# Timeline of numerals and arithmetic

Timeline of numerals and arithmetic

A timeline of numerals and arithmetic

Before 2000 BC

* ca. 20,000 BCNile Valley, Ishango Bone: possibly the earliest reference to prime numbers and Egyptian multiplication.
* ca. 3400 BCMesopotamia, the Sumerians invent the first numeral system, and a system of weights and measures.
* ca. 3100 BCEgypt, earliest known decimal system allows indefinite counting by way of introducing new symbols, [http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptpapyrus.html#berlin] .
* ca. 2800 BCIndus Valley Civilization on the Indian subcontinent, earliest use of decimal ratios in a uniform system of ancient weights and measures, the smallest unit of measurement used is 1.704 millimetres and the smallest unit of mass used is 28 grams.
* ca. 2000 BCMesopotamia, the Babylonians use a base-60 decimal system, and compute the first known approximate value of π at 3.125.

1st millennium BC

* ca 1000 BCVulgar fractions used by the Egyptians.
* ca 8th century BC — the Yajur Veda, one of the four Hindu Vedas, contains the earliest concept of infinity, and states that “if you remove a part from infinity or add a part to infinity, still what remains is infinity.”
* second half of 1st millennium BC — The Lo Shu Square, the unique normal magic square of order three, was discovered in China.
* ca. 400 BCJaina mathematicians in India write the “Surya Prajinapti”, a mathematical text which classifies all numbers into three sets: enumerable, innumerable and infinite. It also recognises five different types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually.
* 300s BCIndian texts use the Sanskrit word “Shunya” to refer to the concept of ‘void’ (zero.)
* ca. 300 BCBrahmi numerals are conceived in India.
* 300 BCMesopotamia, the Babylonians invent the earliest calculator, the abacus.
* ca. 300 BC — Indian mathematician Pingala writes the “Chhandah-shastra”, which contains the first Indian use of 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.
* ca. 250 BC — late Olmecs had already begun to use a true zero (a shell glyph) several centuries before Ptolemy in the New World. See 0 (number).
* 150 BCJain mathematicians in 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 BCIndian numerals, the first positional notation base-10 numeral system, begins developing in India.

1st millennium

* 300 — the earliest known use of zero as a decimal digit is introduced by Indian mathematicians.
* ca. 400 — the “Bakhshali manuscript” is written by Jaina mathematicians, which describes a theory of the infinite containing different levels of infinity, shows an understanding of indices, as well as logarithms to base 2, and computes square roots of numbers as large as a million correct to at least 11 decimal places.
* 550Hindu mathematicians give zero a numeral representation in the positional notation Indian numeral system.
* 628Brahmagupta writes the "Brahma-sphuta-siddhanta", where zero is clearly explained, and where the modern place-value 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 and quadratic equations, and rules for summing series, Brahmagupta's identity, and the Brahmagupta theorem.
* 940Abu'l-Wafa al-Buzjani extracts 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 system led to its standard use throughout the region and the world.

1000&ndash;1500

*ca. 1000Pope Sylvester II introduces the abacus using the Hindu-Arabic numeral system to Europe.
* 1030Ali 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. [MacTutor|id=Al-Nasawi|title=Abu l'Hasan Ali ibn Ahmad Al-Nasawi]
* 1100sIndian numerals have been modified by Arab mathematicians to form the modern Hindu-Arabic numeral system (used universally in the modern world.)
* 1100s — the Hindu-Arabic numeral system reaches Europe through the Arabs.
* 1202 — Leonardo Fibonacci demonstrates the utility of Hindu-Arabic numerals in his "Book of the Abacus".
* ca. 1400Ghiyath al-Kashi “contributed to the development of decimal fractions not only for approximating algebraic numbers, but also for real numbers 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 roots which is a special case of the methods given many centuries later by Ruffini and 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".
* 1400sIbn al-Banna and al-Qalasadi introduced symbolic notation for algebra and for mathematics in general.
* 1427Al-Kashi completes "The Key to Arithmetic" containing work of great depth on decimal fractions. 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 discusses Napierian logarithms in "Mirifici Logarithmorum Canonis Descriptio",
* 1617 - Henry Briggs discusses decimal logarithms in "Logarithmorum Chilias Prima",
* 1618 - John Napier publishes the first references to "e" in a work on logarithms.

18th century

* 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 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 64-node supercomputer.

* Timeline of algorithms

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