Editing Additive inverse (section)

{{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>