User:Phlsph7/Arithmetic - Definition, etymology, and related fields
Definition, etymology, and related fields
[edit]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, division, exponentiation, and logarithm.[1][2][3][4] The term "arithmetic" has its root in the Latin term "arithmetica" which derives from the Ancient Greek words ἀριθμός (arithmos), meaning "number", and ἀριθμητική τέχνη (arithmetike tekhne), meaning "the art of counting".[5][6][7]
There are disagreements about its precise definition. According to a narrow characterization, arithmetic deals only with natural numbers.[8][9] However, the more common view is to include operations on integers, rational numbers, real numbers, and sometimes also complex numbers in its scope.[1][2][3][4] Some definitions restrict arithmetic to the field of numerical calculations.[10] 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.[4][11][12]
Arithmetic is closely related to number theory and some authors use the terms as synonyms.[13][14] 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.[15][16][17][18] Traditionally, it is known as higher arithmetic.[19][18] Arithmetic is intimately connected to many branches of mathematics that depend on numerical operations. Algebra relies on arithmetic principles to solve equations using variables. These principles also play a key role in calculus in its attempt to determine rates of change and areas under curves. Geometry uses arithmetic operations to measure the properties of shapes while statistics utilizes them to analyze numerical data.[20][21][22][23]
- Smith, David E. (1 June 1958). History of Mathematics. Courier Corporation. ISBN 978-0-486-20430-7.
- Peirce, Charles S. (28 April 2015). Arithmetic. Walter de Gruyter GmbH & Co KG. ISBN 978-3-11-086970-5.
- EoM staff (2020a). "Arithmetic". Encyclopedia of Mathematics. Springer. Retrieved 23 October 2023.
- EoM staff (2020b). "Number theory". Encyclopedia of Mathematics. Springer. Retrieved 23 October 2023.
- Wilson, Robin (2020). Number Theory: A Very Short Introduction. Oxford University Press. ISBN 978-0-19-879809-5.
- HS staff (2022). "The American Heritage Dictionary entry: arithmetic". www.ahdictionary.com. HarperCollins. Retrieved 19 October 2023.
- MW staff (13 September 2023). "Definition of Arithmetic". www.merriam-webster.com. Retrieved 19 October 2023.
- Kleiner, Israel (2 February 2012). Excursions in the History of Mathematics. Springer Science & Business Media. ISBN 978-0-8176-8268-2.
- Marcus, Russell; McEvoy, Mark (11 February 2016). An Historical Introduction to the Philosophy of Mathematics: A Reader. Bloomsbury Publishing. ISBN 978-1-4725-3291-6.
- Monahan, John F. (6 July 2012). "2. Basic Computational Algorithms". In Gentle, James E.; Härdle, Wolfgang Karl; Mori, Yuichi (eds.). Handbook of Computational Statistics: Concepts and Methods. Springer Science & Business Media. ISBN 978-3-642-21551-3.
- Oliver, Alexander D. (2005). "arithmetic, foundations of". In Honderich, Ted (ed.). The Oxford Companion to Philosophy. Oxford University Press. ISBN 9780199264797.
- Hofweber, Thomas (2016). "The Philosophy of Arithmetic". Ontology and the Ambitions of Metaphysics. Oxford University Press. ISBN 978-0-19-876983-5.
- Stevenson, Angus; Waite, Maurice (18 August 2011). Concise Oxford English Dictionary: Luxury Edition. OUP Oxford. ISBN 978-0-19-960111-0.
- Sophian, Catherine (25 September 2017). The Origins of Mathematical Knowledge in Childhood. Routledge. ISBN 978-1-351-54175-6.
- Duverney, Daniel (2010). Number Theory: An Elementary Introduction Through Diophantine Problems. World Scientific. ISBN 978-981-4307-46-8.
- Lozano-Robledo, Álvaro (21 March 2019). Number Theory and Geometry: An Introduction to Arithmetic Geometry. American Mathematical Soc. ISBN 978-1-4704-5016-8.
- Campbell, Stephen R. (12 October 2012). "Understanding Elementary Number Theory in Relation to Arithmetic and Algebra". In Zazkis, Rina; Campbell, Stephen R. (eds.). Number Theory in Mathematics Education: Perspectives and Prospects. Routledge. ISBN 978-1-136-50143-2.
- Robbins, Neville (2006). Beginning Number Theory. Jones & Bartlett Learning. ISBN 978-0-7637-3768-9.
- Nagel, Ernest; Newman, James Roy (1 January 2008). Godel's Proof. NYU Press. ISBN 978-0-8147-5837-3.
- Waite, Maurice (9 May 2013). Pocket Oxford English Dictionary. OUP Oxford. ISBN 978-0-19-966615-7.
- ^ a b Romanowski 2008, pp. 302–303.
- ^ a b HS staff 2022.
- ^ a b MW staff 2023.
- ^ a b c EoM staff 2020a.
- ^ Peirce 2015, p. 109.
- ^ Waite 2013, p. 42.
- ^ Smith 1958, p. 7.
- ^ Oliver 2005, p. 58.
- ^ Hofweber 2016, p. 153.
- ^ Sophian 2017, p. 84.
- ^ Stevenson & Waite 2011, p. 70.
- ^ Romanowski 2008, pp. 303–304.
- ^ Lozano-Robledo 2019, p. xiii.
- ^ Nagel & Newman 2008, p. 4.
- ^ Wilson 2020, pp. 1–2.
- ^ EoM staff 2020b.
- ^ Campbell 2012, p. 33.
- ^ a b Robbins 2006, p. 1.
- ^ Duverney 2010, p. v.
- ^ Musser, Peterson & Burger 2013, p. 17.
- ^ Kleiner 2012, p. 255.
- ^ Marcus & McEvoy 2016, p. 285.
- ^ Monahan 2012, 2. Basic Computational Algorithms.