View source for Timeline of numerals and arithmetic

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{{See also|History of arithmetic|History of ancient numeral systems}}
A [[timeline]] of '''numerals''' and '''[[arithmetic]]'''.

==Before 2000 BC==
{{Further|Prehistoric numerals}}
* c. [[Upper Paleolithic|20,000 BC]] — [[Nile Valley]], [[Ishango Bone]]: suggested, though disputed, as the earliest reference to [[prime number]]s as also a common number.<ref>{{cite book|last=Rudman|first=Peter Strom|title=How Mathematics Happened: The First 50,000 Years|year=2007|publisher=Prometheus Books|isbn=978-1-59102-477-4|page=[https://archive.org/details/howmathematicsha0000rudm/page/64 64]}}</ref>
* c. 3400 BC — the [[Sumer]]ians invent the first so-known [[numeral system]],{{Dubious|date=July 2009}} and a system of [[Ancient Mesopotamian units of measurement|weights and measures]].
* c. 3100 BC — [[Egypt]], earliest known [[decimal|decimal system]] allows indefinite counting by way of introducing new symbols, [http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptpapyrus.html#berlin].{{Citation needed|date=July 2009}}
* c. 2800 BC — [[Indus Valley civilization]] on the [[Indian subcontinent]], earliest use of [[decimal|decimal ratios]] in a uniform system of [[Ancient Indus Valley units of measurement|ancient weights and measures]], the smallest unit of measurement used is 1.704 millimetres and the smallest unit of mass used is 28&nbsp;grams.{{Citation needed|date=July 2009}}
* c. 2000 BC — [[Mesopotamia]], the [[Babylonians]] use a base-60 decimal system, and compute the first known approximate value of [[pi|π]] at 3.125.{{Citation needed|date=July 2009}}

==1st millennium BC==
* c. 1000 BC — [[Vulgar fraction]]s used by the [[Egyptians]].
* second half of 1st millennium BC — The [[Lo Shu Square]], the unique normal [[magic square]] of order three, was discovered in [[China]].
* c. 400 BC — [[Jain]]a mathematicians in [[History of India|India]] write the “Surya Prajinapti”, a mathematical text which classifies all numbers into three sets: enumerable, innumerable and [[Infinity|infinite]]. It also recognises five different types of [[infinity]]: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually.
* c. 300 BC — [[Brahmi numeral]]s are conceived in [[History of India|India]].
* 300 BC — [[Mesopotamia]], the [[Babylonians]] invent the earliest calculator, the [[abacus]].<ref>{{cite web |last1=Fernandes |first1=Luis |title=The Abacus: A Brief History |url=https://www.ee.torontomu.ca/~elf/abacus/history.html |publisher=Toronto Metropolitan University |access-date=5 February 2025 |archive-url=https://archive.today/20250205201918/https://www.ee.torontomu.ca/~elf/abacus/history.html |archive-date=5 February 2025 |quote=The oldest surviving counting board is the Salamis tablet (originally thought to be a gaming board), used by the Babylonians circa 300 B.C., discovered in 1846 on the island of Salamis. |url-status=live}}</ref>
* c. 300 BC — [[Indian mathematicians|Indian mathematician]] [[Pingala]] writes the “Chhandah-shastra”, which contains the first Indian use of [[0 (number)|zero]] as a digit (indicated by a dot) and also presents a description of a [[binary numeral system]], along with the first use of [[Fibonacci numbers]] and [[Pascal's triangle]].
* c. 250 BC — late [[Olmec]]s had already begun to use a true [[0 (number)|zero]] (a shell glyph) several centuries before [[Ptolemy]] in the New World. See [[0 (number)]].
* 150 BC — [[Jainism|Jain]] mathematicians in [[History of India|India]] write the “Sthananga Sutra”, which contains work on the theory of numbers, arithmetical operations, [[geometry]], operations with [[fractions]], simple equations, [[cubic equations]], quartic equations, and [[permutations]] and [[combinations]].
* 50 BC — [[Indian numerals]], the first [[positional notation]] [[base-10]] [[numeral system]], begins developing in [[History of India|India]].

==1st millennium AD==
* 300 — the earliest known use of [[0 (number)|zero]] as a decimal digit in the Old World is introduced by [[Indian mathematicians]].
* c. 400 — the [[Bakhshali manuscript]] uses numerals with a place-value system, using a dot as a place holder for zero .
* 550 — [[Hindu]] mathematicians give [[0 (number)|zero]] a numeral representation in the [[positional notation]] [[Indian numerals|Indian numeral]] system.
* 628 — [[Brahmagupta]] writes the ''[[Brahmasphutasiddhanta|Brahma-sphuta-siddhanta]]'', where zero is clearly explained, and where the modern [[place-value]] [[Indian numerals|Indian numeral]] system is fully developed. It also gives rules for manipulating both [[negative and positive numbers]], methods for computing [[square roots]], methods of solving [[linear equation|linear]] and [[quadratic equation]]s, and rules for summing [[series (mathematics)|series]], [[Brahmagupta's identity]], and the [[Brahmagupta theorem]].
* 940 — [[Abu'l-Wafa al-Buzjani]] extracts [[root of a function|roots]] using the Indian numeral system.
* 953 — The [[arithmetic]] of the [[Hindu–Arabic numeral system]] at first required the use of a dust board (a sort of handheld [[blackboard]]) because “the methods required moving the numbers around in the calculation and rubbing some out as the calculation proceeded.” [[Al-Uqlidisi]] modified these methods for [[pen]] and [[paper]] use. Eventually the advances enabled by the [[Decimal|decimal system]] led to its standard use throughout the region and the world.

==1000–1500==
*c. 1000 — [[Pope Sylvester II]] introduces the [[abacus]] using the [[Hindu–Arabic numeral system]] to Europe.
* 1030 — [[Ali Ahmad Nasawi]] writes a treatise on the [[decimal]] and [[sexagesimal]] number systems. His arithmetic explains the division of fractions and the extraction of square and cubic roots (square root of 57,342; cubic root of 3, 652, 296) in an almost modern manner.<ref>{{MacTutor|id=Al-Nasawi|title=Abu l'Hasan Ali ibn Ahmad Al-Nasawi}}</ref>
* 12th century — [[Indian numerals]] have been modified by [[Persian people|Persian]] mathematicians [[al-Khwārizmī]] to form the modern [[Arabic numerals]] (used universally in the modern world.)
* 12th century — the [[Arabic numerals]] reach [[Europe]] through the [[Arabs]].
* 1202 — [[Leonardo of Pisa|Leonardo Fibonacci]] demonstrates the utility of [[Hindu–Arabic numeral system]] in his ''Book of the Abacus''.
* c. 1400 — [[Ghiyath al-Kashi]] “contributed to the development of [[decimal fraction]]s not only for approximating [[algebraic number]]s, but also for [[real number]]s such as [[pi]]. His contribution to decimal fractions is so major that for many years he was considered as their inventor.  Although not the first to do so, al-Kashi gave an algorithm for calculating [[nth root]]s which is a special case of the methods given many centuries later by [[Paolo Ruffini (mathematician)|Ruffini]] and [[William George Horner|Horner]].” He is also the first to use the [[decimal point]] notation in [[arithmetic]] and [[Arabic numerals]]. His works include ''The Key of arithmetics, Discoveries in mathematics, The Decimal point'', and ''The benefits of the zero''. The contents of the ''Benefits of the Zero'' are an introduction followed by five essays: “On whole number arithmetic”, “On fractional arithmetic”, “On astrology”, “On areas”, and “On finding the unknowns [unknown variables]”. He also wrote the ''Thesis on the sine and the chord'' and ''Thesis on finding the first degree sine''.
* 15th century — [[Ibn al-Banna]] and [[al-Qalasadi]] introduced [[Mathematical notation|symbolic notation]] for algebra and for mathematics in general.<ref name=MacTutor>{{MacTutor|class=HistTopics|id=Arabic_mathematics|title=Arabic mathematics: forgotten brilliance?|year=1999}}</ref>
* 1427 — [[Al-Kashi]] completes ''The Key to Arithmetic'' containing work of great depth on [[decimal fraction]]s. It applies arithmetical and algebraic methods to the solution of various problems, including several geometric ones.
* 1478 — An anonymous author writes the [[Treviso Arithmetic]].

==17th century==
* 1614 — [[John Napier]] publishes a table of Napierian [[logarithm]]s in ''[[Mirifici Logarithmorum Canonis Descriptio]]'',
* 1617 — [[Henry Briggs (mathematician)|Henry Briggs]] discusses decimal logarithms in ''Logarithmorum Chilias Prima'',
* 1618 — [[John Napier]] publishes the first references to [[E (mathematical constant)|''e'']] in a work on [[logarithms]].

==18th century==
* 1758 — ''Arithmetika Horvatzka'', [[Croatia]]'s first arithmetic textbook is published in [[Zagreb]] by [[Mihalj Šilobod Bolšić]] (1724–1787).
* 1794 — [[Jurij Vega]] publishes ''[[Thesaurus Logarithmorum Completus]]''.

==Calculation of Pi==
* 1706 — [[John Machin]] develops a quickly converging inverse-tangent series for π and computes π to 100 decimal places.
* 1789 — [[Jurij Vega]] improves Machin's formula and computes π to 140 decimal places.
* 1949 — John von Neumann computes π to 2,037 decimal places using [[ENIAC]].
* 1961 — [[Daniel Shanks]] and [[John Wrench]] compute π to 100,000 decimal places using an inverse-tangent identity and an IBM-7090 computer.
* 1987 — [[Yasumasa Kanada]], [[David H. Bailey (mathematician)|David Bailey]], [[Jonathan Borwein]], and [[Peter Borwein]] use iterative modular equation approximations to elliptic integrals and a [[NEC SX-2]] [[supercomputer]] to compute π to 134 million decimal places.
* 2002 — [[Yasumasa Kanada]], Y. Ushiro, [[Hisayasu Kuroda]], [[Makoto Kudoh]] and a team of nine more compute π to 1241.1 billion digits using a [[Hitachi, Ltd.|Hitachi]] 64-node [[supercomputer]].

==See also==

* {{annotated link|Abacus}}
* {{annotated link|Alphabetic numeral system}}
* {{annotated link|Attic numerals}}
* {{annotated link|Australian Aboriginal enumeration}}
* {{annotated link|Counting rods}}
* {{annotated link|History of ancient numeral systems}}
* {{annotated link|History of arithmetic}}
* {{annotated link|History of mathematics}}
* {{annotated link|History of numbers}}
* {{annotated link|History of the Hindu–Arabic numeral system}}
* {{annotated link|Jeton}}
* {{annotated link|List of numeral system topics}}
* {{annotated link|List of numeral systems}}
* {{annotated link|Number theory}}
* {{annotated link|Prehistoric counting}}
* {{annotated link|Relationship between mathematics and physics}}
* {{annotated link|Roman numerals}}
* {{annotated link|Timeline of algorithms}}
* {{annotated link|Timeline of mathematics}}

==References==

{{reflist}}
{{History of mathematics}}
[[Category:Arithmetic]]
[[Category:History of mathematics]]
[[Category:Mathematics timelines|Numerals and arithmetic]]
[[Category:Numerals]]