❪∂₁ₐ❫ Arithmetic Operations ○｜Definition｜1st｜20261005152145-00-⌔

Arithmetic - Wikipedia#Operations

Operations

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Arithmetic operations underlie many everyday occurrences, like when putting four apples from one bag together with three apples from another bag (top image) or when distributing nine apples equally among three children (bottom image).

Arithmetic operations are ways of combining, transforming, or manipulating numbers. They are functions that have numbers both as input and output.¹ The most important operations in arithmetic are addition, subtraction, multiplication, and division.² Further operations include exponentiation, extraction of roots, and logarithm.³ If these operations are performed on variables rather than numbers, they are sometimes referred to as algebraic operations.⁴

Two important concepts in relation to arithmetic operations are identity elements and inverse elements. The identity element or neutral element of an operation does not cause any change if it is applied to another element. For example, the identity element of addition is 0 since any sum of a number and 0 results in the same number. The inverse element is the element that results in the identity element when combined with another element. For instance, the additive inverse of the number 6 is -6 since their sum is 0.⁵

There are not only inverse elements but also inverse operations. In an informal sense, one operation is the inverse of another operation if it undoes the first operation. For example, subtraction is the inverse of addition since a number returns to its original value if a second number is first added and subsequently subtracted, as in $13+4-4=13$. Defined more formally, the operation "$\star$" is an inverse of the operation "$\circ$" if it fulfills the following condition: $t\star s=r$ if and only if $r\circ s=t$.⁶

Commutativity and associativity are laws governing the order in which some arithmetic operations can be carried out. An operation is commutative if the order of the arguments can be changed without affecting the results. This is the case for addition, for instance, $7+9$ is the same as $9+7$. Associativity is a rule that affects the order in which a series of operations can be carried out. An operation is associative if, in a series of two operations, it does not matter which operation is carried out first. This is the case for multiplication, for example, since $(5\times 4)\times 2$ is the same as $5\times (4\times 2)$.⁷

Printed 2026-10-05.

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Footnotes

1. Nagel 2002, p. 179 Husserl & Willard 2012, pp. XLIV–XLV O’Leary 2015, p. 190 ↩

2. Rising et al. 2021, p. 110 Bukhshtab & Pechaev 2020 Nagel 2002, pp. 177, 179–180 ↩

3. Bukhshtab & Pechaev 2020 Burgin 2022, pp. 57, 77 Adamowicz 1994, p. 299 Nagel 2002, pp. 177, 179–180 ↩

4. Khan & Graham 2018, pp. 9–10 Smyth 1864, p. 55 ↩

5. Tarasov 2008, pp. 57–58 Mazzola, Milmeister & Weissmann 2004, p. 66 Krenn & Lorünser 2023, p. 8 ↩

6. Kay 2021, pp. 44–45 Wright, Ellemor-Collins & Tabor 2011, p. 136 ↩

7. Krenn & Lorünser 2023, p. 8 Mazzola, Milmeister & Weissmann 2004, p. 66 ↩