Editing Multiplication (section)

==Properties==
[[Image:Multiplication chart.svg|thumb|right|upright 1.0|Multiplication of numbers 0–10. Line labels = multiplicand. ''X''&nbsp;axis = multiplier. ''Y''&nbsp;axis = product.<br>Extension of this pattern into other quadrants gives the reason why a negative number times a negative number yields a positive number.<br>Note also how multiplication by zero causes a reduction in dimensionality, as does multiplication by a [[singular matrix]] where the [[determinant]] is 0. In this process, information is lost and cannot be regained.]]
For [[real number|real]] and [[complex number|complex]] numbers, which includes, for example, [[natural number]]s, [[integer]]s, and [[rational number|fractions]], multiplication has certain properties:

;[[Commutative property]]
:The order in which two numbers are multiplied does not matter:<ref name=":0">{{Cite web |title=Multiplication |website=Encyclopedia of Mathematics |url=https://encyclopediaofmath.org/wiki/Multiplication |access-date=2021-12-29}}</ref><ref name=":1">{{Cite book |last=Biggs |first=Norman L. |title=Discrete Mathematics |publisher=Oxford University Press |date=2002 |isbn=978-0-19-871369-2 |pages=25 |language=en}}</ref>
::<math>x \cdot y = y \cdot x.</math>

;[[Associative property]]
:Expressions solely involving multiplication or addition are invariant with respect to the [[order of operations]]:<ref name=":0"/><ref name=":1"/>
::<math>(x \cdot y) \cdot z = x \cdot (y \cdot z).</math>

;[[Distributive property]]
:Holds with respect to multiplication over addition. This identity is of prime importance in simplifying algebraic expressions:<ref name=":0"/><ref name=":1"/>
::<math>x \cdot(y + z) = x \cdot y + x \cdot z.</math>

;[[Identity element]]
:The multiplicative identity is 1; anything multiplied by 1 is itself. This feature of 1 is known as the '''identity property''':<ref name=":0"/><ref name=":1"/>
::<math>x \cdot 1 = x.</math>

;[[Absorbing element|Property of 0]]
:Any number multiplied by 0 is 0. This is known as the '''zero property''' of multiplication:<ref name=":0"/>
::<math>x \cdot 0 = 0.</math>

;[[Additive inverse|Negation]]
:−1 times any number is equal to the '''[[additive inverse]]''' of that number:
::<math>(-1) \cdot x = (-x)</math>, where <math>(-x) + x = 0.</math>

:−1 times −1 is 1:
::<math>(-1) \cdot (-1) = 1.</math>

;[[Inverse element]]
:Every number ''x'', [[division by zero|except 0]], has a '''[[multiplicative inverse]]''', <math>\frac{1}{x}</math>, such that <math>x \cdot \left(\frac{1}{x}\right) = 1</math>.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Multiplicative Inverse |url=https://mathworld.wolfram.com/ |access-date=2022-04-19 |website=Wolfram MathWorld |language=en}}</ref>

;[[Order theory|Order]] preservation
:Multiplication by a positive number preserves the [[Order theory|order]]:
::For {{nowrap|''a'' > 0}}, if {{nowrap|''b'' > ''c'',}} then {{nowrap|''ab'' > ''ac''}}.
:Multiplication by a negative number reverses the order:
::For {{nowrap|''a'' < 0}}, if {{nowrap|''b'' > ''c'',}} then {{nowrap|''ab'' < ''ac''}}.
:The [[complex number]]s do not have an ordering that is compatible with both addition and multiplication.<ref>{{Cite web |last=Angell |first=David |title=ORDERING COMPLEX NUMBERS... NOT* |url=https://web.maths.unsw.edu.au/~angell/articles/complexorder.pdf |access-date=29 December 2021 |publisher=UNSW Sydney, School of Mathematics and Statistics}}</ref>

Other mathematical systems that include a multiplication operation may not have all these properties. For example, multiplication is not, in general, commutative for [[Matrix (mathematics)|matrices]] and [[quaternion]]s.<ref name=":0"/> [[Hurwitz's theorem (composition algebras)|Hurwitz's theorem]] shows that for the [[hypercomplex number]]s of [[dimension]] 8 or greater, including the [[octonion]]s, [[sedenion]]s, and [[trigintaduonion]]s, multiplication is generally not associative.<ref>{{cite arXiv | last1=Cawagas | first1=Raoul E. | last2=Carrascal | first2=Alexander S. | last3=Bautista | first3=Lincoln A. | last4=Maria | first4=John P. Sta. | last5=Urrutia | first5=Jackie D. | last6=Nobles | first6=Bernadeth | title=The Subalgebra Structure of the Cayley-Dickson Algebra of Dimension 32 (trigintaduonion) | date=2009 | class=math.RA | eprint=0907.2047v3}}</ref>