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{{main|Hindu-Arabic numeral system}} 
{{Numeral systems}}
The '''[[Hindu-Arabic numeral system]]''' is a decimal [[place-value]] numeral system that uses a [[0 (number)|zero]] glyph as in "205".<ref>[http://www.scit.wlv.ac.uk/university/scit/modules/mm2217/han.htm Hindu-Arabic Numerals]</ref> <!-- shd be moved down? this ref means little here -Mukerjee --> 

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).<ref>{{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
 |accessdate=2007-01-12 |work= }}</ref>  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.

==Decimal System==
Historians trace modern numerals in most languages to the [[Brahmi numeral]]s, which were in use around the middle of the 3rd century BC.<ref name=ind-num/> The [[place value]] system, however, evolved 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.<ref name=ind-num>
{{cite 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
 |date=November 2000
 | accessdate =   2007-07-24
}}</ref>

<!-- Unsourced image removed: [[File:Indian numerals 100AD.gif|frame|left|float|[[Brahmi numeral]]s in the 1st century AD]] -->
During the [[Gupta empire|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.<ref name=ind-num/>  Beginning around 7th century, the Gupta numerals evolved into the Nagari numerals.

== Similarity with Chinese numeral system ==
[[File:Kushyar ibn Labban division.GIF|thumb|right|160px|<math>\tfrac{5625}{243}=23\tfrac{36}{243}</math><br>
10th century Kushyar ibn Labban division]]
[[File:Sunzi division.GIF|thumb|left|160px|Sunzi division algorithm of 400AD<br><math>\tfrac{6561}{9}=729</math><ref>[[Lam Lay Yong]]: "The Development of Hindu-Arabic and Traditional Chinese Arithmetic", ''Chinese Science'' 13 (1996) p45 diagram i to viii</ref>]]
[[File:AL Khwarizmi division.GIF|thumb|left|160px|<math>\tfrac{46468}{324}=143\tfrac{136}{324}</math><br>[[Muhammad ibn Mūsā al-Khwārizmī|Khwarizmi]] division of 825AD, completely identical to Sun Zi division algorithm<br><ref>Lam Lay Yong, "The Development of Hindu-Arabic and Traditional Chinese Arithmetic", ''Chinese Science'', 1996 p38, Kurt Vogel notation</ref>]]

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 people|Persian]] mathematician [[Muhammad ibn Mūsā al-Khwārizmī|Khwarizmi]] (825), are almost identical to algorithms for square root extraction,<ref>Lam Lay Yong, ''Fleeting Footsteps'' ISBN 981-02-3696-4</ref> multiplication and division<ref>Lam Lay Yong, An Tian Se, ''Fleeting Footsteps'', p42-44</ref> in the [[rod calculus]] described in ''[[The Mathematical Classic of Sunzi|Mathematical Classic of Sun Zi]]'', which was written five centuries earlier.  The Chinese origin of Hindu Arabic numerals is promoted.

==Positional notation==
{{further|positional notation}}
There is indirect evidence that the Indians developed a positional number system as early as the 1st century CE.<ref name="ind-num"/>  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.<ref name=ind-num/>  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.<ref>[http://www-gap.dcs.st-and.ac.uk/%7Ehistory/HistTopics/Indian_numerals.html Indian numerals<!-- Bot generated title -->]</ref>  Indian books dated to this period are able to denote numbers in the hundred thousands using a place value system.<ref name="Hindu-Arabic Numerals">[http://www.scit.wlv.ac.uk/university/scit/modules/mm2217/han.htm Hindu-Arabic Numerals<!-- Bot generated title -->]</ref> 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,<ref name=ind-num/> 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.<ref name=ind-num/> 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 [[0 (number)|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.).<ref name=ind-num/><ref>[http://web.archive.org/web/20090305211345/http://www.uni-tuebingen.de/sinologie/eastm/back/cs13/cs13-3-lam.pdf Lamfin.Pdf<!-- Bot generated title -->]</ref> Documents on [[Indian copper plate inscriptions|copper plates]], with the same small o in them, dated back as far as the 6th century AD, abound.<ref>Kaplan, Robert. (2000). ''The Nothing That Is: A Natural History of Zero''. Oxford: Oxford University Press.</ref>

The oldest known text to use zero is the Jain text from [[India]] entitled the '''[[Lokavibhaga]]''', dated 458 AD.<ref>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</ref> 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.<ref>Ifrah,p416</ref>

==Adoption by the Arabs==
Before the rise of the [[Caliphate|Arab Empire]], the Hindu-Arabic numeral system was already moving West and was mentioned in [[Syria]] in 662 AD by the [[Nestorian Christian|Nestorian]] scholar [[Severus 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 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. [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 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."''[http://www.uni-tuebingen.de/uni/ans/eastm/back/cs13/cs13-3-lam.pdf]

[[Al-Uqlidisi]] developed a notation to represent decimal fractions.<ref>[http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Al-Uqlidisi.html Al-Uqlidisi biography] by J. J. O'Connor and E. F. Robertson</ref><ref>[http://jeff560.tripod.com/fractions.html Earliest Uses of Symbols for Fractions] by Jeff Miller</ref>
The numerals came to fame due to their use in the pivotal work of the [[Persian people|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 [2]) "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.

== 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:

{| border="1"
|-----
|
[[File:Apices du moyen-âge.PNG|left|300px|Table of apices]]
|
[[File:EuropeanFormOfArabianDigits.png|right|400px|Table of numerals]]
|-----
|}

== The abacus versus the Hindu-Arabic numeral system in medieval pictures ==
<gallery>
Image:Gregor Reisch, Margarita Philosophica, 1508 (1230x1615).png
Image:Rechentisch.png
Image:Rechnung auff der Linihen und Federn.JPG
Image:Köbel Böschenteyn 1514.jpg
Image:Rechnung auff der linihen 1525 Adam Ries.PNG
Image:1543 Robert Recorde.PNG
Image:Peter Apian 1544.PNG
Image:Adam riesen.jpg
</gallery>

==Adoption in Europe==
{{main|Arabic numerals}}
*'''976'''. The first Arabic numerals in Europe appeared in the ''[[Codex Vigilanus]]'' in the year 976.

*'''1202'''. [[Leonardo of Pisa|Fibonacci]], an [[Italy|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 [http://bell.lib.umn.edu/map/PTO/TOUR/1482u.html 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.

{| border="1"
|-----
|
[[File:Codex Vigilanus Primeros Numeros Arabigos.jpg|300px|thumb|The first Arabic numerals in Europe appeared in the ''[[Codex Vigilanus]]'' in the year 976.]]
|
[[File:Cosmographia 1482.JPG|thumb|Medieval Arabic Numbers at World map from Ptolemy, Cosmographia. Ulm: Lienhart Holle, 1482]]
|
[[File:JuanDeYciar.jpg|thumb|Libro Intitulado Arithmetica Practica, 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]] and [[mathematics]].

==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 of Perga|Apollonius]], two of the greatest minds produced by antiquity." ''

Tobias Dantzig, the father of [[George Dantzig]], had this to say in ''Number'':<ref>[http://books.google.co.in/books?id=GV65dVUX2HoC&pg=PA9&dq=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&hl=en&ei=R9PtTdTUFMyxrAeH5rzjAw&sa=X&oi=book_result&ct=book-preview-link&resnum=3&ved=0CDgQuwUwAg#v=onepage&q=Dantzig&f=false Essays on ancient India] By Raj Kumar - Discovery Publishing House, 2003.</ref><ref>[http://books.google.co.in/books?id=qvnjXOCjv7EC&pg=PA197&dq=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&hl=en&ei=R9PtTdTUFMyxrAeH5rzjAw&sa=X&oi=book_result&ct=book-preview-link&resnum=1&ved=0CDAQuwUwAA#v=onepage&q=%22Dantzig%2C%20in%20his%20Number%2C%20writes%22&f=false Geometry] By Roger Fenn, Springer, 2001</ref>

:''"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."''

==See also==
*[[Table of mathematical symbols by introduction date]]

==Notes==
<references />

==References==
*[http://sciences.aum.edu/~sbrown/Hindu%20Arabic%20and%20Chinese.pdf "The Development of Hindu-Arabic and Traditional Chinese Arithmetic" by Professor Lam Lay Yong, 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]
*{{Citation
 | last1=Filliozat
 | first1=Pierre-Sylvain 
 | year=2004
 | chapter=Ancient Sanskrit Mathematics: An Oral Tradition and a Written Literature
 | chapter-url=http://www.springerlink.com/content/x0000788497q4858/
 | 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=978-1-4020-2320-0
 | url=
 }}.

{{DEFAULTSORT:History Of The Hindi-Arabic Numeral System}}
[[Category:Numeral systems]]
[[Category:Elementary mathematics]]
[[Category:Arabic language]]
[[Category:History of Hinduism]]