Editing Arithmetic (section)

== History ==
[[File:Ishango bone (cropped).jpg|thumb|alt=Photo of the Ishango bone|upright=0.6|Some historians interpret the [[Ishango bone]] as one of the earliest arithmetic artifacts.]]

The earliest forms of arithmetic are sometimes traced back to [[counting]] and [[tally marks]] used to keep track of quantities. Some historians suggest that the [[Lebombo bone]] (dated about 43,000 years ago) and the [[Ishango bone]] (dated about 22,000 to 30,000 years ago) are the oldest arithmetic artifacts but this interpretation is disputed.<ref>{{multiref | {{harvnb|Burgin|2022|pp=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA2 2–3]}} | {{harvnb|Ore|1948|pp=1, 6, 8, 10}} | {{harvnb|Thiam|Rochon|2019|p=[https://books.google.com/books?id=EWSsDwAAQBAJ&pg=PA164 164]}} }}</ref> However, a basic [[number sense|sense of numbers]] may predate these findings and might even have existed before the development of language.<ref>{{multiref | {{harvnb|Burgin|2022|p=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA3 3]}} | {{harvnb|Ponticorvo|Schmbri|Miglino|2019|p=[https://books.google.com/books?id=zSiXDwAAQBAJ&pg=PA33 33]}} }}</ref>

It was not until the emergence of [[ancient civilizations]] that a more complex and structured approach to arithmetic began to evolve, starting around 3000 BCE. This became necessary because of the increased need to keep track of stored items, manage land ownership, and arrange exchanges.<ref>{{multiref | {{harvnb|Burgin|2022|pp=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA4 4–6]}} | {{harvnb|Ang|Lam|2004|p=[https://books.google.com/books?id=GxDJCgAAQBAJ&pg=PA170 170]}} }}</ref> All the major ancient civilizations developed non-positional numeral systems to facilitate the representation of numbers. They also had symbols for operations like addition and subtraction and were aware of fractions. Examples are [[Egyptian hieroglyphics]] as well as the numeral systems invented in [[Sumeria]], [[Ancient China|China]], and [[Ancient India|India]].<ref>{{multiref | {{harvnb|Burgin|2022|pp=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA5 5–7, 9–11]}} | {{harvnb|Ore|1948|pp=10–15}} | {{harvnb|Nagel|2002|p=178}} | {{harvnb|Hindry|2011|p=ix}} }}</ref> The first [[positional numeral system]] was developed by the [[Babylonians]] starting around 1800 BCE. This was a significant improvement over earlier numeral systems since it made the representation of large numbers and calculations on them more efficient.<ref>{{multiref | {{harvnb|Burgin|2022|pp=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA6 6–7, 9]}} | {{harvnb|Ore|1948|pp=16–18}} | {{harvnb|ITL Education Solutions Limited|2011|p=[https://books.google.com/books?id=CsNiKdmufvYC&pg=PA28 28]}} }}</ref> [[Abacus]]es have been utilized as hand-operated calculating tools since ancient times as efficient means for performing complex calculations.<ref>{{multiref | {{harvnb|Ore|1948|p=15}} | {{harvnb|Yadin|2016|p=[https://books.google.com/books?id=KzeLDQAAQBAJ&pg=PT24 24]}} }}</ref>

Early civilizations primarily used numbers for concrete practical purposes, like commercial activities and tax records, but lacked an abstract concept of number itself.<ref>{{multiref | {{harvnb|Burgin|2022|pp=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA3 4–5]}} | {{harvnb|Brown|2010|p=[https://books.google.com/books?id=TzrNgAsJY1MC&pg=PA184 184]}} }}</ref> This changed with the [[ancient Greek mathematicians]], who began to explore the abstract nature of numbers rather than studying how they are applied to specific problems.<ref>{{multiref | {{harvnb|Burgin|2022|p=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA15 15]}} | {{harvnb|Brown|2010|p=[https://books.google.com/books?id=TzrNgAsJY1MC&pg=PA184 184]}} | {{harvnb|Romanowski|2008|p=303}} | {{harvnb|Nagel|2002|p=178}} }}</ref> Another novel feature was their use of [[Mathematical proof|proofs]] to establish mathematical truths and validate theories.<ref>{{multiref | {{harvnb|Burgin|2022|p=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA15 15]}} | {{harvnb|Madden|Aubrey|2017|p=[https://books.google.com/books?id=6EkzDwAAQBAJ&pg=PP18 xvii]}} }}</ref> A further contribution was their distinction of various classes of numbers, such as [[even numbers]], odd numbers, and [[prime numbers]].<ref>{{multiref | {{harvnb|Burgin|2022|p=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA31 31]}} | {{harvnb|Payne|2017|p=[https://books.google.com/books?id=qMU2DwAAQBAJ&pg=PA202 202]}} }}</ref> This included the discovery that numbers for certain geometrical lengths are [[Irrational number|irrational]] and therefore cannot be expressed as a fraction.<ref>{{multiref | {{harvnb|Burgin|2022|pp=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA20 20–21]}} | {{harvnb|Bloch|2011|p=[https://books.google.com/books?id=vXw_AAAAQBAJ&pg=PA52 52]}} }}</ref> The works of [[Thales of Miletus]] and [[Pythagoras]] in the 7th and 6th centuries BCE are often regarded as the inception of Greek mathematics.<ref>{{multiref | {{harvnb|Burgin|2022|p=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA16 16]}} | {{harvnb|Lützen|2023|p=[https://books.google.com/books?id=joikEAAAQBAJ&pg=PA19 19]}} }}</ref> [[Diophantus]] was an influential figure in Greek arithmetic in the 3rd century BCE because of his numerous contributions to [[number theory]] and his exploration of the application of arithmetic operations to [[algebraic equations]].<ref>{{multiref | {{harvnb|Burgin|2022|pp=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA29 29–31]}} | {{harvnb|Klein|2013a|p=[https://books.google.com/books?id=EkPDAgAAQBAJ&pg=PT12 12]}} }}</ref>

The ancient Indians were the first to develop the concept of [[zero]] as a number to be used in calculations. The exact rules of its operation were written down by [[Brahmagupta]] in around 628 CE.<ref>{{multiref | {{harvnb|Burgin|2022|pp=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA36 36–37]}} | {{harvnb|Bradley|2006|pp=[https://books.google.com/books?id=EIdtVPeD7GcC&pg=PA82 82–83]}} | {{harvnb|Conradie|Goranko|2015|p=[https://books.google.com/books?id=D3jCCAAAQBAJ&pg=PA268 268]}} }}</ref> The concept of zero or none existed long before, but it was not considered an object of arithmetic operations.<ref>{{multiref | {{harvnb|Burgin|2022|pp=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA35 35–36]}} | {{harvnb|Cai|2023|p=[https://books.google.com/books?id=DFLNEAAAQBAJ&pg=PA110 110]}} }}</ref> Brahmagupta further provided a detailed discussion of calculations with [[negative numbers]] and their application to problems like credit and debt.<ref>{{multiref | {{harvnb|Burgin|2022|pp=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA37 37, 40]}} | {{harvnb|Bradley|2006|pp=[https://books.google.com/books?id=EIdtVPeD7GcC&pg=PA82 82–83]}} | {{harvnb|Conradie|Goranko|2015|p=[https://books.google.com/books?id=D3jCCAAAQBAJ&pg=PA268 268]}} }}</ref> The concept of negative numbers itself is significantly older and was [[The Nine Chapters on the Mathematical Art|first explored]] in [[Chinese mathematics]] in the first millennium BCE.<ref>{{multiref | {{harvnb|Hua|Feng|2020|pp=[https://books.google.com/books?id=6_sOEAAAQBAJ&pg=PA119 119–120]}} | {{harvnb|Chemla|Keller|Proust|2023|p=[https://books.google.com/books?id=afKkEAAAQBAJ&pg=PA47 47]}} }}</ref>

Indian mathematicians also developed the [[Hindu–Arabic numeral system|positional decimal]] system used today, in particular the concept of a zero digit instead of empty or missing positions.<ref>{{multiref | {{harvnb|Burgin|2022|pp=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA13 13, 34–35]}} | {{harvnb|Conradie|Goranko|2015|p=[https://books.google.com/books?id=D3jCCAAAQBAJ&pg=PA268 268]}} }}</ref> For example, [[Aryabhatiya|a detailed treatment]] of its operations was provided by [[Aryabhata]] around the turn of the 6th century CE.<ref>{{multiref | {{harvnb|Burgin|2022|pp=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA13 13, 34]}} | {{harvnb|Conradie|Goranko|2015|p=[https://books.google.com/books?id=D3jCCAAAQBAJ&pg=PA268 268]}} }}</ref> The Indian decimal system was further refined and expanded to non-integers during the [[Islamic Golden Age]] by Middle Eastern mathematicians such as [[Al-Khwarizmi]]. His work was influential in introducing the decimal numeral system to the Western world, which at that time relied on the [[Roman numeral system]].<ref>{{multiref | {{harvnb|Burgin|2022|pp=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA38 38, 43–46]}} | {{harvnb|Conradie|Goranko|2015|p=[https://books.google.com/books?id=D3jCCAAAQBAJ&pg=PA268 268]}} }}</ref> There, it was popularized by mathematicians like [[Leonardo Fibonacci]], who lived in the 12th and 13th centuries and also developed the [[Fibonacci sequence]].<ref>{{multiref | {{harvnb|Burgin|2022|p=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA56 56]}} | {{harvnb|Oakes|2020|p=[https://books.google.com/books?id=RtyPEAAAQBAJ&pg=PA330 330]}} }}</ref> During the [[Middle Ages]] and [[Renaissance]], many popular textbooks were published to cover the practical calculations for commerce. The use of abacuses also became widespread in this period.<ref>{{multiref | {{harvnb|Burgin|2022|p=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA55 55]}} | {{harvnb|Wedell|2015|pp=[https://books.google.com/books?id=Uh5pCgAAQBAJ&pg=PA1235 1235–1236]}} }}</ref> In the 16th century, the mathematician [[Gerolamo Cardano]] conceived the concept of [[complex numbers]] as a way to solve [[cubic equations]].<ref>{{multiref | {{harvnb|Burgin|2022|p=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA62 62]}} | {{harvnb|Lützen|2023|p=[https://books.google.com/books?id=joikEAAAQBAJ&pg=PA124 124]}} }}</ref>

[[File:Leibniz Stepped Reckoner.png|thumb|alt=Photo of Leibniz's stepped reckoner|Leibniz's [[stepped reckoner]] was the first calculator that could perform all four arithmetic operations.<ref>{{harvnb|Vullo|2020|p=[https://books.google.com/books?id=g67QDwAAQBAJ&pg=PA140 140]}}</ref>]]

The first [[mechanical calculator]]s were developed in the 17th century and greatly facilitated complex mathematical calculations, such as [[Blaise Pascal]]'s [[Pascal's calculator|calculator]] and [[Gottfried Wilhelm Leibniz]]'s [[stepped reckoner]].<ref>{{multiref | {{harvnb|Cignoni|Cossu|2016|p=[https://books.google.com/books?id=LridDQAAQBAJ&pg=PA103 103]}} | {{harvnb|Koetsier|2018|p=[https://books.google.com/books?id=9jF7DwAAQBAJ&pg=PA255 255]}} | {{harvnb|Igarashi|Altman|Funada|Kamiyama|2014|pp=[https://books.google.com/books?id=58ySAwAAQBAJ&pg=PA87 87–89]}} }}</ref> The 17th century also saw the discovery of the [[logarithm]] by [[John Napier]].<ref>{{multiref | {{harvnb|Burgin|2022|p=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA77 77]}} | {{harvnb|Eriksson|Estep|Johnson|2013|p=[https://books.google.com/books?id=FD8mBQAAQBAJ&pg=PA474 474]}} }}</ref>

In the 18th and 19th centuries, mathematicians such as [[Leonhard Euler]] and [[Carl Friedrich Gauss]] laid the foundations of modern number theory.<ref>{{multiref | {{harvnb|Burgin|2022|pp=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA68 68–72]}} | {{harvnb|Weil|2009|p=[https://books.google.com/books?id=Ar7gBwAAQBAJ&pg=PR9 ix]}} }}</ref> Another development in this period concerned work on the formalization and foundations of arithmetic, such as [[Georg Cantor]]'s [[set theory]] and the [[Dedekind–Peano axioms]] used as an axiomatization of natural-number arithmetic.<ref>{{multiref | {{harvnb|Burgin|2022|pp=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA2 2, 88, 95–97]}} | {{harvnb|Wang|1997|p=[https://books.google.com/books?id=pckvCy6L_ocC&pg=PA334 334]}} }}</ref> [[Computers]] and [[electronic calculators]] were first developed in the 20th century. Their widespread use revolutionized both the accuracy and speed with which even complex arithmetic computations can be calculated.<ref>{{multiref | {{harvnb|Burgin|2022|pp=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA119 119, 124]}} | {{harvnb|Curley|2011|pp=[https://books.google.com/books?id=EdGbAAAAQBAJ&pg=PA5 5, 19]}} | {{harvnb|Igarashi|Altman|Funada|Kamiyama|2014|p=[https://books.google.com/books?id=58ySAwAAQBAJ&pg=PA149 149]}} }}</ref>