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 A and B is denoted with an equals sign as A = B, and read "A equals B". 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 ("nothing else"), 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 (a statement that may have free variables) with the reflexive property (called the law of identity), 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 (specifically Zermelo–Fraenkel 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.
Contents
Etymology
In English, the word equal is derived from the Latin aequālis ('like', 'comparable', 'similar'), which itself stems from aequus ('level', 'just'). The word entered Middle English around the 14th century, borrowed from Old French equalité (modern égalité). More generally, the interlingual synonyms of equal 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 as aequales, aequantur, esgale, faciunt, ghelijck, or gleich, and sometimes by the abbreviated form aeq, or simply ⟨æ⟩ and ⟨œ⟩. Diophantus's use of ⟨ἴσ⟩, short for ἴσος (ísos 'equals'), in Arithmetica (c. 250 AD) 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 (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 Latin gemellus ('twin'), 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 (61 years later), 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
For every a, one has a = a.
Symmetry
For every a and b, if a = b, then b = a.
Transitivity
For every a, b, and c, if a = b and b = c, then a = c.
Substitution
Informally, this just means that if a = b, then a can replace b in any mathematical expression or formula without changing its meaning. (For a formal explanation, see § Axioms) For example:
Function application
For every a and b, with some function
f
(
x
)
,
{\displaystyle f(x),}
if a = b, then
f
(
a
)
=
f
(
b
)
.
{\displaystyle f(a)=f(b).}
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 unknown, for which the specified equality is true. Each value of the unknown for which the equation holds is called a solution of the given equation; also stated as satisfying the equation. For example, the equation
x
2
−
6
x
+
5
=
0
{\displaystyle x^{2}-6x+5=0}
has the values
x
=
1
{\displaystyle x=1}
and
x
=
5
{\displaystyle x=5}
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.
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 specifies a subset of the variable space to be the subset where the equation is true. An example is
(
x
+
1
)
(
x
+
1
)
=
x
2
+
2
x
+
1
,
{\displaystyle \left(x+1\right)\left(x+1\right)=x^{2}+2x+1,}
which is true for each real number
x
.
{\displaystyle x.}
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:
(
x
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 (
:=
{\displaystyle :=}
). It is similar to the concept of assignment of a variable in computer science. For example,
e
:=
∑
n
=
0
∞
1
n
!
{\textstyle \mathbb {e} :=\sum _{n=0}^{\infty }{\frac {1}{n!}}}
defines Euler's number, and
i
2
=
−
1
{\displaystyle i^{2}=-1}
is the defining property of the imaginary number
i
.
{\displaystyle i.}
In mathematical logic, this is called an extension by definition (by equality) 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 (1894) 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 (c. 350 BC) defines the notion of quantity in terms of a more primitive equality (distinct from identity or similarity), 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. ― (translated by E. M. Edghill)
Aristotle had separate categories for quantities (number, length, volume) and qualities (temperature, density, pressure), 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 (e.g., warming vs boiling). 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 (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
x
=
y
,
{\displaystyle x=y,}
Axioms
Law of identity: Stating that each thing is identical with itself, without restriction. That is, for every
a
,
{\displaystyle a,}
a
=
a
.
{\displaystyle a=a.}
It is the first of the traditional three laws of thought.The above can be stated symbolically as:
∀
a
(
a
=
a
)
.
{\displaystyle \forall a(a=a).}
Substitution property: Generally stating that if two things are equal, then any property of one must be a property of the other. It is sometimes referred to as "Leibniz's law".It can be stated formally as: for every a and b, and any formula
ϕ
(
x
)
,
{\displaystyle \phi (x),}
with a free variable x, if
a
Derivations of basic properties
Reflexivity: Given any expression
a
,
{\displaystyle a,}
by the law of identity,
a
=
a
.
{\displaystyle a=a.}
Symmetry: Given
a
=
b
,
{\displaystyle a=b,}
take the formula
ϕ
(
x
)
:
x
=
a
.
{\displaystyle \phi (x):x=a.}
Accordingly,
(
a
=
b
)
⟹
In set theory
Set theory is the branch of mathematics that studies sets, which can be informally described as "collections of objects". 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 their elements; this means that two sets that have precisely the same elements are equal (they are the same set). In a formalized set theory, this is usually defined by an axiom called the Axiom of extensionality.
For example, using set builder notation, the following states that "The set of all integers
(
Z
)
{\displaystyle (\mathbb {Z} )}
greater than 0 but not more than 3 is equal to the set containing only 1, 2, and 3", despite the differences in formulation.
{
x
∈
Z
∣
0
<
x
≤
3
}
=
{
1
,
2
Background
Around the turn of the 20th century, mathematics faced several paradoxes and counter-intuitive results. For example, Russell's paradox showed a contradiction of naive set theory, it was shown that the parallel postulate cannot be proved, the existence of mathematical objects that cannot be computed or explicitly described, and the existence of theorems of arithmetic that cannot be proved with Peano arithmetic. The result was a foundational crisis of mathematics.
The resolution of this crisis involved the rise of a new mathematical discipline called mathematical logic, which studies formal 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 of axiomatic method and on set theory, specifically Zermelo–Fraenkel set theory, developed by Ernst Zermelo and Abraham Fraenkel. This set theory (and set theory in general) is now considered the most common foundation of mathematics.
Set equality based on first-order logic with equality
In first-order logic with equality (see § Axioms), the axiom of extensionality states that two sets that contain the same elements are the same set.
Logic axiom:
x
=
y
⟹
∀
z
,
(
z
∈
x
⟺
z
∈
y
)
{\displaystyle x=y\implies \forall z,(z\in x\iff z\in y)}
Logic axiom:
x
=
y
⟹
∀
z
,
(
x
∈
z
⟺
y
Set equality based on first-order logic without equality
In first-order logic without equality, two sets are defined to be equal if they contain the same elements. Then the axiom of extensionality states that two equal sets are contained in the same sets.
Set theory definition:
(
x
=
y
)
:=
∀
z
,
(
z
∈
x
⟺
z
∈
y
)
{\displaystyle (x=y)\ :=\ \forall z,(z\in x\iff z\in y)}
Set theory axiom:
x
=
y
⟹
∀
z
,
(
x
∈
Proof of basic properties
Reflexivity: Given a set
X
,
{\displaystyle X,}
assume
z
∈
X
,
{\displaystyle z\in X,}
it follows trivially that
z
∈
X
,
{\displaystyle z\in X,}
and the same follows in reverse, thus
∀
z
,
(
z
∈
X
⟺
z
∈
X
)
,
{\displaystyle \forall z,(z\in X\iff z\in X),}
therefore
X
=
Similar relations
Approximate equality
Numerical analysis is the study of constructive methods and algorithms to find numerical approximations (as opposed to symbolic manipulations) of solutions to problems in mathematical analysis. Especially those which cannot be solved analytically.
Calculations are likely to involve rounding errors and other approximation 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.
If approximate equality is viewed as a binary relation (denoted by the symbol
≈
{\displaystyle \approx }
) between real 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 a fuzzy relation.
In computer science, equality is expressed using relational operators. On computers, physical constraints fundamentally limit the level of precision with which numbers can be represented. Thus, the real numbers are often approximated by floating-point numbers. Each floating-point number is represented as a significand—comprising some fixed-length sequence of digits in a given base—which is scaled by some integer exponent of said base, in effect enabling the radix 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. In order to avoid losing precision, it is common to represent real numbers on computers in the form of an expression that denotes the real number. However, the equality of two real numbers given by an expression is known to be undecidable (specifically, real numbers defined by expressions involving the integers, the basic arithmetic operations, the logarithm and the exponential function). In other words, there cannot exist any algorithm for deciding such an equality (see Richardson's theorem).
Equivalence relation
An equivalence relation is a mathematical relation that generalizes the idea of similarity or sameness. It is defined on a set
X
{\displaystyle X}
as a binary relation
∼
{\displaystyle \sim }
that satisfies the three properties: reflexivity, symmetry, and transitivity. Reflexivity means that every element in
X
{\displaystyle X}
is equivalent to itself (
a
∼
a
{\displaystyle a\sim a}
for all
a
∈
X
{\displaystyle a\in X}
). Symmetry requires that if one element is equivalent to another, the reverse also holds (
a
∼
b
⟹
b
∼
a
{\displaystyle a\sim b\implies b\sim a}
Isomorphism
In mathematics, especially in abstract algebra and category theory, it is common to deal with objects that already have some internal structure. An isomorphism describes a kind of structure-preserving correspondence between two objects, establishing them as essentially identical in their structure or properties.
More formally, an isomorphism is a bijective mapping (or morphism)
f
{\displaystyle f}
between two sets or structures
A
{\displaystyle A}
and
B
{\displaystyle B}
such that
f
{\displaystyle f}
and its inverse
f
−
1
{\displaystyle f^{-1}}
preserve the operations, relations, or functions defined on those structures. This means that any operation or relation valid in
A
{\displaystyle A}
corresponds precisely to the operation or relation in
B
{\displaystyle B}
under the mapping. For example, in group theory, a group isomorphism
Geometry
In geometry, formally, two figures are equal if they contain exactly the same points. However, historically, geometric-equality has always been taken to be much broader. Euclid and Archimedes used "equal" (ἴσος 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." (See Area of a circle § Rearrangement proof.)
This notion persisted until Adrien-Marie Legendre introduced the term "equivalent" in 1867 to describe figures of equal area, and reserved "equal" to mean "congruent"—the same shape and size, or if one has the same shape and size as the mirror image of the other. Euclid's terminology continued in the work of David Hilbert in his Grundlagen 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.
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 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.
While Euclid addressed proportionality and figures of the same shape, it was not until the 17th century that the concept of 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 and congruence. Later a concept of equality of directed line segments, equipollence, was advanced by Giusto Bellavitis in 1835.