Editing Positional notation (section)

== History ==
[[File:abacus 6.png|thumb|[[Suanpan]] (the number represented in the picture is 6,302,715,408)]]
Today, the base-10 ([[decimal]]) system, which is presumably motivated by counting with the ten [[finger]]s, is ubiquitous. Other bases have been used in the past, and some continue to be used today. For example, the [[Babylonian numerals|Babylonian numeral system]], credited as the first positional numeral system, was [[base-60]]. However, it lacked a real [[zero]]. Initially inferred only from context, later, by about 700&nbsp;BC, zero came to be indicated by a "space" or a "punctuation symbol" (such as two slanted wedges) between numerals.<ref name="multiref1">{{cite book | last = Kaplan | first = Robert | year = 2000 | title = The Nothing That Is: A Natural History of Zero | url = https://archive.org/details/nothingthatisnat00kapl | url-access = registration |via=archive.org | location = Oxford | publisher = Oxford University Press |pages=11–12 }}</ref> It was a [[variable (mathematics)|placeholder]] rather than a true zero because it was not used alone or at the end of a number. Numbers like 2 and 120 (2×60) looked the same because the larger number lacked a final placeholder. Only context could differentiate them.

The polymath [[Archimedes]] (ca. 287–212 BC) invented a decimal positional system based on 10<sup>8</sup> in his [[The Sand Reckoner|Sand Reckoner]];<ref name="Greek numerals">{{Cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Greek_numbers.html |title=Greek numerals |access-date=31 May 2016 |archive-url=https://web.archive.org/web/20161126013536/http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Greek_numbers.html |archive-date=26 November 2016 |url-status=dead }}</ref> 19th century German mathematician [[Carl Friedrich Gauss|Carl Gauss]] lamented how science might have progressed had Archimedes only made the leap to something akin to the modern decimal system.<ref>[[Karl Menninger (mathematics)|Menninger, Karl]]: ''Zahlwort und Ziffer. Eine Kulturgeschichte der Zahl'', Vandenhoeck und Ruprecht, 3rd. ed., 1979, {{isbn|3-525-40725-4}}, pp. 150–153</ref> [[Hellenistic period|Hellenistic]] and [[Roman Empire|Roman]] astronomers used a base-60 system based on the Babylonian model (see {{slink|Greek numerals|Zero}}).

Before positional notation became standard, simple additive systems ([[sign-value notation]]) such as [[Roman numerals]] or [[Chinese numerals]] were used, and accountants in the past used the [[abacus]] or stone counters to do arithmetic until the introduction of positional notation.<ref>Ifrah, page 187</ref>

[[File:Chounumerals.svg|thumb|right|280px|Chinese [[rod numerals]]; Upper row vertical form{{br}} Lower row horizontal form]]
[[Counting rods]] and most abacuses have been used to represent numbers in a positional numeral system. With counting rods or [[abacus]] to perform arithmetic operations, the writing of the starting, intermediate and final values of a calculation could easily be done with a simple additive system in each position or column. This approach required no memorization of tables (as does positional notation) and could produce practical results quickly.

The oldest extant positional notation system is either that of Chinese [[rod numerals]], used from at least the early 8th century, or perhaps [[Khmer numerals]], showing possible usages of positional-numbers in the 7th century. Khmer numerals and other [[Indian numerals]] originate with the [[Brahmi numerals]] of about the 3rd century BC, which symbols were, at the time, not used positionally. Medieval Indian numerals are positional, as are the derived [[Arabic numerals]], recorded from the 10th century.

After the [[French Revolution]] (1789–1799), the new French government promoted the extension of the decimal system.<ref>L. F. Menabrea.
Translated by Ada Augusta, Countess of Lovelace.
[http://www.fourmilab.ch/babbage/sketch.html "Sketch of The Analytical Engine Invented by Charles Babbage"] {{Webarchive|url=https://web.archive.org/web/20080915134651/http://www.fourmilab.ch/babbage/sketch.html |date=15 September 2008 }}.
1842.</ref> Some of those pro-decimal efforts—such as [[decimal time]] and the [[decimal calendar]]—were unsuccessful. Other French pro-decimal efforts—currency [[decimalisation]] and the [[metrication]] of weights and measures—spread widely out of France to almost the whole world.

=== History of positional fractions ===
{{Main|Decimal}}

Decimal fractions were first developed and used by the Chinese in the form of [[Rod calculus#Fractions|rod calculus]] in the 1st century BC, and then spread to the rest of the world. <ref>[[Lam Lay Yong]], "The Development of Hindu-Arabic and Traditional Chinese Arithmetic", ''Chinese Science'', 1996 p38, Kurt Vogel notation</ref><ref name=jnfractn1>{{Cite book | author=[[Joseph Needham]] | chapter = Decimal System | title = [[Science and Civilisation in China|Science and Civilisation in China, Volume III, Mathematics and the Sciences of the Heavens and the Earth]] | year = 1959 | publisher = Cambridge University Press}}</ref> J. Lennart Berggren notes that positional decimal fractions were first used in the Arab by mathematician [[Abu'l-Hasan al-Uqlidisi]] as early as the 10th century.<ref name=Berggren>{{cite book | first=J. Lennart | last=Berggren | title=The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook | chapter=Mathematics in Medieval Islam | publisher=Princeton University Press | year=2007 | isbn=978-0-691-11485-9 | page=518 }}</ref> The Jewish mathematician [[Immanuel Bonfils]] used decimal fractions around 1350, but did not develop any notation to represent them.<ref>[[Solomon Gandz|Gandz, S.]]: The invention of the decimal fractions and the application of the exponential calculus by Immanuel Bonfils of Tarascon (c. 1350), Isis 25 (1936), 16–45.</ref> The Persian mathematician [[Jamshīd al-Kāshī]] made the same discovery of decimal fractions in the 15th century.<ref name=Berggren /> [[Al Khwarizmi]] introduced fractions to Islamic countries in the early 9th century; his fraction presentation was similar to the traditional Chinese mathematical fractions from [[Sunzi Suanjing]].<ref name=Lam>[[Lam Lay Yong]], "The Development of Hindu-Arabic and Traditional Chinese Arithmetic", ''Chinese Science'', 1996, p. 38, Kurt Vogel notation</ref> This form of fraction with numerator on top and denominator at bottom without a horizontal bar was also used by 10th century [[Abu'l-Hasan al-Uqlidisi]] and 15th century [[Jamshīd al-Kāshī]]'s work "Arithmetic Key".<ref name=Lam/><ref>{{cite journal | last1 = Lay Yong | first1 = Lam | author-link = Lam Lay Yong | title = A Chinese Genesis, Rewriting the history of our numeral system | journal = Archive for History of Exact Sciences | volume = 38 | pages = 101–108 }}</ref>
{| class="wikitable floatright"
! style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: right; " class="table-rh" | Number
|184.54290
|-
! style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: right; " class="table-rh" | Simon Stevin's notation
|184⓪5①4②2③9④0
|}
The adoption of the [[decimal representation]] of numbers less than one, a [[fraction (mathematics)|fraction]], is often credited to [[Simon Stevin]] through his textbook [[De Thiende]];<ref name=van>{{Cite book | author = B. L. van der Waerden | author-link = Bartel Leendert van der Waerden | year = 1985 | title = A History of Algebra. From Khwarizmi to Emmy Noether | url = https://archive.org/details/historyofalgebra0000waer| url-access = registration| publisher = Springer-Verlag | place = Berlin}}</ref> but both Stevin and [[E. J. Dijksterhuis]] indicate that [[Regiomontanus]] contributed to the European adoption of general [[decimal]]s:<ref name=EJD>[[E. J. Dijksterhuis]] (1970) ''Simon Stevin: Science in the Netherlands around 1600'', [[Martinus Nijhoff Publishers]], Dutch original 1943</ref>
: European mathematicians, when taking over from the Hindus, ''via'' the Arabs, the idea of positional value for integers, neglected to extend this idea to fractions. For some centuries they confined themselves to using common and [[sexagesimal]] fractions&nbsp;... This half-heartedness has never been completely overcome, and sexagesimal fractions still form the basis of our trigonometry, astronomy and measurement of time. ¶ ... Mathematicians sought to avoid fractions by taking the radius ''R'' equal to a number of units of length of the form 10<sup>''n''</sup> and then assuming for ''n'' so great an integral value that all occurring quantities could be expressed with sufficient accuracy by integers. ¶ The first to apply this method was the German astronomer Regiomontanus. To the extent that he expressed goniometrical line-segments in a unit ''R''/10<sup>''n''</sup>, Regiomontanus may be called an anticipator of the doctrine of decimal positional fractions.<ref name=EJD/>{{rp|17,18}}

In the estimation of Dijksterhuis, "after the publication of [[De Thiende]] only a small advance was required to establish the complete system of decimal positional fractions, and this step was taken promptly by a number of writers ... next to Stevin the most important figure in this development was Regiomontanus." Dijksterhuis noted that [Stevin] "gives full credit to Regiomontanus for his prior contribution, saying that the trigonometric tables of the German astronomer actually contain the whole theory of 'numbers of the tenth progress'."<ref name=EJD/>{{rp|19}}