Equality (mathematics)


In mathematics, equality is a relationship between two quantities or expressions, stating that they have the same value, or represent the same mathematical object. Equality between and is denoted with an equals sign as, and read " equals ". A written expression of equality is called an equation or identity depending on the context. Two objects that are not equal are said to be distinct.
Equality is often considered a primitive 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, reflecting a general conceptual difficulty in fully characterizing the concept. Basic properties about equality like reflexivity, symmetry, and transitivity have been understood intuitively since at least the ancient Greeks, but were not symbolically stated as general properties of relations until the late 19th century by Giuseppe Peano. Other properties like substitution and function application weren't formally stated until the development of symbolic logic.
There are generally two ways that equality is formalized in mathematics: through logic or through set theory. In logic, equality is a primitive predicate with the reflexive property, and the substitution property. From those, one can derive the rest of the properties usually needed for equality. After the foundational crisis in mathematics at the turn of the 20th century, set theory became the most common foundation of mathematics. In set theory, any two sets are defined to be equal if they have all the same members. This is called the axiom of extensionality.

Etymology

In English, the word equal is derived from the Latin aequālis, which itself stems from aequus. The word entered Middle English around the 14th century, borrowed from Old French equalité. More generally, the interlingual synonyms of equal have been used more broadly throughout history.
Before the 16th century, there was no common symbol for equality, and equality was usually expressed with a word, such as aequales, aequantur, esgale, faciunt, ghelijck, or gleich, and sometimes by the abbreviated form aeq, or simply and. Diophantus's use of, short for ἴσος, in Arithmetica is considered one of the first uses of an equals sign.
The sign, now universally accepted in mathematics for equality, was first recorded by Welsh mathematician Robert Recorde in The Whetstone of Witte, 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 Latin gemellus, using two parallel lines to represent equality because he believed that "no two things could be more equal."
Recorde's symbol was not immediately popular. After its introduction, it wasn't used again in print until 1618, in an anonymous Appendix in Edward Wright's English translation of Descriptio, by John 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, both Isaac Newton and Gottfried Leibniz, and due to the prevalence of calculus at the time, it quickly spread throughout the rest of Europe.

Basic properties

; Reflexivity
; Symmetry
; Transitivity
; Substitution
; Function application
The first three properties are generally attributed to Giuseppe Peano for being the first to explicitly state these as fundamental properties of equality in his Arithmetices principia, nova methodo exposita. However, the basic notions have always existed; for example, in Euclid's Elements, he includes 'common notions': "Things that are equal to the same thing are also equal to one another", "Things that coincide with one another are equal to one another", along with some function-application properties for addition and subtraction. The function-application property was also stated in Peano's Arithmetices principia, however, it had been common practice in algebra since at least Diophantus. The substitution property is generally attributed to Gottfried Leibniz, and often called Leibniz's Law.

Equations

An equation is a symbolic equality of two mathematical expressions connected with an equals sign. Algebra is the branch of mathematics concerned with equation solving: the problem of finding values of some variable, called, for which the specified equality is true. Each value of the unknown for which the equation holds is called a of the given equation; also stated as the equation. For example, the equation has the values and as its only solutions. The terminology is used similarly for equations with several unknowns. The set of solutions to an equation or system of equations is called its solution set.
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 diagrams. 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.
Often, equations are considered to be a statement, or relation, which can be true or false. For example, is true, and is false. Equations with unknowns are considered conditionally true; for example, is true when or and false otherwise. There are several different terminologies for this. In mathematical logic, an equation is a binary predicate which satisfies certain properties. In computer science, an equation is defined as a boolean-valued expression, or relational operator, which returns 1 and 0 for true and false respectively.

Identities

An identity is an equality that is true for all values of its variables in a given domain. An "equation" may sometimes mean an identity, but more often than not, it a subset of the variable space to be the subset where the equation is true. An example is which is true for each real 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. Sometimes, but not always, an identity is written with a triple bar: This notation was introduced by Bernhard Riemann in his 1857 Elliptische Funktionen lectures.
Alternatively, identities may be viewed as an equality of functions, where instead of writing one may simply write This is called the extensionality of functions. In this sense, the function-application property refers to operators, operations on a function space like composition or the derivative, commonly used in operational calculus. An identity can contain functions as "unknowns", which can be solved for similarly to a regular equation, called a functional equation. A functional equation involving derivatives is called a differential equation.

Definitions

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. It is similar to the concept of assignment of a variable in computer science. For example, defines Euler's number, and is the defining property of the imaginary number
In mathematical logic, this is called an extension by definition which is a conservative extension to a formal system. This is done by taking the equation defining the new constant symbol as a new axiom of the theory. The first recorded symbolic use of "Equal by definition" appeared in Logica Matematica by Cesare Burali-Forti, an Italian mathematician. Burali-Forti, in his book, used the notation.

In logic

History

Equality is often considered a primitive notion, informally said to be "a relation each thing bears to itself and to no other thing". This tradition can be traced at least as far back as Aristotle, who in his Categories defines the notion of quantity in terms of a more primitive equality, stating:
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. ― 

Aristotle had separate categories for quantities and qualities, now called intensive and extensive properties. The Scholastics, particularly Richard Swineshead and other Oxford Calculators in the 14th century, began seriously thinking about kinematics and quantitative treatment of qualities. For example, two flames have the same heat-intensity if they produce the same effect on water. Since two intensities could be shown to be equal, and equality was considered the defining feature of quantities, it meant those intensities were quantifiable.
The precursor to the substitution property of equality was first formulated by Gottfried Leibniz in his Discourse on Metaphysics, stating, roughly, that "No two distinct things can have all properties in common." This has since broken into two principles, the substitution property, and its converse, the identity of indiscernibles.
Around the turn of the 20th century, it would become necessary to have a more concrete description of equality. In 1879 Gottlob Frege would publish his pioneering text Begriffsschrift, which would shift the focus of logic from Aristotelian logic, focused on classes of objects, to being property-based, with what would grow to become modern predicate logic. This was followed by a movement for describing mathematics in logical foundations, called logicism. This trend lead to the axiomatization of equality through the law of identity and the substitution property especially in mathematical logic and analytic philosophy.
Later, Frege's Foundations of Arithmetic and Basic Laws of Arithmetic would attempt to derive the foundations of mathematics from the logical system developed in his Begriffsschrift. This would eventually be shown to be flawed by allowing Russell's paradox, and would contribute to the foundational crisis of mathematics. The work of Frege would eventually be resolved by a three volume work by Bertrand Russell and Alfred Whitehead known as Principia Mathematica. Russell and Whitehead's work would also introduce and formalize the Leibniz' Law to symbolic logic, wherein they claim it follows from their axiom of reducibility, but credit Leibniz for the idea.