Editing Equality (mathematics) (section)

== Similar relations ==

=== Approximate equality ===
{{Main|Approximation#Mathematics}}
[[File:Archimedes pi.svg|alt=diagram of a hexagon and pentagon circumscribed outside a circle|right|thumb|upright=1.35|The sequence given by the [[perimeter]]s of regular ''n''-sided [[polygon]]s that [[circumscribe]] the [[unit circle]] approximates <math>2\pi</math>]]
[[Numerical analysis]] is the study of [[Constructive proof|constructive]] methods and [[algorithms]] to find numerical [[approximation]]s (as opposed to [[Symbolic computation|symbolic manipulations]]) of solutions to problems in [[mathematical analysis]]. Especially those which cannot be [[Analytic solution|solved analytically]].<ref>{{Cite book |last=Kress |first=Rainer |date=1998 |title=Numerical Analysis |publisher=Springer |series=Graduate Texts in Mathematics |volume=181 |location=New York |pages=1–4 |doi=10.1007/978-1-4612-0599-9 |isbn=978-1-4612-6833-8 |issn=0072-5285}}</ref>

Calculations are likely to involve [[Round-off error|rounding errors]] and other [[approximation error]]s. [[Logarithm|Log tables]], slide rules, and calculators produce approximate answers to all but the simplest calculations. The results of computer calculations are normally an approximation, expressed in a limited number of significant digits, although they can be programmed to produce more precise results.<ref>{{Cite web |title=Numerical Computation Guide |url=http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html |url-status=dead |archive-url=https://web.archive.org/web/20160406101256/http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html |archive-date=2016-04-06 |access-date=2013-06-16}}</ref>

If viewed as a [[binary relation]], (denoted by the symbol <math>\approx</math>) between [[real number]]s or other things, if precisely defined, is not an equivalence relation since it's not transitive, even if modeled as a [[fuzzy relation]].<ref>{{Cite web |last1=Kerre |first1=Etienne E. |last2=De Cock |first2=Martine |date=2001 |title=Approximate Equality is no Fuzzy Equality |url=https://faculty.washington.edu/mdecock/papers/mdecock2001a.pdf}}</ref>

In [[computer science]], equality is given by some [[relational operator]]. Real numbers are often approximated by [[floating-point numbers]] (A sequence of some fixed number of digits of a given base, scaled by an integer [[exponent]] of that base),<ref>{{cite book |last1=Sterbenz |first1=Pat H. |url=https://archive.org/details/SterbenzFloatingPointComputation/mode/2up |title=Floating-Point Computation |date=1974 |publisher=Prentice-Hall |isbn=0-13-322495-3 |location=Englewood Cliffs, New Jersey}}</ref> thus it is common to store an [[Expression (computer science)|expression]] that denotes the real number as to not lose precision. However, the equality of two real numbers given by an expression is known to be [[undecidable problem|undecidable]] (specifically, real numbers defined by expressions involving the [[integer]]s, the basic [[arithmetic operation]]s, the [[logarithm]] and the [[exponential function]]). In other words, there cannot exist any [[algorithm]] for deciding such an equality (see [[Richardson's theorem]]).<ref>{{cite journal |last=Richardson |first=Daniel |year=1968 |title=Some Undecidable Problems Involving Elementary Functions of a Real Variable |journal=Journal of Symbolic Logic |volume=33 |issue=4 |pages=514–520 |jstor=2271358 |zbl=0175.27404}}</ref>

=== Equivalence relation ===
{{Main|Equivalence relation}}
[[File:Equivalentie.svg|thumb|upright=0.8|Graph of an example equivalence with 7 classes]]
An [[equivalence relation]] is a [[mathematical relation]] that generalizes the idea of similarity or sameness. It is defined on a [[Set (mathematics)|set]] <math>X</math> as a [[binary relation]] <math>\sim</math> that satisfies the three properties: [[Reflexive relation|reflexivity]], [[Symmetric relation|symmetry]], and [[Transitive relation|transitivity]]. Reflexivity means that every element in <math>X</math> is equivalent to itself (<math>a \sim a</math> for all <math>a \in X</math>). Symmetry requires that if one element is equivalent to another, the reverse also holds (<math>a \sim b \implies b \sim a</math>). Transitivity ensures that if one element is equivalent to a second, and the second to a third, then the first is equivalent to the third (<math>a \sim b</math> and <math>b \sim c \implies a \sim c</math>).{{Snf|Stoll|1963|p=29}} These properties are enough to [[Partition of a set|partition a set]] into disjoint [[equivalence class]]es. Conversely, every partition defines an equivalence class.{{Sfn|Stoll|1963|p=31}}

The equivalence relation of equality is a special case, as, if restricted to a given set <math>S,</math> it is the strictest possible equivalence relation on <math>S</math>; specifically, equality partitions a set into equivalence classes consisting of all [[singleton set]]s.{{Sfn|Stoll|1963|p=31}} Other equivalence relations, since they're less restrictive, generalize equality by identifying elements based on shared properties or transformations, such as [[Modular arithmetic#Congruence|congruence in modular arithmetic]] or [[Similarity (geometry)|similarity in geometry]].<ref>{{Cite book |last=Stark |first=Harold M. |url=https://mitpress.mit.edu/9780262690607/an-introduction-to-number-theory/ |title=An Introduction to Number Theory |date=May 30, 1978 |publisher=MIT Press |isbn=978-0-262-69060-7 |edition= |location=Cambridge, Massachusetts |pages=51–54}}</ref><ref>{{Cite web |date=2020-02-10 |title=2.2.1: Similarity |url=https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_II_(Illustrative_Mathematics_-_Grade_8)/02:_Dilations_Similarity_and_Introducing_Slope/2.02:_New_Page/2.2.1:_Similarity |access-date=2025-03-24 |website=Mathematics LibreTexts}}</ref>

==== Congruence relation ====
{{Main|Congruence relation}}
In [[abstract algebra]], a [[congruence relation]] extends the idea of an equivalence relation to include the [[Equality (mathematics)#Basic properties|operation-application property]]. That is, given a set <math>X,</math> and a set of operations on <math>X,</math> then a congruence relation <math>\sim</math> has the property that <math>a \sim b \implies f(a) \sim f(b)</math> for all operations <math>f</math> (here, written as unary to avoid cumbersome notation, but <math>f</math> may be of any [[arity]]). A congruence relation on an [[algebraic structure]] such as a [[Group (mathematics)|group]], [[Ring (mathematics)|ring]], or [[Module (mathematics)|module]] is an equivalence relation that respects the operations defined on that structure.<ref>{{Cite book |last=Hungerford |first=Thomas W. |date=1974 |title=Algebra |publisher=Springer |location=New York |series=Graduate Texts in Mathematics |volume=73 |doi=10.1007/978-1-4612-6101-8 |isbn=978-1-4612-6103-2 |issn=0072-5285}}</ref>

=== Isomorphism ===
{{Main|Isomorphism}}

In mathematics, especially in [[abstract algebra]] and [[category theory]], it is common to deal with objects that already have some internal [[Mathematical structure|structure]].  An [[isomorphism]] describes a kind of structure-preserving correspondence between two objects, establishing them as essentially identical in their structure or properties.<ref name="Isomorphism2024" /><ref>{{Citation |last=Leinster |first=Tom |title=Basic Category Theory |date=2016-12-30 |page=12 |arxiv=1612.09375}}</ref>

More formally, an isomorphism is a bijective [[Map (mathematics)|mapping]] (or [[morphism]]) <math>f</math> between two [[Set (mathematics)|sets]] or structures <math>A</math> and <math>B</math> such that <math>f</math> and its inverse <math>f^{-1}</math> preserve the [[Operation (mathematics)|operations]], [[Relation (mathematics)|relations]], or [[Function (mathematics)|functions]] defined on those structures.<ref name="Isomorphism2024">{{Cite encyclopedia |date=2024-11-25 |title=Isomorphism |encyclopedia=Encyclopædia Britannica |url=https://www.britannica.com/science/isomorphism-mathematics |access-date=2025-01-12}}</ref> This means that any operation or relation valid in <math>A</math> corresponds precisely to the operation or relation in <math>B</math> under the mapping. For example, in [[group theory]], a [[group isomorphism]] <math>f: G \mapsto H </math> satisfies <math>f(a * b) = f(a) * f(b)</math> for all elements <math>a, b,</math> where <math>*</math> denotes the group operation.<ref>{{harvnb|Pinter|2010|p=94}}.</ref>

When two objects or systems are isomorphic, they are considered indistinguishable in terms of their internal structure, even though their elements or representations may differ. For instance, all [[cyclic groups]] of order <math>\infty</math> are isomorphic to the integers, <math>\Z,</math> with addition.<ref>{{harvnb|Pinter|2010|p=114}}.</ref> Similarly, in [[linear algebra]], two [[vector spaces]] are isomorphic if they have the same [[Dimension (vector space)|dimension]], as there exists a [[Linear isomorphism|linear bijection]] between their elements.<ref>{{Cite book |last=Axler |first=Sheldon |url=https://linear.axler.net/LADR4e.pdf |title=Linear Algebra Done Right |publisher=[[Springer (publisher)|Springer]] |page=86}}</ref>

The concept of isomorphism extends to numerous branches of mathematics, including [[graph theory]] ([[graph isomorphism]]), [[topology]] ([[homeomorphism]]), and algebra (group and [[Ring isomorphism|ring isomorpisms]]), among others. Isomorphisms facilitate the classification of mathematical entities and enable the transfer of results and techniques between similar systems. Bridging the gap between isomorphism and equality was one motivation for the development of [[category theory]], as well as for [[homotopy type theory]] and [[univalent foundations]].<ref>{{cite journal |last1=Eilenberg |first1=S. |last2=Mac Lane |first2=S. |date=1942 |title=Group Extensions and Homology |journal=Annals of Mathematics |volume=43 |issue=4 |pages=757–831 |issn=0003-486X |jstor=1968966}}</ref><ref>{{cite encyclopedia |last=Marquis |first=Jean-Pierre |date=2019 |title=Category Theory |url=https://plato.stanford.edu/entries/category-theory/ |access-date=26 September 2022 |encyclopedia=[[Stanford Encyclopedia of Philosophy]] |publisher=Department of Philosophy, [[Stanford University]]}}</ref><ref>{{cite book |last1=Hofmann |first1=Martin |title=Twenty Five Years of Constructive Type Theory |last2=Streicher |first2=Thomas |author2-link=Thomas Streicher |date=1998 |publisher=Clarendon |isbn=978-0-19-158903-4 |editor1-last=Sambin |editor1-first=Giovanni |series=Oxford Logic Guides |volume=36 |pages=83–111 |chapter=The groupoid interpretation of type theory |mr=1686862 |editor2-last=Smith |editor2-first=Jan M. |chapter-url=https://books.google.com/books?id=pLnKggT_In4C&pg=PA83}}</ref>

=== Geometry ===
[[File:Congruent non-congruent triangles.svg|thumb|upright=1.35|The two triangles on the left are [[Congruence (geometry)|congruent]]. The third is [[Similarity (geometry)|similar]] to them. The last triangle is neither congruent nor similar to any of the others. ]]
In [[geometry]], formally, two figures are equal if they contain exactly the same [[Point (geometry)|points]]. However, historically, geometric-equality has always been taken to be much broader. [[Euclid]] and [[Archimedes]] used "equal" ({{lang|grc|ἴσος}} {{tlit|grc|isos}}) often referring to figures with the same area or those that could be cut and rearranged to form one another. For example, Euclid stated the [[Pythagorean theorem]] as "the square on the hypotenuse is equal to the squares on the sides, taken together"; and Archimedes said that "a circle is equal to the rectangle whose sides are the radius and half the circumference."<ref>{{Cite journal |last=Beeson |first=Michael |date=2023-09-01 |title=On the notion of equal figures in Euclid |journal=Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry |volume=64 |issue=3 |pages=581–625 |doi=10.1007/s13366-022-00649-9 |arxiv=2008.12643 |issn=2191-0383}}</ref>

This notion persisted until [[Adrien-Marie Legendre]], who introduced the term "equivalent" to describe figures of equal area and restricted "equal" to what we now call "[[Congruence (geometry)|congruent]]"—the same [[shape]] and [[size]], or if one has the same shape and size as the [[mirror image]] of the other.<ref>{{Cite book |last=Legendre |first=Adrien Marie |url=https://archive.org/details/cu31924001166341/page/n77/mode/2up |title=Elements of geometry |date=1867 |publisher=Baltimore, Kelly & Piet |others=Cornell University Library |page=68}}</ref><ref>{{cite dictionary |last1=Clapham |first1=C. |last2=Nicholson |first2=J. |year=2009 |dictionary=Oxford Concise Dictionary of Mathematics |entry=Congruent Figures |url=http://web.cortland.edu/matresearch/OxfordDictionaryMathematics.pdf |url-status=dead |archive-url=https://web.archive.org/web/20131029203826/http://web.cortland.edu/matresearch/OxfordDictionaryMathematics.pdf |archive-date=29 October 2013 |access-date=2 June 2017 |publisher=Addison-Wesley |page=167}}</ref> Euclid's terminology continued in the work of [[David Hilbert]] in his ''{{lang|De|[[Grundlagen der Geometrie]]}}'', who further refined Euclid's ideas by introducing the notions of polygons being "divisibly equal" ({{lang|de|zerlegungsgleich}}) if they can be cut into finitely many triangles which are congruent, and "equal in content" ({{lang|de|inhaltsgleichheit}}) if one can add finitely many divisibly equal polygons to each such that the resulting polygons are divisibly equal.<ref>{{Cite book |last=Hilbert |first=David |url=https://archive.org/details/grundlagendergeo00hilb/page/40/mode/2up |title=Grundlagen der Geometrie |date=1899 |publisher=B. G. Teubner |others=Wellesley College Library |page=40 |language=de}}</ref>

After the rise of set theory, around the 1960s, there was a push for a reform in [[mathematics education]] called [[New Math]], following [[Andrey Kolmogorov]], who, in an effort to restructure Russian geometry courses, proposed presenting geometry through the lens of [[Transformation geometry|transformations]] and set theory. Since a figure was seen as a set of points, it could only be equal to itself, as a result of Kolmogorov, the term "congruent" became standard in schools for figures that were previously called "equal", which popularized the term.<ref>[https://books.google.com/books?id=qwyBPybT4oMC Alexander Karp & Bruce R. Vogeli – Russian Mathematics Education: Programs and Practices, Volume 5], pp. 100–102</ref>

While Euclid addressed [[Proportionality (mathematics)|proportionality]] and figures of the same shape, it was not until the 17th century that the concept of [[Similarity (geometry)|similarity]] was formalized in the modern sense. Similar figures are those that have the same shape but can differ in size; they can be transformed into one another by [[Scaling (geometry)|scaling]] and congruence.<ref>{{Cite book |date=2020-02-10 |title=PreAlgebra |chapter=2.2.1: Similarity |chapter-url=https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_II_(Illustrative_Mathematics_-_Grade_8)/02:_Dilations_Similarity_and_Introducing_Slope/2.02:_New_Page/2.2.1:_Similarity |access-date=2025-03-04 |publisher=Mathematics LibreTexts}}</ref> Later a concept of equality of [[directed line segment]]s, [[Equipollence (geometry)|equipollence]], was advanced by [[Giusto Bellavitis]] in 1835.<ref>{{Cite web |title=Giusto Bellavitis – Biography |url=https://mathshistory.st-andrews.ac.uk/Biographies/Bellavitis/ |access-date=2025-03-04 |website=Maths History}}</ref>