Editing Arithmetic (section)

== Definition, etymology, and related fields ==
Arithmetic is the fundamental branch of [[mathematics]] that studies numbers and their operations. In particular, it deals with numerical calculations using the arithmetic operations of [[addition]], [[subtraction]], [[multiplication]], and [[Division (mathematics)|division]].<ref>{{multiref | {{harvnb|Romanowski|2008|pp=302–303}} | {{harvnb|HC staff|2022b}} | {{harvnb|MW staff|2023}} | {{harvnb|Bukhshtab|Pechaev|2020}} }}</ref> In a wider sense, it also includes [[exponentiation]], extraction of [[nth root|roots]], and [[logarithm]].<ref>{{multiref | {{harvnb|Bukhshtab|Pechaev|2020}} | {{harvnb|Burgin|2022|pp=57, 77}} | {{harvnb|Adamowicz|1994|p=299}} }}</ref> The term ''arithmetic'' has its root in the Latin term {{wikt-lang|la|arithmetica}} which derives from the Ancient Greek words {{wikt-lang|grc|ἀριθμός}} (''arithmos''), meaning {{gloss|number}}, and {{wikt-lang|grc|ἀριθμητική}} {{wikt-lang|grc|τέχνη}} (''arithmetike tekhne''), meaning {{gloss|the art of counting}}.<ref>{{multiref | {{harvnb|Peirce|2015|p=[https://books.google.com/books?id=oI_yCQAAQBAJ&pg=PA109 109]}} | {{harvnb|Waite|2013|p=[https://books.google.com/books?id=xqKcAQAAQBAJ&pg=PA42 42]}} | {{harvnb|Smith|1958|p=[https://books.google.com/books?id=uTytJGnTf1kC&pg=PA7 7]}} }}</ref>

There are disagreements about its precise definition. According to a narrow characterization, arithmetic deals only with [[natural numbers]].<ref>{{multiref | {{harvnb|Oliver|2005|p=58}} | {{harvnb|Hofweber|2016|p=153}} }}</ref> However, the more common view is to include operations on [[integers]], [[rational numbers]], [[real numbers]], and sometimes also [[complex numbers]] in its scope.<ref>{{multiref | {{harvnb|Romanowski|2008|pp=302–303}} | {{harvnb|HC staff|2022b}} | {{harvnb|MW staff|2023}} | {{harvnb|Bukhshtab|Pechaev|2020}} }}</ref> Some definitions restrict arithmetic to the field of numerical calculations.<ref>{{harvnb|Sophian|2017|p=[https://books.google.com/books?id=8FYPEAAAQBAJ&pg=PA84 84]}}</ref> When understood in a wider sense, it also includes the study of how the concept of [[numbers]] developed, the analysis of properties of and relations between numbers, and the examination of the axiomatic structure of arithmetic operations.<ref>{{multiref | {{harvnb|Bukhshtab|Pechaev|2020}} | {{harvnb|Stevenson|Waite|2011|p=[https://books.google.com/books?id=sYScAQAAQBAJ&pg=PA70 70]}} | {{harvnb|Romanowski|2008|pp=303–304}} }}</ref>

Arithmetic is closely related to [[number theory]] and some authors use the terms as synonyms.<ref>{{multiref | {{harvnb|Lozano-Robledo|2019|p=[https://books.google.com/books?id=ESiODwAAQBAJ&pg=PR13 xiii]}} | {{harvnb|Nagel|Newman|2008|p=[https://books.google.com/books?id=WgwUCgAAQBAJ&pg=PA4 4]}} }}</ref> However, in a more specific sense, number theory is restricted to the study of integers and focuses on their properties and relationships such as [[divisibility]], [[factorization]], and [[primality]].<ref>{{multiref | {{harvnb|Wilson|2020|pp=[https://books.google.com/books?id=fcDgDwAAQBAJ&pg=PA1 1–2]}} | {{harvnb|Karatsuba|2020}} | {{harvnb|Campbell|2012|p=[https://books.google.com/books?id=yoEFp-Q2OXIC&pg=PT33 33]}} | {{harvnb|Robbins|2006|p=[https://books.google.com/books?id=TtLMrKDsDuIC&pg=PR12-IA1 1]}} }}</ref> Traditionally, it is known as higher arithmetic.<ref>{{multiref | {{harvnb|Duverney|2010|p=[https://books.google.com/books?id=sr5S9oN1xPAC&pg=PR5 v]}} | {{harvnb|Robbins|2006|p=[https://books.google.com/books?id=TtLMrKDsDuIC&pg=PR12-IA1 1]}} }}</ref>