Editing Equality (mathematics) (section)

== Equations ==
{{Main|Equation}}
[[File:Balance scale.svg|alt=Diagram of a balance scale|class=skin-invert-image|thumb|upright=1.2|[[Balance scales]], used in algebra education to help students visualize how equations can be transformed to determine unknown values.]]
An [[equation]] is a [[Glossary of mathematical symbols|symbolic]] equality of two [[Expression (mathematics)|mathematical expressions]] connected with an [[equals sign]] (=).<ref>{{Cite dictionary |title=Equation (n.), sense III.6.a |dictionary=[[Oxford English Dictionary]] |date=2023 |doi=10.1093/OED/2918848458 |quote=A formula affirming the equivalence of two quantitative expressions, which are for this purpose connected by the sign =.}}</ref> [[Algebra]] is the branch of mathematics concerned with [[equation solving]]: the problem of finding values of some [[Variable (mathematics)|variable]], called {{em|unknown}}, for which the specified equality is true. Each value of the unknown for which the equation holds is called a {{em|solution}} of the given equation; also stated as {{em|satisfying}} the equation. For example, the equation <math>x^2 - 6x + 5=0</math> has the values <math>x=1</math> and <math>x=5</math> as its only solutions. The terminology is used similarly for equations with several unknowns.<ref>Sobolev, S. K. (originator). "[https://encyclopediaofmath.org/wiki/Equation Equation]". ''[[Encyclopedia of Mathematics]]''. [[Springer Publishing|Springer]]. {{ISBN|1402006098}}.</ref> The set of solutions to an equation or [[system of equations]] is called its [[solution set]].<ref>{{Cite dictionary |date=2025-02-24 |title=Solution set |url=https://www.merriam-webster.com/dictionary/solution%20set |access-date=2025-03-01 |dictionary=Merriam-Webster}}</ref>

In [[mathematics education]], students are taught to rely on concrete models and visualizations of equations, including geometric analogies, manipulatives including sticks or cups, and "function machines" representing equations as [[flow diagram]]s. One method uses [[balance scales]] as a pictorial approach to help students grasp basic problems of algebra. The mass of some objects on the scale is unknown and represents variables. Solving an equation corresponds to adding and removing objects on both sides in such a way that the sides stay in balance until the only object remaining on one side is the object of unknown mass.<ref>{{Cite book |last1=Gardella |first1=Francis |url=https://books.google.com/books?id=HBXFDwAAQBAJ&pg=PA19 |title=Algebra for the Middle Grades |last2=DeLucia |first2=Maria |date=2020 |publisher=IAP |isbn=978-1-64113-847-5 |page=19}}</ref>

Often, equations are considered to be a statement, or [[Relation (mathematics)|relation]], which can be [[true or false]]. For example, <math>1+1=2</math> is true, and <math>1+1=3</math> is false. Equations with unknowns are considered [[Truth condition|conditionally true]]; for example, <math>x^2 - 6x + 5=0</math> is true when <math>x=1</math> or <math>x=5,</math> and false otherwise.<ref>{{Cite book |last=Levin |first=Oscar |url=https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf |title=Discrete Mathematics: An Open Introduction |date=2021 |isbn=978-1-79290-169-0 |page=5 |publisher=Oscar Levin |edition=3rd}}</ref> There are several different terminologies for this. In [[mathematical logic]], an equation is a binary [[Predicate (logic)|predicate]] (i.e. a [[Statement (logic)|logical statement]], that can have [[free variables]]) which satisfies [[Equality (mathematics)#Axioms|certain properties]].<ref name="Mendelson1964" /> In [[computer science]], an equation is defined as a [[Boolean data type|boolean]]-valued [[Expression (computer science)|expression]], or [[relational operator]], which returns 1 and 0 for true and false respectively.<ref>{{Cite web |title=Equality and inequality operators == != |url=https://www.ibm.com/docs/en/xl-c-and-cpp-aix/16.1?topic=expressions-equality-inequality-operators |date=2025-02-25 |access-date=2025-03-24 |website=XL C/C++ for AIX Documentation |publisher=IBM}}</ref>

=== 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>

=== Definitions ===
Equations are often used to introduce new terms or symbols for constants, [[Judgment (mathematical logic)|assert]] equalities, and introduce shorthand for complex expressions, which is called "equal by definition", and often denoted with (<math>:=</math>).<ref>{{Cite web |last1=Lankham |first1=Isaiah |last2=Nachtergaele |first2=Bruno |last3=Schilling |first3=Anne |date=January 21, 2007 |title=Some Common Mathematical Symbols and Abbreviations (with History) |url=https://www.math.ucdavis.edu/~anne/WQ2007/mat67-Common_Math_Symbols.pdf |publisher=[[University of California, Davis]]}}</ref> It is similar to the concept of [[Assignment (computer science)|assignment]] of a [[Variable (computer science)|variable in computer science]]. For example, <math display = "inline">\mathbb{e} := \sum_{n=0}^\infty \frac{1}{n!}</math> defines [[Euler's number]],<ref>{{Cite encyclopedia |title=e |url=https://www.britannica.com/science/e-mathematics |access-date=2025-01-13 |encyclopedia=Encyclopædia Britannica}}</ref> and <math>i^2 = -1</math> is the defining property of the [[imaginary number]] <math>i.</math><ref>{{Cite book |last1=Marecek |first1=Lynn |last2=Mathis |first2=Andrea Honeycutt |date=2020-05-06 |title=Intermediate Algebra 2e |chapter=8.8 Use the Complex Number System |chapter-url=https://openstax.org/books/intermediate-algebra-2e/pages/8-8-use-the-complex-number-system |access-date=2025-03-04 |publisher=OpenStax |isbn=978-1-975076-49-8}}</ref>

In [[mathematical logic]], this is called an [[extension by definition]] (by equality) which is a [[conservative extension]] to a [[formal system]].{{Sfn|Mendelson|1964|pp=82–83}} This is done by taking the equation defining the new constant symbol as a new [[axiom]] of the [[Theory (mathematical logic)|theory]]. The first recorded symbolic use of "Equal by definition" appeared in ''Logica Matematica'' (1894) by [[Cesare Burali-Forti]], an Italian mathematician. Burali-Forti, in his book, used the notation (<math>=_\text{Def}   </math>).<ref>{{Cite book |last=Burali-Forti |first=Cesare |author-link=Cesare Burali-Forti |url=https://books.google.com/books?id=F5xJAAAAIAAJ |title=Logica matematica |date=1894 |publisher=[[Ulrico Hoepli]] |others=University of California |page=120 |language=it |trans-title=Mathematical logic |archive-url=https://archive.org/details/logicamatematic01buragoog/page/n131/mode/2up?q=def |archive-date=2009-08-01}}</ref><ref>{{Cite book |date=2013-11-07 |last1=Lankham |first1=Isaiah |last2=Nachtergaele |first2=Bruno |last3=Schilling |first3=Anne |title=Linear Algebra |chapter=13.3: Some Common Mathematical Symbols and Abbreviations |chapter-url=https://math.libretexts.org/Bookshelves/Linear_Algebra/Book:_Linear_Algebra_(Schilling_Nachtergaele_and_Lankham)/13:_Appendices/13.03:_Some_Common_Mathematical_Symbols_and_Abbreviations#:~:text=:=%20(the%20equal%20by%20definition,especially%20common%20in%20applied%20mathematics. |access-date=2025-03-04 |publisher=Mathematics LibreTexts, University of California, Davis}}</ref>