Editing Additive inverse (section)

== 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>