Inmathematics,equality is a relationship between twoquantities orexpressions, stating that they have the same value, or represent the samemathematical object.[1][2] Equality betweenA andB is denoted with anequals sign asA = B, and read "A equalsB". A written expression of equality is called anequation oridentity depending on the context. Two objects that arenot equal are said to bedistinct.[3]
Equality is often considered aprimitive notion, meaning it is not formally defined, but rather informally said to be "a relation each thing bears to itself and nothing else". This characterization is notably circular ("nothingelse"), reflecting a general conceptual difficulty in fully characterizing the concept. Basic properties about equality likereflexivity,symmetry, andtransitivity have been understood intuitively since at least theancient Greeks, but were not symbolically stated as general properties of relations until the late 19th century byGiuseppe Peano. Other properties likesubstitution andfunction application weren't formally stated until the development ofsymbolic logic.
There are generally two ways that equality is formalized in mathematics: throughlogic or throughset theory. In logic, equality is a primitivepredicate (astatement that may havefree variables) with the reflexive property (called thelaw of identity), and the substitution property. From those, one can derive the rest of the properties usually needed for equality. After thefoundational crisis in mathematics at the turn of the 20th century, set theory (specificallyZermelo–Fraenkel set theory) became the most commonfoundation of mathematics. In set theory, any twosets are defined to be equal if they have all the samemembers. This is called theaxiom of extensionality.
In English, the wordequal is derived from theLatinaequālis ('like', 'comparable', 'similar'), which itself stems fromaequus ('level', 'just').[5] The word enteredMiddle English around the 14th century, borrowed fromOld Frenchequalité (modernégalité).[6] More generally, the interlingual synonyms ofequal have been used more broadly throughout history (see§ Geometry).
Before the 16th century, there was no common symbol for equality, and equality was usually expressed with a word, such asaequales, aequantur, esgale, faciunt, ghelijck, orgleich, and sometimes by the abbreviated formaeq, or simply⟨æ⟩ and⟨œ⟩.[7]Diophantus's use of⟨ἴσ⟩, short forἴσος (ísos 'equals'), inArithmetica (c. 250 AD) is considered one of the first uses of anequals sign.[8]
The sign=, now universally accepted in mathematics for equality, was first recorded by Welsh mathematicianRobert Recorde inThe Whetstone of Witte (1557), just one year before his death. The original form of the symbol was much wider than the present form. In his book, Recorde explains his symbol as "Gemowe lines", from the Latingemellus ('twin'), using twoparallel lines to represent equality because he believed that "no two things could be more equal."[4][7]
Recorde's symbol was not immediately popular. After its introduction, it wasn't used again in print until 1618 (61 years later), in an anonymous Appendix inEdward Wright's English translation ofDescriptio, byJohn Napier. It wasn't until 1631 that it received more than general recognition in England, being adopted as the symbol for equality in a few influential works. Later used by several influential mathematicians, most notably, bothIsaac Newton andGottfried Leibniz, and due to the prevalence ofcalculus at the time, it quickly spread throughout the rest of Europe.[7]
The first three properties are generally attributed toGiuseppe Peano for being the first to explicitly state these as fundamental properties of equality in hisArithmetices principia (1889).[14][15] However, the basic notions have always existed; for example, inEuclid'sElements (c. 300 BC),he includes 'common notions': "Things that are equal to the same thing are also equal to one another" (transitivity), "Things that coincide with one another are equal to one another" (reflexivity), along with some function-application properties for addition and subtraction.[16] The function-application property was also stated in Peano'sArithmetices principia,[14] however, it had been common practice inalgebra since at least Diophantus (c. 250 AD).[17] The substitution property is generally attributed toGottfried Leibniz (c. 1686), and often calledLeibniz's Law.[11][18]
Anequation is asymbolic equality of twomathematical expressions connected with anequals sign (=).[19]Algebra is the branch of mathematics concerned withequation solving: the problem of finding values of somevariable, calledunknown, for which the specified equality is true. Each value of the unknown for which the equation holds is called asolution of the given equation; also stated assatisfying the equation. For example, the equation has the values and as its only solutions. The terminology is used similarly for equations with several unknowns.[20] The set of solutions to an equation orsystem of equations is called itssolution set.[21]
Inmathematics 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 asflow diagrams. One method usesbalance 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.[22]
Often, equations are considered to be a statement, orrelation, which can betrue or false. For example, is true, and is false. Equations with unknowns are consideredconditionally true; for example, is true when or and false otherwise.[23] There are several different terminologies for this. Inmathematical logic, an equation is a binarypredicate (i.e. alogical statement, that can havefree variables) which satisfiescertain properties.[24] Incomputer science, an equation is defined as aboolean-valuedexpression, orrelational operator, which returns 1 and 0 for true and false respectively.[25]
Anidentity is an equality that is true for all values of its variables in a given domain.[26][27] An "equation" may sometimes mean an identity, but more often than not, itspecifies a subset of the variable space to be the subset where the equation is true. An example is which is true for eachreal number 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.[28] Sometimes, but not always, an identity is written with atriple bar:[29] This notation was introduced byBernhard Riemann in his 1857Elliptische Funktionen lectures (published in 1899).[30][31][32]
Alternatively, identities may be viewed as an equality offunctions, where instead of writing one may simply write[33][34] This is called theextensionality of functions.[35][36] In this sense, the function-application property refers tooperators, operations on afunction space (functions mapping between functions) likecomposition[37] or thederivative, commonly used inoperational calculus.[38] An identity can contain functions as "unknowns", which can be solved for similarly to a regular equation, called afunctional equation.[39] A functional equation involving derivatives is called adifferential equation.[40]
Equations are often used to introduce new terms or symbols for constants,assert equalities, and introduce shorthand for complex expressions, which is called "equal by definition", and often denoted with ().[41] It is similar to the concept ofassignment of avariable in computer science. For example, definesEuler's number,[42] and is the defining property of theimaginary number[43]
Inmathematical logic, this is called anextension by definition (by equality) which is aconservative extension to aformal system.[44] This is done by taking the equation defining the new constant symbol as a newaxiom of thetheory. The first recorded symbolic use of "Equal by definition" appeared inLogica Matematica (1894) byCesare Burali-Forti, an Italian mathematician. Burali-Forti, in his book, used the notation ().[45][46]
Equality is often considered aprimitive notion, informally said to be "a relation each thing bears to itself and to no other thing".[47] This tradition can be traced at least as far back asAristotle, who in hisCategories (c. 350 BC) defines the notion ofquantity in terms of a more primitiveequality (distinct fromidentity orsimilarity), stating:[48]
The most distinctive mark of quantity is that equality and inequality are predicated of it. Each of the aforesaid quantities is said to be equal or unequal. For instance, one solid is said to be equal or unequal to another; number, too, and time can have these terms applied to them, indeed can all those kinds of quantity that have been mentioned.
That which is not a quantity can by no means, it would seem, be termed equal or unequal to anything else. One particular disposition or one particular quality, such as whiteness, is by no means compared with another in terms of equality and inequality but rather in terms of similarity. Thus it is the distinctive mark of quantity that it can be called equal and unequal. ― (translated byE. M. Edghill)
Aristotle had separate categories forquantities (number, length, volume) andqualities (temperature, density, pressure), now calledintensive and extensive properties. TheScholastics, particularlyRichard Swineshead and otherOxford Calculators in the 14th century, began seriously thinking aboutkinematics and quantitative treatment of qualities. For example, two flames have the same heat-intensity if they produce the same effect on water (e.g, warming vsboiling). Since two intensities could be shown to be equal, and equality was considered the defining feature of quantities, it meant those intensities were quantifiable.[49][50]
The precursor to the substitution property of equality was first formulated byGottfried Leibniz in hisDiscourse on Metaphysics (1686), stating, roughly, that "No two distinct things can have all properties in common." This has since broken into two principles, the substitution property (if then any property of is a property of), and itsconverse, theidentity of indiscernibles (if and have all properties in common, then).[51]
Around the turn of the 20th century, it would become necessary to have a more concrete description of equality. In 1879Gottlob Frege would publish his pioneering textBegriffsschrift, which would shift the focus of logic fromAristotelian logic, focused on classes of objects, to being property-based, with what would grow to become modernpredicate logic. This was followed by a movement for describing mathematics in logical foundations, calledlogicism. This trend lead to the axiomatization of equality through thelaw of identity and thesubstitution property especially inmathematical logic[11][24] andanalytic philosophy.[52]
Later, Frege'sFoundations of Arithmetic (1884) andBasic Laws of Arithmetic (1893, 1903) would attempt to derive the foundations of mathematics from the logical system developed in hisBegriffsschrift. This would eventually be shown to be flawed by allowingRussell's paradox, and would contribute to thefoundational crisis of mathematics. The work of Frege would eventually be resolved by a three volume work byBertrand Russell andAlfred Whitehead known asPrincipia Mathematica (1910–1913). Russell and Whitehead's work would also introduce and formalize the Leibniz' Law to symbolic logic, wherein they claim it follows from theiraxiom of reducibility, but credit Leibniz for the idea.[53]
Function application is also sometimes included in the axioms of equality,[13] but isn't necessary as it can be deduced from the other two axioms, and similarly for symmetry and transitivity (see§ Derivations of basic properties). Infirst-order logic, these areaxiom schemas (usually, see below), each of which specify an infinite set of axioms.[56] If a theory has a predicate that satisfies the law of identity and substitution property, it is common to say that it "has equality", or is "a theory with equality".[44]
The use of "equality" here somewhat of amisnomer in that any system with equality can be modeled by a theory without standard identity, and withindiscernibles.[57][56] Those two axioms are strong enough, however, to be isomorphic to a model with identity; that is, if a system has a predicate satisfying those axiomswithout standard equality, there is a model of that systemwith standard equality.[56] This can be done by defining a newdomain whose objects are theequivalence classes of the original "equality".[58] If a model is interpreted to have equality then those properties are enough, since if has all the same properties as and has the property of being equal to then has the property of being equal to[53][59]
As axioms, one candeduce from the first usinguniversal instantiation, and the from second, given and by usingmodus ponens twice. Alternatively, each of these may be included in logic asrules of inference.[56] The first called "equality introduction", and the second "equality elimination"[60] (also calledparamodulation), used by sometheoretical computer scientists likeJohn Alan Robinson in their work onresolution andautomated theorem proving.[61]
The substitution property can produce false statements when applied naively. For example, if denotes "the number of planets in the solar system," then the statement "Johannes Kepler did not know that" is true, sinceUranus andNeptune were discovered after his death. However, since, applying the substitution property gives the statement "Johannes Kepler did not know that" which is false.[62] The difference here is that while the expressions "the number of planets" and "8" refer to the same object (theirextension), they have different meanings (theirintension). Thus, the substitution property can only be guaranteed inextensional contexts, which is guaranteed in modern mathematics by theaxiom of extensionality.[63]
Set theory is the branch of mathematics that studiessets, which can be informally described as "collections of objects".[65] Although objects of any kind can be collected into a set, set theory—as a branch of mathematics—is mostly concerned with those that are relevant to mathematics as a whole.Sets are uniquely characterized by theirelements; this means that two sets that have precisely the same elements are equal (they are the same set).[66] In aformalized set theory, this is usually defined by anaxiom called theAxiom of extensionality.[67]
For example, usingset builder notation, the following states that "The set of allintegers greater than 0 but not more than 3 is equal to the set containing only 1, 2, and 3", despite the differences in formulation.
The termextensionality, as used in'Axiom of Extensionality' has its roots in logic and grammar (cf.Extension (semantics)). In grammar, anintensional definition describes thenecessary and sufficient conditions for a term to apply to an object. For example: "APlatonic solid is aconvex,regular polyhedron inthree-dimensional Euclidean space." An extensional definition instead lists all objects where the term applies. For example: "A Platonic solid is one of the following:Tetrahedron,Cube,Octahedron,Dodecahedron, orIcosahedron." In logic, theextension of apredicate is the set of all objects for which the predicate is true.[68] Further, the logical principle ofextensionality judges two objects to be equal if they satisfy the same external properties. Since, by the axiom, two sets are defined to be equal if they satisfymembership, sets are extentional.[69]
José Ferreirós creditsRichard Dedekind for being the first to explicitly state the principle, although he does not assert it as a definition:[70]
It very frequently happens that different things a, b, c... considered for any reason under a common point of view, are collected together in the mind, and one then says that they form a system S; one calls the things a, b, c... the elements of the system S, they are contained in S; conversely, S consists of these elements. Such a system S (or a collection, a manifold, a totality), as an object of our thought, is likewise a thing; it is completely determined when, for every thing, it is determined whether it is an element of S or not.
— Richard Dedekind, 1888 (translated by José Ferreirós)
Around the turn of the 20th century, mathematics faced severalparadoxes and counter-intuitive results. For example,Russell's paradox showed a contradiction ofnaive set theory, it was shown that theparallel postulate cannot be proved, the existence ofmathematical objects that cannot be computed or explicitly described, and the existence of theorems of arithmetic that cannot be proved withPeano arithmetic. The result was afoundational crisis of mathematics.[72]
The resolution of this crisis involved the rise of a new mathematical discipline calledmathematical logic, which studiesformal logic within mathematics. Discoveries made during the 20th century stabilized the foundations of mathematics, and produced a coherent framework valid for all branches of the discipline. This framework is based on a systematic use ofaxiomatic method and on set theory, specificallyZermelo–Fraenkel set theory, developed byErnst Zermelo andAbraham Fraenkel. This set theory (and set theory in general) is now considered the most commonfoundation of mathematics.[73]
In first-order logic with equality (see§ Axioms), the axiom of extensionality states that two sets thatcontain the same elements are the same set.[74]
The first two are given by the substitution property of equality from first-order logic; the last is a new axiom of the theory. Incorporating half of the work into the first-order logic may be regarded as a mere matter of convenience, as noted byAzriel Lévy:
The reason why we take up first-order predicate calculuswith equality is a matter of convenience; by this, we save the labor of defining equality and proving all its properties; this burden is now assumed by the logic.[75]
In first-order logic without equality, two sets aredefined to be equal if they contain the same elements. Then the axiom of extensionality states that two equal setsare contained in the same sets.[76]
Or, equivalently, one may choose to define equality in a way that mimics, the substitution property explicitly, as theconjunction of allatomic formulas:[77]
In either case, the axiom of extensionality based on first-order logic without equality states that sets which contain the same elements are always contained in the same sets:
Numerical analysis is the study ofconstructive methods andalgorithms to find numericalapproximations (as opposed tosymbolic manipulations) of solutions to problems inmathematical analysis. Especially those which cannot besolved analytically.[80]
Calculations are likely to involverounding errors and otherapproximation errors.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.[81]
If approximate equality is viewed as abinary relation (denoted by the symbol) betweenreal numbers or other things, any rigorous definition of it will not be an equivalence relation, due to its not being transitive. This is the case even when it is modeled as afuzzy relation.[82]
Incomputer science, equality is expressed usingrelational operators. On computers, physical constraints fundamentally limit the level of precision with which numbers can be represented. Thus, the real numbers are often approximated byfloating-point numbers. Each floating-point number is represented as asignificand—comprising some fixed-length sequence of digits in a given base—which is scaled by some integerexponent of said base, in effect enabling theradix point to "float" between each possible location in the significand. This allows numbers spanning many orders of magnitude to be represented, but only as fuzzy ranges of values that become less precise as they increase in magnitude.[83] In order to avoid losing precision, it is common to represent real numbers on computers in the form of anexpression that denotes the real number. However, the equality of two real numbers given by an expression is known to beundecidable (specifically, real numbers defined by expressions involving theintegers, the basicarithmetic operations, thelogarithm and theexponential function). In other words, there cannot exist anyalgorithm for deciding such an equality (seeRichardson's theorem).[84]
Anequivalence relation is amathematical relation that generalizes the idea of similarity or sameness. It is defined on aset as abinary relation that satisfies the three properties:reflexivity,symmetry, andtransitivity. Reflexivity means that every element in is equivalent to itself ( for all). Symmetry requires that if one element is equivalent to another, the reverse also holds (). 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 ( and).[85] These properties are enough topartition a set into disjointequivalence classes. Conversely, every partition defines an equivalence class.[86]
The equivalence relation of equality is a special case, as, if restricted to a given set it is the strictest possible equivalence relation on; specifically, equality partitions a set into equivalence classes consisting of allsingleton sets.[86] Other equivalence relations, since they're less restrictive, generalize equality by identifying elements based on shared properties or transformations, such ascongruence in modular arithmetic orsimilarity in geometry.[87][88]
Inabstract algebra, acongruence relation extends the idea of an equivalence relation to include thefunction-application property. That is, given a set and a set of operations on then a congruence relation has the property that for all operations (here, written as unary to avoid cumbersome notation, but may be of anyarity). A congruence relation on analgebraic structure such as agroup,ring, ormodule is an equivalence relation that respects the operations defined on that structure.[89]
In mathematics, especially inabstract algebra andcategory theory, it is common to deal with objects that already have some internalstructure. Anisomorphism describes a kind of structure-preserving correspondence between two objects, establishing them as essentially identical in their structure or properties.[90][91]
More formally, an isomorphism is a bijectivemapping (ormorphism) between twosets or structures and such that and its inverse preserve theoperations,relations, orfunctions defined on those structures.[90] This means that any operation or relation valid in corresponds precisely to the operation or relation in under the mapping. For example, ingroup theory, agroup isomorphism satisfies for all elements where denotes the group operation.[92]
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, allcyclic groups of order are isomorphic to the integers, with addition.[93] Similarly, inlinear algebra, twovector spaces are isomorphic if they have the samedimension, as there exists alinear bijection between their elements.[94]
The concept of isomorphism extends to numerous branches of mathematics, includinggraph theory (graph isomorphism),topology (homeomorphism), and algebra (group andring isomorphisms), 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 ofcategory theory, as well as forhomotopy type theory andunivalent foundations.[95][96][97]
Ingeometry, formally, two figures are equal if they contain exactly the samepoints. However, historically, geometric-equality has always been taken to be much broader.Euclid andArchimedes used "equal" (ἴσοςisos) often referring to figures with the same area or those that could be cut and rearranged to form one another. For example, Euclid stated thePythagorean 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."[98] (SeeArea of a circle § Rearrangement proof.)
This notion persisted untilAdrien-Marie Legendre introduced the term "equivalent" in 1867 to describe figures of equal area, and reserved "equal" to mean "congruent"—the sameshape andsize, or if one has the same shape and size as themirror image of the other.[99][100] Euclid's terminology continued in the work ofDavid Hilbert in hisGrundlagen der Geometrie, who further refined Euclid's ideas by introducing the notions of polygons being "divisibly equal" (zerlegungsgleich) if they can be cut into finitely many triangles which are congruent, and "equal in content" (inhaltsgleichheit) if one can add finitely many divisibly equal polygons to each such that the resulting polygons are divisibly equal.[101]
After the rise of set theory, around the 1960s, there was a push for a reform inmathematics education called "New Math", followingAndrey Kolmogorov, who, in an effort to restructure Russian geometry courses, proposed presenting geometry through the lens oftransformations 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.[102]
While Euclid addressedproportionality and figures of the same shape, it was not until the 17th century that the concept ofsimilarity 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 byscaling and congruence.[103] Later a concept of equality ofdirected line segments,equipollence, was advanced byGiusto Bellavitis in 1835.[104]
A relation between two quantities or other mathematical expressions stating that the two are the same; (also) an expression of such a relation by means of symbols, an equation.
A formula affirming the equivalence of two quantitative expressions, which are for this purpose connected by the sign =.
A set is a collection of definite distinct objects of our perception or of our thought, which are called elements of the set.
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