Editing Additive inverse

{{Short description|Number that, when added to the original number, yields the additive identity}}
{{Redirect|Opposite number|other uses|Analog (disambiguation){{!}}analog|and|counterpart (disambiguation){{!}}counterpart}}

In mathematics, the '''additive inverse''' of an [[Element (mathematics)|element]] {{Mvar|x}}, denoted {{Mvar|−x}},<ref>{{Cite book |last=Gallian |first=Joseph A. |title=Contemporary abstract algebra |date=2017 |publisher=Cengage Learning |isbn=978-1-305-65796-0 |edition=9th |location=Boston, MA |page=52}}</ref> is the element that when [[Addition|added]] to {{Mvar|x}}, yields the [[additive identity]], [[0|0 (zero)]].<ref>{{Cite book |last=Fraleigh |first=John B. |title=A first course in abstract algebra |date=2014 |publisher=Pearson |isbn=978-1-292-02496-7 |edition=7th |location=Harlow |pages=169–170}}</ref> In the most familiar cases, this is the [[number]] 0, but it can also refer to a more generalized [[zero element]].

In [[elementary mathematics]], the additive inverse is often referred to as the '''opposite''' number,<ref>{{Cite web |last=Mazur |first=Izabela |date=March 26, 2021 |title=2.5 Properties of Real Numbers -- Introductory Algebra |url=https://pressbooks.bccampus.ca/intermediatedevelopmentalmath/chapter/properties-of-real-numbers/ |url-status= |access-date=August 4, 2024}}</ref><ref>{{Cite web |title=Standards::Understand p + q as the number located a distance {{!}}q{{!}} from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. |url=https://learninglab.si.edu/standards/CCSS.Math.Content.7.NS.A.1b/340 |access-date=2024-08-04 |website=learninglab.si.edu}}</ref> or its '''negative'''.<ref> {{MathWorld |title=Negative |id=Negative |access-date=2025-01-04}}</ref> The [[unary operation]] of '''arithmetic negation'''<ref>{{Cite book |last1=Kinard |first1=James T. |url=https://books.google.com/books?id=BCSuwlwt5NAC |title=Rigorous Mathematical Thinking: Conceptual Formation in the Mathematics Classroom |last2=Kozulin |first2=Alex |date=2008-06-02 |publisher=Cambridge University Press |isbn=978-1-139-47239-5 |language=en}}</ref> is closely related to ''[[subtraction]]''<ref>{{Cite web |last=Brown |first=Christopher |title=SI242: divisibility |url=https://www.usna.edu/Users/cs/wcbrown/courses/F23SI242/lec/l25/lec.html |access-date=2024-08-04 |website=www.usna.edu}}</ref> and is important in [[Elementary algebra|solving algebraic equations]].<ref name=":0">{{Cite web |date=2020-07-21 |title=2.2.5: Properties of Equality with Decimals |url=https://k12.libretexts.org/Bookshelves/Mathematics/Algebra/02%3A_Linear_Equations/2.02%3A_One-Step_Equations_and_the_Properties_of_Equality/2.2.05%3A_Properties_of_Equality_with_Decimals |access-date=2024-08-04 |website=K12 LibreTexts |language=en}}</ref> Not all [[Set (mathematics)|sets]] where addition is defined have an additive inverse, such as the [[Natural number|natural numbers]].<ref name=":1">{{Cite book |last=Fraleigh |first=John B. |title=A first course in abstract algebra |date=2014 |publisher=Pearson |isbn=978-1-292-02496-7 |edition=7th |location=Harlow |pages=37–39}}</ref>

== Common examples ==
When working with [[integer]]s, [[rational number]]s, [[real number]]s, and [[complex number]]s, the additive inverse of any number can be found by multiplying it by [[−1]].<ref name=":0" />[[Image:NegativeI2Root.svg|thumb|right|These complex numbers, two of eight values of [[root of unity|{{radic|1|8}}]], are mutually opposite]]
{| class="wikitable"
|+ Simple cases of additive inverses
! <math>n</math>
! <math>-n</math>
|-
| <math>7</math>
| <math>-7</math>
|-
| <math>0.35</math>
| <math>-0.35</math>
|-
| <math>\frac{1}{4}</math>
| <math>-\frac{1}{4}</math>
|-
| <math>\pi</math>
| <math>-\pi</math>
|-
| <math>1 + 2i</math> 
| <math>-1 - 2i</math>
|}

The concept can also be extended to algebraic expressions, which is often used when balancing [[equation]]s.
{| class="wikitable"
|+ Additive inverses of algebraic expressions
! <math>n</math>
! <math>-n</math>
|-
| <math>a - b</math>
| <math>-(a - b) = -a + b</math>
|-
| <math>2x^2 + 5</math>
| <math>-(2x^2 + 5) = -2x^2 - 5</math> 
|-
| <math>\frac{1}{x + 2}</math>
| <math>-\frac{1}{x+2}</math>
|-
| <math>\sqrt{2}\sin{\theta} - \sqrt{3}\cos{2\theta}</math>
|<math>-(\sqrt{2}\sin{\theta} - \sqrt{3}\cos{2\theta}) = -\sqrt{2}\sin{\theta} + \sqrt{3}\cos{2\theta}</math>
|}

== Relation to subtraction ==
The additive inverse is closely related to [[subtraction]], which can be viewed as an addition using the inverse:
:{{math|1=''a'' − ''b''  =  ''a'' + (−''b'')}}.
Conversely, the additive inverse can be thought of as subtraction from zero:
:{{math|1=−''a'' = 0 − ''a''}}.
This connection lead to the minus sign being used for both opposite magnitudes and subtraction as far back as the 17th century. While this notation is standard today, it was met with opposition at the time, as some mathematicians felt it could be unclear and lead to errors.<ref>{{Cite book |last=Cajori |first=Florian |title=A History of Mathematical Notations: two volume in one |date=2011 |publisher=Cosimo Classics |isbn=978-1-61640-571-7 |location=New York |pages=246–247}}</ref>

== Formal definition ==
Given an algebraic structure defined under addition <math>(S, +)</math> with an additive identity <math>e \in S</math>, an element <math>x \in S</math> has an additive inverse <math>y</math> if and only if <math>y \in S</math>, <math>x + y = e</math>, and <math>y + x = e</math>.<ref name=":1" />

Addition is typically only used to refer to a [[Commutative property|commutative]] operation, but it is not necessarily [[Associative property|associative]]. When it is associative, so <math>(a + b) + c = a + (b + c)</math>, the left and right inverses, if they exist, will agree, and the additive inverse will be unique. In non-associative cases, the left and right inverses may disagree, and in these cases, the inverse is not considered to exist.

The definition requires [[Closure (mathematics)|closure]], that the additive element <math>y</math> be found in <math>S</math>. However, despite being able to add the natural numbers together, the set of natural numbers does not include the additive inverse values. This is because the additive inverse of a natural number (e.g., <math>-3</math> for <math>3</math>) is not a natural number; it is an [[integer]]. Therefore, the natural numbers in set <math>S</math> do have additive inverses and their associated inverses are [[negative number]]s.

== Further examples ==
* In a [[vector space]], the additive inverse {{math|−'''v'''}} (often called the ''[[opposite vector]]'' of {{math|'''v'''}}) has the same [[norm (mathematics)|magnitude]] as {{math|'''v'''}} and but the opposite direction.<ref>{{Citation |last=Axler |first=Sheldon |title=Vector Spaces |date=2024 |work=Linear Algebra Done Right |series=Undergraduate Texts in Mathematics |pages=1–26 |editor-last=Axler |editor-first=Sheldon |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-031-41026-0_1 |isbn=978-3-031-41026-0|doi-access=free }}</ref>
* In [[modular arithmetic]], the '''modular additive inverse''' of {{mvar|x}} is the number {{mvar|a}} such that {{math|1={{mvar|a}} + {{mvar|x}} ≡ 0 (mod {{mvar|n}})}} and always exists. For example, the inverse of 3 modulo 11 is 8, as {{math|1= 3 + 8 ≡ 0 (mod 11)}}.<ref>{{Cite book |last=Gupta |first=Prakash C. |title=Cryptography and network security |date=2015 |publisher=PHI Learning Private Limited |isbn=978-81-203-5045-8 |series=Eastern economy edition |location=Delhi |page=15}}</ref>
* In a [[Boolean ring]], which has elements <math>\{0, 1\}</math> addition is often defined as the [[symmetric difference]]. So <math>0 + 0 = 0</math>, <math>0 + 1 = 1</math>, <math>1 + 0 = 1</math>, and <math>1 + 1 = 0</math>. Our additive identity is 0, and both elements are their own additive inverse as <math>0 + 0 = 0</math> and <math>1 + 1 = 0</math>.<ref>{{Cite journal |last1=Martin |first1=Urusula |last2=Nipkow |first2=Tobias |date=1989-03-01 |title=Boolean unification — The story so far |url=https://www.sciencedirect.com/science/article/pii/S0747717189800136 |journal=Journal of Symbolic Computation |series=Unification: Part 1 |volume=7 |issue=3 |pages=275–293 |doi=10.1016/S0747-7171(89)80013-6 |issn=0747-7171}}</ref>

== See also ==
* [[Absolute value]] (related through the identity {{math|1={{!}}−''x''{{!}} = {{!}}''x''{{!}}}}).
* [[Monoid]]
* [[Inverse function]]
* [[Involution (mathematics)]]
* [[Multiplicative inverse]]
* [[Reflection (mathematics)]]
* [[Reflection symmetry]]
* [[Semigroup]]

== Notes and references ==
{{reflist}}

[[Category:Abstract algebra]]
[[Category:Arithmetic]]
[[Category:Elementary algebra]]