Editing Equality (mathematics) (section)

=== Identities ===
{{Main|Identity (mathematics)}}
An [[Identity (mathematics)|identity]] is an equality that is true for all values of its variables in a given domain.<ref>{{Cite encyclopedia |title=Equation |encyclopedia=[[Encyclopedia of Mathematics]] |publisher=[[Springer-Verlag]] |url=http://encyclopediaofmath.org/index.php?title=Equation&oldid=32613 |last=Grishin |first=V. N. |isbn=1-4020-0609-8}}</ref><ref>{{Cite book |last=Hall |first=Henry Sinclair |url=https://archive.org/details/algebraforbeginn00hall/ |title=Algebra for Beginners |last2=Algebra for Beginners |first2=Samuel Ratcliffe |date=1895 |publisher=[[Macmillan & Co]] |location=New York |page=52}}</ref> An "equation" may sometimes mean an identity, but more often than not, it {{em|specifies}} a subset of the variable space to be the subset where the equation is true. An example is <math>\left(x + 1\right)\left(x + 1\right) = x^2 + 2 x + 1,</math> which is true for each [[real number]] <math>x.</math> There is no standard notation that distinguishes an equation from an identity, or other use of the equality relation: one has to guess an appropriate interpretation from the semantics of expressions and the context.<ref>{{cite web |last1=Marcus |first1=Solomon |author1-link=Solomon Marcus |last2=Watt |first2=Stephen M. |title=What is an Equation? |url=https://www.academia.edu/3287674 |access-date=2019-02-27 |at=Section V. ''Types of Equations and Terminology in Various Languages''}}</ref> Sometimes, but not always, an identity is written with a [[triple bar]]: <math>\left(x + 1\right)\left(x + 1\right) \equiv x^2 + 2 x + 1.</math><ref>{{Cite dictionary |editor-last1=Earl |editor-first1=Richard |dictionary=The Concise Oxford Dictionary of Mathematics |entry=Identity |editor-last2=Nicholson |editor-first2=James |date=2021 |publisher=Oxford University Press |doi=10.1093/acref/9780198845355.001.0001 |isbn=978-0-19-884535-5 |edition=6th |last1=Earl |first1=Richard |last2=Nicholson |first2=James}}</ref> This notation was introduced by [[Bernhard Riemann]] in his 1857 ''{{lang|de|Elliptische Funktionen}}'' lectures (published in 1899).{{Sfn|Cajori|1928|p=417}}<ref>{{Cite book |last=Kronecker |first=Leopold |url=https://archive.org/details/vorlesungenberz00krongoog/page/86/mode/2up?q=Biemann |title=Vorlesungen über Zahlentheorie |date=1978 |publisher=Springer |isbn=978-3-662-22798-5 |page=86 |doi=10.1007/978-3-662-24731-0 |orig-year=1901}}</ref><ref>{{Cite book |last1=Riemann |first1=Bernhard |url=https://archive.org/details/elliptischefunc00riemgoog/page/n17/mode/2up?q=%22Aus+der+letzten+Gleichung%22 |title=Elliptische functionen |last2=Stahl |first2=Hermann |date=1899 |publisher=B. G. Teubner |language=de}}</ref>

Alternatively, identities may be viewed as an equality of [[Function (mathematics)|functions]], where instead of writing <math>f(a) = g(a) \text{ for all } a,</math> one may simply write <math>f = g.</math><ref>{{Cite book |last=Tao |first=Terence |date=2022 |title=Analysis I |series=Texts and Readings in Mathematics |volume=37 |publisher=Springer |location=Singapore |pages=42–43 |doi=10.1007/978-981-19-7261-4 |isbn=978-981-19-7261-4 |issn=2366-8717}}</ref>{{Sfn|Krabbe|1975|p=7}} This is called the [[extensionality]] of functions.<ref>{{Cite web |title=function extensionality in nLab |url=https://ncatlab.org/nlab/show/function+extensionality |access-date=2025-03-01 |website=ncatlab.org}}</ref>{{sfn|Lévy|2002|p=27}} In this sense, the operation-application property refers to [[Operator (mathematics)|operators]], operations on a [[function space]] (functions mapping between functions) like [[Function composition|composition]]<ref>{{Cite book |last1=Malik |first1=D. S. |url=https://archive.org/details/fundamentals-of-abstract-algebra-d.-s.-malik-j.-m.-mordeson-m.-k.-sen/page/83/mode/2up |title=Fundamentals of Abstract Algebra |last2=Mordeson |first2=J. M. |last3=Sen |first3=M. K. |publisher=[[McGraw-Hill]] |year=1997 |isbn=0-07-040035-0 |location=New York |page=83}}</ref> or the [[derivative]], commonly used in [[operational calculus]].{{Sfn|Krabbe|1975|pp=2–3}} An identity can contain an functions as "unknowns", which can be solved for similarly to a regular equation, called a [[functional equation]].<ref>{{Cite book |date=2007 |editor-last=Small |editor-first=Christopher G. |title=Functional Equations and How to Solve Them |publisher=Springer |location=New York |series=Problem Books in Mathematics |page=1 |doi=10.1007/978-0-387-48901-8 |isbn=978-0-387-34534-5 |issn=0941-3502}}</ref> A functional equation involving derivatives is called a [[differential equation]].<ref>{{Cite book |last1=Adkins |first1=William A. |last2=Davidson |first2=Mark G. |date=2012 |title=Ordinary Differential Equations |publisher=Springer |location=New York |series=Undergraduate Texts in Mathematics |pages=2–5 |doi=10.1007/978-1-4614-3618-8 |isbn=978-1-4614-3617-1 |issn=0172-6056}}</ref>