- History of the Hindu-Arabic numeral system
The

is aHindu-Arabic numeral system place-value numeral system: the value of a digit depends on the place where it appears; the '2' in 205 is ten times greater than the '2' in 25. It requires a zero to handle the empty powers of ten (as in "205"). [*[*]*http://www.scit.wlv.ac.uk/university/scit/modules/mm2217/han.htm Hindu-Arabic Numerals*]The numeral system was developed in ancient India, and was well established by the time of the

Bakhshali manuscript (ca. 3d c. CE). Despite its Indian origins it was initially known in the West as "Arabic numerals" because of its introduction to Europe through Arabic texts such asAl-Khwarizmi 's "On the Calculation with Hindu Numerals" (ca.825 ), andAl-Kindi 's four volume work "On the Use of the Indian Numerals" (ca.830 ) [*cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Al-Kindi.html |title=Abu Yusuf Yaqub ibn Ishaq al-Sabbah Al-Kindi*] . Today the name "Hindu-Arabic numerals" is usually used.

accessdate=2007-01-12 |format= HTML|work=**Decimal System**An early decimal system was clearly in use by the inhabitants of the

Indus valley civilization by3000 BC. Excavations at bothHarappa andMohenjo Daro reveal decimal weights belonging to"two series both being decimal in nature with each decimal number multipliedand divided by two, giving for the main series ratios of 0.05, 0.1, 0.2, 0.5,1, 2, 5, 10, 20, 50, 100, 200, and 500."cite web

title = Early Indian culture - Indus civilisation

author = Ian Pearce

publisher = The MacTutor History of Mathematics archive

url = http://www-history.mcs.st-andrews.ac.uk/~history/Miscellaneous/Pearce/Lectures/Ch3.html

month = May | year = 2002

accessdate = 2007-07-24] Also, marked rulers atLodhar (?Lothal) reveal gradations of 1.32 inches (3.35 centimetres), ten of which are 13.2 inches, possibly something akin to a "foot" (similar measures exist in other parts of Asia and beyond). Markings on these and other texts reveal a number systemwith symbols for the numbers one through nine, and separate symbols for 10, 20, 100; thus the decimal system is highly developed thoughplace-value is not used.Linguistic comparison among

Indo-European languages (ca. 3000 BC), showsa decimal enumeration system [*[*] .In early Vedic texts, composed between 2500 BC and 1800 BC, we find*http://www.scit.wlv.ac.uk/university/scit/modules/mm2217/han.htm Hindu-Arabic Numerals*]

Sanskrit number words not only for counting numbers in very large ranges, ranging up to 10^{19}, with somepuranas referring to numbers as large as 10^{62}[*G Ifrah: A universal history of numbers: From prehistory to the invention of the computer (London : Harvill Press, 1998). ISBN 1-86046-324-X*] .Historians trace modern numerals in most languages to the

Brahmi numeral s, which were in use around the middle of the third century BC. Theplace value system, however, evolved later. The Brahmi numerals have been found in inscriptions in caves and on coins in regions nearPune ,Mumbai , andUttar Pradesh . These numerals (with slight variations) were in use over quite a long time span up to the 4th century ADcite web

title = Indian numerals

author = John J O'Connor and Edmund F Robertson

publisher = The MacTutor History of Mathematics archive

url = http://www-gap.dcs.st-and.ac.uk/%7Ehistory/HistTopics/Indian_numerals.html

month = November | year = 2000

accessdate = 2007-07-24] .During the Gupta period (early 4th century AD to the late 6th century AD), 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 evolved into the Nagari numerals.

**Positional notation**There is indirect evidence that the

Babylonian s had a place value system as early as the 19th century BC, to the base 60, with a separator mark in empty places. This separator mark never was used at the end of a number, and it was not possible to tell the difference between 2 and 20. This innovation was brought about by Brahmagupta of India. Further, the Babylonian place value marker did not stand alone, as per the Indian "0"Fact|date=June 2007.There is indirect evidence that the Indians developed a positional number system as early as the

first century CE . TheBakhshali manuscript (c. 3d c. 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 theChhedi calendar, which translates to594 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 the6th century . [*[*] . Indian books dated to this period are able to denote numbers in the hundred thousands using a place value system. [*http://www-gap.dcs.st-and.ac.uk/%7Ehistory/HistTopics/Indian_numerals.html Indian numerals*]*[*] 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.*http://www.scit.wlv.ac.uk/university/scit/modules/mm2217/han.htm Hindu-Arabic Numerals*]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. [*http://www-gap.dcs.st-and.ac.uk/%7Ehistory/HistTopics/Indian_numerals.html*] . 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 are 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 theVikrama calendar (876 CE.) [*[*] .*http://www.uni-tuebingen.de/sinologie/eastm/back/cs13/cs13-3-lam.pdf Lamfin.Pdf*]The oldest known text to use zero is the Jain text from

India entitled the**Lokavibhaaga**, dated 458 AD. [*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*]The first indubitable appearance of a symbol for zero appears in 876 in India on a stone tablet in

Gwalior . Documents on copper plates, with the same small o in them, dated back as far as the sixth century AD, abound. [*Kaplan, Robert. (2000). "The Nothing That Is: A Natural History of Zero". Oxford: Oxford University Press.*]**Rational numbers**In

Sanskrit literature ,fractions , orrational numbers were always expressed by an integer followed by a fraction. When the integer is written on a line, the fraction is placed below it and is itself written on two lines, thenumerator called "amsa" part on the first line, the denominator called "cheda" “divisor” on the second below. If the fraction is written without any particular additional sign, one understands that it is added to the integer above it. If it is marked by a small circle or a cross (the shape of the “plus” sign in the West) placed on its right, one understands that it is subtracted from the integer. For example,Bhaskara I writesHarv|Filliozat|2004|p=152]६ १ २ १ १ १

_{०}४ ५ ९That is,

6 1 2 1 1 1

_{०}4 5 9to denote 6+1/4, 1+1/5, and 2–1/9

**Adoption by the Arabs**Before the rise of the

Arab Empire , the Hindu-Arabic numeral system was already moving West and was mentioned inSyria in662 AD by theNestorian scholarSeverus Sebokht who wrote the following::"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." [

*http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Arabic_numerals.html*]According to al-Qifti's chronology of the scholars [

*http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Arabic_numerals.html*] ::"... 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

Arab s a solid base for calculating the movements of the planets ..."The work was most likely to have been

Brahmagupta 's "Brahmasphutasiddhanta " (Ifrah) [*http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Arabic_numerals.html*] (The Opening of the Universe) which was written in 628 [*http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Arabic_numerals.html*] . Irrespective of whether Ifrah is right, since all Indian texts afterAryabhata '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. [*http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Arabic_numerals.html*] 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 seventh 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." [*http://www.uni-tuebingen.de/uni/ans/eastm/back/cs13/cs13-3-lam.pdf*]Al-Uqlidisi developed a notation to represent decimal fractions. [*http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Al-Uqlidisi.html*] [*http://members.aol.com/jeff570/fractions.html*] The numerals came to fame due to their use in the pivotal work of the Persian mathematicianAl-Khwarizmi , whose book "On the Calculation with Hindu Numerals" was written about825 , and theArab mathematicianAl-Kindi , who wrote four volumes (see [2] ) "On the Use of the Indian Numerals" (Ketab fi Isti'mal al-'Adad al-Hindi) about830 . They, amongst other works, contributed to the diffusion of the Indian system of numeration in theMiddle-East and the West.**Evolution of symbols**The evolution of the numerals in early Europe is shown below:

The French scholar J.E. Montucla created this table “Histoire de la Mathematique”, published in 1757:

**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 inBéjaïa (Bougie), Algeria, promoted the Arabic numeral system inEurope with his book "Liber Abaci ", which was published in1202 .

*"1482"The system did not come into wide use in Europe, however, until the invention ofprinting (See, for example, the [*http://bell.lib.umn.edu/map/PTO/TOUR/1482u.html 1482 Ptolemaeus map of the world*] printed byLienhart Holle in Ulm, and other examples in theGutenberg Museum inMainz ,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.

Even in many countries in languages which have their own numeral systems, the European Arabic numerals are widely used in

commerce andmathematics .**The Abacus versus the Hindu-Arabic numeral system in Medieval Pictures****Impact on Mathematics**The significance of the development of the positional number system is probably best 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." "

Tobias Dantzig, the father of

George Dantzig , had this to say in "Number"::"This long period of nearly five thousand years saw the rise and fall of many a civilization, each leaving behind it a heritage of literature, art, philosophy, and religion. But what was the net achievement in the field of reckoning, the earliest art practiced by man? An inflexible numeration so crude as to make progress well nigh impossible, and a calculating device so limited in scope that even elementary calculations called for the services of an expert [...] Man used these devices for thousands of years without contributing a single important idea to the system [...] Even when compared with the slow growth of ideas during the dark ages, the history of reckoning presents a peculiar picture of desolate stagnation. When viewed in this light, the achievements of the unknown Hindu, who some time in the first centuries of our era discovered the principle of position, assumes the importance of a world event."

**ee also***

Table of mathematical symbols by introduction date **Notes****References*** [

*http://www.uni-tuebingen.de/sinologie/eastm/back/cs13/cs13-3-lam.pdf "The Development of Hindu-Arabic and Traditional Chinese Arithmetic" by Professor Lam Lay Yon, member of the International Academy of the History of Science*]

* [*http://www-gap.dcs.st-and.ac.uk/%7Ehistory/HistTopics/Indian_numerals.html Indian numerals by J J O'Connor and E F Robertson*]

* [*http://www-gap.dcs.st-and.ac.uk/%7Ehistory/HistTopics/Arabic_numerals.html Arabic numerals by J J O'Connor and E F Robertson*]

* [*http://www.scit.wlv.ac.uk/university/scit/modules/mm2217/han.htm Hindu-Arabic numerals*]

* [*http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Arabic_numerals.html The Arabic numeral system by: J J O'Connor and E F Robertson*]

*Harvard reference

last1=Filliozat

first1=Pierre-Sylvain

year=2004

chapter= [*http://www.springerlink.com/content/x0000788497q4858/ Ancient Sanskrit Mathematics: An Oral Tradition and a Written Literature*]

pages=360-375

editor1-last=Chemla

editor1-first=Karine

editor2-last=Cohen

editor2-first=Robert S.

editor3-last=Renn

editor3-first=Jürgen

editor4-last=Gavroglu

editor4-first=Kostas

title=History of Science, History of Text (Boston Series in the Philosophy of Science)

place=

publisher=Dordrecht: Springer Netherlands, 254 pages, pp. 137-157

pages=137-157

isbn=9781402023200

url= .

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