespective of whether Ifrah is right, since all Indian texts after Aryabhata's Aryabhatiya used the Indian number system, certainly from this time the Arabs had a translation of a text written in the Indian number system. 
In his text The Arithmetic of Al-Uqlîdisî (Dordrecht: D. Reidel, 1978), A.S. Saidan's studies were unable to answer in full how the numerals reached the Arab world:
- "It seems plausible that it drifted gradually, probably before the 7th century, through two channels, one starting from Sind, undergoing Persian filtration and spreading in what is now known as the Middle East, and the other starting from the coasts of the Indian Ocean and extending to the southern coasts of the Mediterranean."
Al-Uqlidisi developed a notation to represent decimal fractions. The numerals came to fame due to their use in the pivotal work of the Persian mathematician Al-Khwarizmi, whose book On the Calculation with Hindu Numerals was written about 825, and the Arab mathematician Al-Kindi, who wrote four volumes (see ) "On the Use of the Indian Numerals" (Ketab fi Isti'mal al-'Adad al-Hindi) about 830. They, amongst other works, contributed to the diffusion of the Indian system of numeration in the Middle-East and the West.
Development of symbols
The Development of the numerals in early Europe is shown below:
The abacus versus the Hindu–Arabic numeral system in medieval pictures
Adoption in Europe
- 976. The first Arabic numerals in Europe appeared in the Codex Vigilanus in the year 976.
- 1202. Fibonacci, an Italian mathematician who had studied in Béjaïa (Bougie), Algeria, promoted the Arabic numeral system in Europe with his book Liber Abaci, which was published in 1202.
- 1482. The system did not come into wide use in Europe, however, until the invention of printing. (See, for example, the 1482 Ptolemaeus map of the world printed by Lienhart Holle in Ulm, and other examples in the Gutenberg Museum in Mainz, Germany.)
- 1549. These are correct format and sequence of the “modern numbers” in titlepage of the Libro Intitulado Arithmetica Practica by Juan de Yciar, the Basque calligrapher and mathematician, Zaragoza 1549.
In the last few centuries, the European variety of Arabic numbers was spread around the world and gradually became the most commonly used numeral system in the world.
Impact on arithmetic
The significance of the development of the positional number system is described by the French mathematician Pierre Simon Laplace (1749–1827) who wrote:
- "It is India that gave us the ingenuous method of expressing all numbers by the means of ten symbols, each symbol 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, but its very simplicity, the great ease which it has lent to all computations, puts our arithmetic in the first rank of useful inventions, and we shall appreciate the grandeur of this achievement when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest minds produced by antiquity."
- Hindu-Arabic Numerals
- "Abu Yusuf Yaqub ibn Ishaq al-Sabbah Al-Kindi". Retrieved 2007-01-12.
- John J O'Connor and Edmund F Robertson (November 2000). "Indian numerals". The MacTutor History of Mathematics archive. Retrieved 2007-07-24.
- Lam Lay Yong: "The Development of Hindu-Arabic and Traditional Chinese Arithmetic", Chinese Science 13 (1996) p45 diagram i to viii
- Lam Lay Yong, "The Development of Hindu-Arabic and Traditional Chinese Arithmetic", Chinese Science, 1996 p38, Kurt Vogel notation
- Lam Lay Yong, Fleeting Footsteps ISBN 981-02-3696-4
- Lam Lay Yong, An Tian Se, Fleeting Footsteps, p47
- Lam Lay Yong, An Tian Se, Fleeting Footsteps, p42-44
- Lam Lay Yong, An Tian Se, Fleeting Footsteps, p143
- Indian numerals
- Hindu-Arabic Numerals
- Kaplan, Robert. (2000). The Nothing That Is: A Natural History of Zero. Oxford: Oxford University Press.
- Ifrah, Georges. 2000. The Universal History of Numbers: From Prehistory to the Invention of the Computer. David Bellos, E. F. Harding, Sophie Wood and Ian Monk, trans. New York: John Wiley & Sons, Inc. Ifrah 2000:417-1 9
- Al-Uqlidisi biography by J. J. O'Connor and E. F. Robertson
- Earliest Uses of Symbols for Fractions by Jeff Miller
- "The Development of Hindu-Arabic and Traditional Chinese Arithmetic" by Professor Lam Lay Yong, member of the International Academy of the History of Science
- Indian numerals by J J O'Connor and E F Robertson
- Arabic numerals by J J O'Connor and E F Robertson
- Hindu-Arabic numerals
- The Arabic numeral system by: J J O'Connor and E F Robertson
- Filliozat, Pierre-Sylvain (2004), "Ancient Sanskrit Mathematics: An Oral Tradition and a Written Literature", in Chemla, Karine; Cohen, Robert S.; Renn, Jürgen et al., History of Science, History of Text (Boston Series in the Philosophy of Science), Dordrecht: Springer Netherlands, 254 pages, pp. 137-157, pp. 137–157, ISBN 978-1-4020-2320-0 .