Editing Arithmetic (section)

== In various fields ==
=== Education ===
{{main|Mathematics education}}
Arithmetic education forms part of [[primary education]]. It is one of the first forms of [[mathematics education]] that children encounter. [[Elementary arithmetic]] aims to give students a basic [[number sense|sense of numbers]] and to familiarize them with fundamental numerical operations like addition, subtraction, multiplication, and division.<ref>{{multiref | {{harvnb|NCTM Staff}} | {{harvnb|Musser|Peterson|Burger|2013|loc=Curriculum Focal Points for Prekindergarten Through Grade 8 Mathematics, p. 44, p. 130}} | {{harvnb|Odom|Barbarin|Wasik|2009|p=[https://books.google.com/books?id=MIuyhc8W-A8C&pg=PA589 589]}} }}</ref> It is usually introduced in relation to concrete scenarios, like counting [[bead]]s, dividing the class into groups of children of the same size, and calculating change when buying items. Common tools in early arithmetic education are [[number lines]], addition and multiplication tables, [[counting blocks]], and abacuses.<ref>{{multiref | {{harvnb|Laski|Jor’dan|Daoust|Murray|2015|pp=1–3}} | {{harvnb|Musser|Peterson|Burger|2013|pp=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA59 59, 90–91, 93–94, 106–108]}} | {{harvnb|Nurnberger-Haag|2017|p=[https://books.google.com/books?id=Uls6DwAAQBAJ&pg=PA215 215]}} }}</ref>

Later stages focus on a more abstract understanding and introduce the students to different types of numbers, such as negative numbers, fractions, real numbers, and complex numbers. They further cover more advanced numerical operations, like exponentiation, extraction of roots, and logarithm.<ref>{{multiref | {{harvnb|NCTM Staff}} | {{harvnb|Musser|Peterson|Burger|2013|loc=Curriculum Focal Points for Prekindergarten Through Grade 8 Mathematics, pp. [https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA208 208, 304, 340, 362]}} }}</ref> They also show how arithmetic operations are employed in other branches of mathematics, such as their application to describe geometrical shapes and the use of variables in algebra. Another aspect is to teach the students the use of [[algorithm]]s and [[calculator]]s to solve complex arithmetic problems.<ref>{{multiref | {{harvnb|NCTM Staff}} | {{harvnb|Musser|Peterson|Burger|2013|loc=Curriculum Focal Points for Prekindergarten Through Grade 8 Mathematics}} | {{harvnb|Carraher|Schliemann|2015|p=[https://books.google.com/books?id=lpFGCgAAQBAJ&pg=PA197 197]}} | {{harvnb|Ruthven|2012|pp=[https://books.google.com/books?id=RjZzAgAAQBAJ&pg=PA435 435, 443–444]}} }}</ref>

=== Psychology ===
The [[psychology]] of arithmetic is interested in how humans and animals learn about numbers, represent them, and use them for calculations. It examines how mathematical problems are understood and solved and how arithmetic abilities are related to [[perception]], [[memory]], [[judgment]], and [[decision making]].<ref>{{multiref | {{harvnb|De Cruz|Neth|Schlimm|2010|pp=[https://kops.uni-konstanz.de/server/api/core/bitstreams/08a99b71-10aa-4c26-8cd0-9fff4e1e5427/content 59–60]}} | {{harvnb|Grice|Kemp|Morton|Grace|2023|loc=Abstract}} }}</ref> For example, it investigates how collections of concrete items are first encountered in perception and subsequently associated with numbers.<ref>{{harvnb|De Cruz|Neth|Schlimm|2010|pp=[https://kops.uni-konstanz.de/server/api/core/bitstreams/08a99b71-10aa-4c26-8cd0-9fff4e1e5427/content 60–62]}}</ref> A further field of inquiry concerns the relation between numerical calculations and the use of language to form representations.<ref>{{harvnb|De Cruz|Neth|Schlimm|2010|p=[https://kops.uni-konstanz.de/server/api/core/bitstreams/08a99b71-10aa-4c26-8cd0-9fff4e1e5427/content 63]}}</ref> Psychology also explores the biological origin of arithmetic as an inborn ability. This concerns pre-verbal and pre-symbolic cognitive processes implementing arithmetic-like operations required to successfully represent the world and perform tasks like spatial navigation.<ref>{{harvnb|Grice|Kemp|Morton|Grace|2023|loc=Abstract}}</ref>

One of the concepts studied by psychology is [[numeracy]], which is the capability to comprehend numerical concepts, apply them to concrete situations, and [[Logical reasoning|reason]] with them. It includes a fundamental number sense as well as being able to estimate and compare quantities. It further encompasses the abilities to symbolically represent numbers in numbering systems, interpret [[numerical data]], and evaluate arithmetic calculations.<ref>{{multiref | {{harvnb|Victoria Department of Education Staff|2023}} | {{harvnb|Askew|2010|pp=[https://books.google.com/books?id=yqbSaXf0RKwC&pg=PA33 33–34]}} | {{harvnb|Dreeben-Irimia|2010|p=[https://books.google.com/books?id=L3gqsXsoiiAC&pg=PA102 102]}} }}</ref> Numeracy is a key skill in many academic fields. A lack of numeracy can inhibit academic success and lead to bad economic decisions in everyday life, for example, by misunderstanding [[mortgage]] plans and [[insurance policies]].<ref>{{multiref | {{harvnb|Victoria Department of Education Staff|2023}} | {{harvnb|Barnes|Rice|Hanoch|2017|p=[https://books.google.com/books?id=yzMlDwAAQBAJ&pg=PA196 196]}} | {{harvnb|Gerardi|Goette|Meier|2013|pp=11267–11268}} | {{harvnb|Jackson|2008|p=[https://books.google.com/books?id=tLuPAgAAQBAJ&pg=PA152 152]}} }}</ref>

=== Philosophy ===
{{main|Philosophy of mathematics}}
The philosophy of arithmetic studies the fundamental concepts and principles underlying numbers and arithmetic operations. It explores the nature and [[Ontology|ontological status]] of numbers, the relation of arithmetic to language and [[logic]], and how it is possible to acquire arithmetic [[knowledge]].<ref>{{multiref | {{harvnb|Hofweber|2016|pp=153–154, 162–163}} | {{harvnb|Oliver|2005|p=58}} | {{harvnb|Sierpinska|Lerman|1996|p=827}} }}</ref>

According to [[Platonism]], numbers have mind-independent existence: they exist as [[abstract objects]] outside spacetime and without causal powers.<ref>{{multiref | {{harvnb|Oliver|2005|p=58}} | {{harvnb|Horsten|2023|loc=§ 3. Platonism}} }}</ref>{{efn|An [[Quine–Putnam indispensability argument|influential argument]] for Platonism, first formulated by [[Willard Van Orman Quine]] and [[Hilary Putnam]], states that numbers exist because they are indispensable to the best scientific theories.{{sfn|Colyvan|2023|loc=Lead Section}}}} This view is rejected by [[Intuitionism|intuitionists]], who claim that mathematical objects are mental constructions.<ref>{{harvnb|Horsten|2023|loc=§ 2.2 Intuitionism}}</ref> Further theories are [[logicism]], which holds that mathematical truths are reducible to [[logical truth]]s,<ref>{{multiref | {{harvnb|Horsten|2023|loc=§ 2.1 Logicism}} | {{harvnb|Hofweber|2016|pp=174–175}} }}</ref> and [[Formalism (philosophy of mathematics)|formalism]], which states that mathematical principles are rules of how symbols are manipulated without claiming that they correspond to entities outside the rule-governed activity.<ref>{{harvnb|Weir|2022|loc=Lead Section}}</ref>

The traditionally dominant view in the [[epistemology]] of arithmetic is that arithmetic truths are knowable [[a priori]]. This means that they can be known by thinking alone without the need to rely on [[sensory experience]].<ref>{{multiref | {{harvnb|Oliver|2005|p=58}} | {{harvnb|Sierpinska|Lerman|1996|p=830}} }}</ref> Some proponents of this view state that arithmetic knowledge is innate while others claim that there is some form of [[rational intuition]] through which mathematical truths can be apprehended.<ref>{{multiref | {{harvnb|Oliver|2005|p=58}} | {{harvnb|Sierpinska|Lerman|1996|pp=827–876}} }}</ref> A more recent alternative view was suggested by [[Naturalism (philosophy)|naturalist]] philosophers like [[Willard Van Orman Quine]], who argue that mathematical principles are high-level generalizations that are ultimately grounded in the sensory world as described by the empirical sciences.<ref>{{multiref | {{harvnb|Horsten|2023|loc=§ 3.2 Naturalism and Indispensability}} | {{harvnb|Sierpinska|Lerman|1996|p=830}} }}</ref>

=== Others ===
Arithmetic is relevant to many fields. In [[daily life]], it is required to calculate change when shopping, manage [[personal finances]], and adjust a cooking recipe for a different number of servings. Businesses use arithmetic to calculate profits and losses and [[Marketing research|analyze market trends]]. In the field of [[engineering]], it is used to measure quantities, calculate loads and forces, and design structures.<ref>{{multiref | {{harvnb|Lockhart|2017|pp=1–2}} | {{harvnb|Bird|2021|p=3}} | {{harvnb|Aubrey|1999|p=[https://books.google.com/books?id=_NhpY_VPdCsC&pg=PA49 49]}} }}</ref> [[Cryptography]] relies on arithmetic operations to protect sensitive information by [[encrypt]]ing data and messages.<ref>{{multiref | {{harvnb|Omondi|2020|p=[https://books.google.com/books?id=_m7NDwAAQBAJ&pg=PR8 viii]}} | {{harvnb|Paar|Pelzl|2009|p=[https://books.google.com/books?id=f24wFELSzkoC&pg=PA13 13]}} }}</ref>

Arithmetic is intimately connected to many branches of mathematics that depend on numerical operations. [[Algebra]] relies on arithmetic principles to solve [[equation]]s using variables. These principles also play a key role in [[calculus]] in its attempt to determine [[rates of change]] and areas under [[curve]]s. [[Geometry]] uses arithmetic operations to measure the properties of shapes while [[statistics]] utilizes them to analyze numerical data.<ref>{{multiref | {{harvnb|Musser|Peterson|Burger|2013|p=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA17 17]}} | {{harvnb|Kleiner|2012|p=[https://books.google.com/books?id=7NcEIteAfyYC&pg=PA255 255]}} | {{harvnb|Marcus|McEvoy|2016|p=[https://books.google.com/books?id=x29VCwAAQBAJ&pg=PA285 285]}} | {{harvnb|Monahan|2012}} }}</ref> Due to the relevance of arithmetic operations throughout mathematics, the influence of arithmetic extends to most sciences such as [[physics]], [[computer science]], and [[economics]]. These operations are used in calculations, [[problem-solving]], [[data analysis]], and algorithms, making them integral to scientific research, technological development, and economic modeling.<ref>{{multiref | {{harvnb|Gallistel|Gelman|2005|pp=559–560}} | {{harvnb|Ali Rahman|Shahrill|Abbas|Tan|2017|pp=373–374}} | {{harvnb|Li|Schoenfeld|2019|loc=Abstract, Introducation}} | {{harvnb|Asano|2013|pp=xiii–xv}} }}</ref>