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The equals sign (
British English British English (BrE, en-GB, or BE) is, according to Oxford Dictionaries, "English as used in Great Britain, as distinct from that used elsewhere". More narrowly, it can refer specifically to the English language in England, or, more broadl ...
,
Unicode Unicode, formally The Unicode Standard,The formal version reference is is an information technology standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems. The standard, ...
) or equal sign (
American English American English, sometimes called United States English or U.S. English, is the set of varieties of the English language native to the United States. English is the most widely spoken language in the United States and in most circumstances ...
), also known as the equality sign, is the
mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formul ...
, which is used to indicate equality in some
well-defined In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A func ...
sense. In an
equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in F ...
, it is placed between two expressions that have the same value, or for which one studies the conditions under which they have the same value. In
Unicode Unicode, formally The Unicode Standard,The formal version reference is is an information technology standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems. The standard, ...
and
ASCII ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding standard for electronic communication. ASCII codes represent text in computers, telecommunications equipment, and other devices. Because ...
, it has the
code point In character encoding terminology, a code point, codepoint or code position is a numerical value that maps to a specific character. Code points usually represent a single grapheme—usually a letter, digit, punctuation mark, or whitespace—but ...
U+003D. It was invented in 1557 by
Robert Recorde Robert Recorde () was an Anglo-Welsh physician and mathematician. He invented the equals sign (=) and also introduced the pre-existing plus sign (+) to English speakers in 1557. Biography Born around 1512, Robert Recorde was the second and las ...
.


History

The etymology of the word "equal" is from the Latin word "''æqualis",'' as meaning "uniform", "identical", or "equal", from ''aequus'' ("level", "even", or "just"). The symbol, now universally accepted in mathematics for equality, was first recorded by Welsh mathematician
Robert Recorde Robert Recorde () was an Anglo-Welsh physician and mathematician. He invented the equals sign (=) and also introduced the pre-existing plus sign (+) to English speakers in 1557. Biography Born around 1512, Robert Recorde was the second and las ...
in ''
The Whetstone of Witte ''The Whetstone of Witte'' is the shortened title of Robert Recorde's mathematics book published in 1557, the full title being ''The whetstone of , is the : The ''Coßike'' practise, with the rule of ''Equation'': and the of ''Surde Nombers. ...
'' (1557). The original form of the symbol was much wider than the present form. In his book Recorde explains his design of the "Gemowe lines" (meaning ''twin'' lines, from the
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
'' gemellus'')See also
geminus Geminus of Rhodes ( el, Γεμῖνος ὁ Ῥόδιος), was a Greek astronomer and mathematician, who flourished in the 1st century BC. An astronomy work of his, the ''Introduction to the Phenomena'', still survives; it was intended as an int ...
and Gemini.
"The symbol was not immediately popular. The symbol was used by some and (or ), from the Latin word meaning equal, was widely used into the 1700s" (''History of Mathematics'',
University of St Andrews (Aien aristeuein) , motto_lang = grc , mottoeng = Ever to ExcelorEver to be the Best , established = , type = Public research university Ancient university , endowment ...
).


Usage in mathematics and computer programming

In mathematics, the equal sign can be used as a simple statement of fact in a specific case (), or to create definitions (), conditional statements (), or to express a universal equivalence (). The first important
computer programming language A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language. The description of a programming l ...
to use the equal sign was the original version of Fortran, FORTRAN I, designed in 1954 and implemented in 1957. In Fortran, serves as an
assignment Assignment, assign or The Assignment may refer to: * Homework * Sex assignment * The process of sending National Basketball Association players to its development league; see Computing * Assignment (computer science), a type of modification to ...
operator: sets the value of to 2. This somewhat resembles the use of in a mathematical definition, but with different semantics: the expression following is evaluated first, and may refer to a previous value of . For example, the assignment increases the value of by 2. A rival programming-language usage was pioneered by the original version of
ALGOL ALGOL (; short for "Algorithmic Language") is a family of imperative computer programming languages originally developed in 1958. ALGOL heavily influenced many other languages and was the standard method for algorithm description used by the ...
, which was designed in 1958 and implemented in 1960. ALGOL included a
relational operator In computer science, a relational operator is a programming language construct or operator that tests or defines some kind of relation between two entities. These include numerical equality (''e.g.'', ) and inequalities (''e.g.'', ). In pr ...
that tested for equality, allowing constructions like with essentially the same meaning of as the conditional usage in mathematics. The equal sign was reserved for this usage. Both usages have remained common in different programming languages into the early 21st century. As well as Fortran, is used for assignment in such languages as C,
Perl Perl is a family of two high-level, general-purpose, interpreted, dynamic programming languages. "Perl" refers to Perl 5, but from 2000 to 2019 it also referred to its redesigned "sister language", Perl 6, before the latter's name was offic ...
, Python, awk, and their descendants. But is used for equality and not assignment in the
Pascal Pascal, Pascal's or PASCAL may refer to: People and fictional characters * Pascal (given name), including a list of people with the name * Pascal (surname), including a list of people and fictional characters with the name ** Blaise Pascal, Frenc ...
family,
Ada Ada may refer to: Places Africa * Ada Foah, a town in Ghana * Ada (Ghana parliament constituency) * Ada, Osun, a town in Nigeria Asia * Ada, Urmia, a village in West Azerbaijan Province, Iran * Ada, Karaman, a village in Karaman Province, T ...
, Eiffel, APL, and other languages. A few languages, such as
BASIC BASIC (Beginners' All-purpose Symbolic Instruction Code) is a family of general-purpose, high-level programming languages designed for ease of use. The original version was created by John G. Kemeny and Thomas E. Kurtz at Dartmouth College ...
and
PL/I PL/I (Programming Language One, pronounced and sometimes written PL/1) is a procedural, imperative computer programming language developed and published by IBM. It is designed for scientific, engineering, business and system programming. I ...
, have used the equal sign to mean both assignment and equality, distinguished by context. However, in most languages where has one of these meanings, a different character or, more often, a sequence of characters is used for the other meaning. Following ALGOL, most languages that use for equality use for assignment, although APL, with its special character set, uses a left-pointing arrow. Fortran did not have an equality operator (it was only possible to compare an expression to zero, using the arithmetic IF statement) until FORTRANIV was released in 1962, since when it has used the four characters to test for equality. The language B introduced the use of with this meaning, which has been copied by its descendant C and most later languages where means assignment. The equal sign is also used in defining attribute–value pairs, in which an attribute is assigned a
value Value or values may refer to: Ethics and social * Value (ethics) wherein said concept may be construed as treating actions themselves as abstract objects, associating value to them ** Values (Western philosophy) expands the notion of value beyo ...
.


Several equal signs

In PHP, the triple equal sign, , denotes value and type equality, meaning that not only do the two expressions evaluate to equal values, but they are also of the same data type. For instance, the expression is true, but is not, because the number 0 is an integer value whereas false is a Boolean value.
JavaScript JavaScript (), often abbreviated as JS, is a programming language that is one of the core technologies of the World Wide Web, alongside HTML and CSS. As of 2022, 98% of websites use JavaScript on the client side for webpage behavior, of ...
has the same semantics for , referred to as "equality without type coercion". However, in JavaScript the behavior of cannot be described by any simple consistent rules. The expression is true, but is false, even though both sides of the act the same in Boolean context. For this reason it is sometimes recommended to avoid the operator in JavaScript in favor of . In Ruby, equality under requires both operands to be of identical type, e.g. is false. The operator is flexible and may be defined arbitrarily for any given type. For example, a value of type is a range of integers, such as . is false, since the types are different (Range vs. Integer); however is true, since on values means "inclusion in the range". Under these semantics,

is non-symmetric; e.g. 1844 

 (1800..1899)
is false, since it is interpreted to mean Integer#

rather than Range#

. In most programming languages,

is used to check equality, so 1844 

1844
will return true.


Other uses


Spelling


Tone letter

The equal sign is also used as a grammatical
tone letter Tone letters are letters that represent the tones of a language, most commonly in languages with contour tones. __TOC__ Chao tone letters (IPA) A series of iconic tone letters based on a musical staff was devised by Yuen Ren Chao in the ...
in the orthographies of Budu in the
Congo-Kinshasa The Democratic Republic of the Congo (french: République démocratique du Congo (RDC), colloquially "La RDC" ), informally Congo-Kinshasa, DR Congo, the DRC, the DROC, or the Congo, and formerly and also colloquially Zaire, is a country in ...
, in Krumen, Mwan and
Dan Dan or DAN may refer to: People * Dan (name), including a list of people with the name ** Dan (king), several kings of Denmark * Dan people, an ethnic group located in West Africa **Dan language, a Mande language spoken primarily in Côte d'Ivoir ...
in the
Ivory Coast Ivory Coast, also known as Côte d'Ivoire, officially the Republic of Côte d'Ivoire, is a country on the southern coast of West Africa. Its capital is Yamoussoukro, in the centre of the country, while its largest city and economic centre i ...
. The Unicode character used for the tone letter (U+A78A) is different from the mathematical symbol (U+003D).


Personal names

A possibly unique case of the equal sign of European usage in a person's name, specifically in a
double-barreled name A double-barrelled name is a type of compound surname, typically featuring two words (occasionally more), often joined by a hyphen. Examples of some notable people with double-barrelled names include Winnie Madikizela-Mandela and Sacha Baron Co ...
, was by pioneer aviator
Alberto Santos-Dumont Alberto Santos-Dumont ( Palmira, 20 July 1873 — Guarujá, 23 July 1932) was a Brazilian aeronaut, sportsman, inventor, and one of the few people to have contributed significantly to the early development of both lighter-than-air and heavie ...
, as he is also known not only to have often used a
double hyphen In Latin script, the double hyphen is a punctuation mark that consists of two parallel hyphens. It was a development of the earlier , which developed from a Central European variant of the virgule slash, originally a form of scratch comma. S ...
resembling an equal sign between his two surnames in place of a hyphen, but also seems to have personally preferred that practice, to display equal respect for his father's French ethnicity and the Brazilian ethnicity of his mother. Instead of a double hyphen, the equal sign is sometimes used in Japanese as a separator between names. In Ojibwe, the readily available equal sign on a keyboard is used as a substitute for a double hyphen.


Linguistics

In linguistic
interlinear gloss In linguistics and pedagogy, an interlinear gloss is a gloss (series of brief explanations, such as definitions or pronunciations) placed between lines, such as between a line of original text and its translation into another language. When gloss ...
es, an equal sign is conventionally used to mark clitic boundaries: the equal sign is placed between the
clitic In morphology and syntax, a clitic (, backformed from Greek "leaning" or "enclitic"Crystal, David. ''A First Dictionary of Linguistics and Phonetics''. Boulder, CO: Westview, 1980. Print.) is a morpheme that has syntactic characteristics of a ...
and the word that the clitic is attached to.


Chemistry

In
chemical formula In chemistry, a chemical formula is a way of presenting information about the chemical proportions of atoms that constitute a particular chemical compound or molecule, using chemical element symbols, numbers, and sometimes also other symbols, ...
s, the two parallel lines denoting a
double bond In chemistry, a double bond is a covalent bond between two atoms involving four bonding electrons as opposed to two in a single bond. Double bonds occur most commonly between two carbon atoms, for example in alkenes. Many double bonds exist betwee ...
are commonly rendered using an equal sign.


LGBT symbol

In recent years, the equal sign has been used to symbolize
LGBT rights Rights affecting lesbian, gay, bisexual, and transgender ( LGBT) people vary greatly by country or jurisdiction—encompassing everything from the legal recognition of same-sex marriage to the death penalty for homosexuality. Notably, ...
. The symbol has been used since 1995 by the
Human Rights Campaign The Human Rights Campaign (HRC) is an American LGBTQ advocacy group. It is the largest LGBTQ political lobbying organization within the United States. Based in Washington, D.C., the organization focuses on protecting and expanding rights for ...
, which lobbies for
marriage equality Same-sex marriage, also known as gay marriage, is the marriage of two people of the same sex or gender. marriage between same-sex couples is legally performed and recognized in 33 countries, with the most recent being Mexico, constituting ...
, and subsequently by the United Nations Free & Equal, which promotes LGBT rights at the United Nations.


Hate speech

The not equal (≠) symbol has been adopted by some
white supremacist White supremacy or white supremacism is the belief that white people are superior to those of other races and thus should dominate them. The belief favors the maintenance and defense of any power and privilege held by white people. White s ...
and other racist groups.


Telegrams and Telex

In
Morse code Morse code is a method used in telecommunication to encode text characters as standardized sequences of two different signal durations, called ''dots'' and ''dashes'', or ''dits'' and ''dahs''. Morse code is named after Samuel Morse, one ...
, the equal sign is encoded by the letters B (-...) and T (-) run together (-...-). The letters BT stand for Break Text, and are put between paragraphs, or groups of paragraphs in messages sent via
Telex The telex network is a station-to-station switched network of teleprinters similar to a telephone network, using telegraph-grade connecting circuits for two-way text-based messages. Telex was a major method of sending written messages electroni ...
, a standardised tele-typewriter. The sign, used to mean Break Text, is given at the end of a
telegram Telegraphy is the long-distance transmission of messages where the sender uses symbolic codes, known to the recipient, rather than a physical exchange of an object bearing the message. Thus flag semaphore is a method of telegraphy, whereas ...
to separate the text of the message from the signature.


Related symbols


Approximately equal

Symbols used to denote items that are '' approximately equal'' include the following: * (,
LaTeX Latex is an emulsion (stable dispersion) of polymer microparticles in water. Latexes are found in nature, but synthetic latexes are common as well. In nature, latex is found as a milky fluid found in 10% of all flowering plants (angiosperms ...
''\approx'') * (, LaTeX ''\simeq''), a combination of and , also used to indicate asymptotic equality * (, LaTeX ''\cong''), another combination of ≈ and =, which is also sometimes used to indicate
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
or congruence * (, LaTeX ''\sim''), which is also sometimes used to indicate proportionality or similarity, being related by an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
, or to indicate that a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
is distributed according to a specific
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
(see also
tilde The tilde () or , is a grapheme with several uses. The name of the character came into English from Spanish, which in turn came from the Latin '' titulus'', meaning "title" or "superscription". Its primary use is as a diacritic (accent) i ...
) * (, LaTex ''\backsim''), which is also used to indicate proportionality * (, LaTeX ''\doteq''), which can also be used to represent the approach of a variable to a
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
* (, LaTeX ''\fallingdotseq''), commonly used in Japan, Taiwan, and Korea. * (, LaTex ''\risingdotseq'') In some areas of East Asia such as Japan, "≒" is used to mean "the two terms are almost equal", but in other areas and specialized literature such as mathematics, "≃" is often used. In addition to its mathematical meaning, it is sometimes used in Japanese sentences with the intention of "almost the same."


Not equal

The symbol used to denote
inequation In mathematics, an inequation is a statement that an inequality holds between two values. It is usually written in the form of a pair of expressions denoting the values in question, with a relational sign between them indicating the specific in ...
(when items are not equal) is a slashed equal sign (U+2260). In
LaTeX Latex is an emulsion (stable dispersion) of polymer microparticles in water. Latexes are found in nature, but synthetic latexes are common as well. In nature, latex is found as a milky fluid found in 10% of all flowering plants (angiosperms ...
, this is done with the "\neq" command. Most programming languages, limiting themselves to the
7-bit ASCII ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding standard for electronic communication. ASCII codes represent text in computers, telecommunications equipment, and other devices. Because of ...
character set Character encoding is the process of assigning numbers to graphical characters, especially the written characters of human language, allowing them to be stored, transmitted, and transformed using digital computers. The numerical values tha ...
and typeable characters, use , , , or to represent their Boolean
inequality operator In computer science, a relational operator is a programming language construct or operator that tests or defines some kind of relation between two entities. These include numerical equality (''e.g.'', ) and inequalities (''e.g.'', ). In p ...
.


Identity

The
triple bar The triple bar, or tribar ≡, is a symbol with multiple, context-dependent meanings. It has the appearance of an equals sign  sign with a third line. The triple bar character in Unicode is code point .. The closely related code point ...
symbol (U+2261, LaTeX ''\equiv'') is often used to indicate an
identity Identity may refer to: * Identity document * Identity (philosophy) * Identity (social science) * Identity (mathematics) Arts and entertainment Film and television * ''Identity'' (1987 film), an Iranian film * ''Identity'' (2003 film), an ...
, a
definition A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definiti ...
(which can also be represented by or ), or a
congruence relation In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done ...
in
modular arithmetic In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his boo ...
.


Isomorphism

The symbol is often used to indicate
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
algebraic structures or
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...
geometric figures.


In logic

Equality of
truth value In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false''). Computing In some pro ...
s (through bi-implication or
logical equivalence In logic and mathematics, statements p and q are said to be logically equivalent if they have the same truth value in every model. The logical equivalence of p and q is sometimes expressed as p \equiv q, p :: q, \textsfpq, or p \iff q, depending o ...
), may be denoted by various symbols including , , and .


Other related symbols

Additional precomposed symbols with
code point In character encoding terminology, a code point, codepoint or code position is a numerical value that maps to a specific character. Code points usually represent a single grapheme—usually a letter, digit, punctuation mark, or whitespace—but ...
s in Unicode for notations related to the equal sign include: * () * () (see also
assignment (computer science) In computer programming, an assignment statement sets and/or re-sets the value stored in the storage location(s) denoted by a variable name; in other words, it copies a value into the variable. In most imperative programming languages, the as ...
for ) * () * () * () * () * () * () * () * () * () * () * () (see also
Backus–Naur form In computer science, Backus–Naur form () or Backus normal form (BNF) is a metasyntax notation for context-free grammars, often used to describe the syntax of languages used in computing, such as computer programming languages, document format ...
for ) * () * ()


Incorrect usage

The equal sign is sometimes used incorrectly within a mathematical argument to connect math steps in a non-standard way, rather than to show equality (especially by early mathematics students). For example, if one were finding the sum, step by step, of the numbers 1, 2, 3, 4, and 5, one might incorrectly write :1 + 2 = 3 + 3 = 6 + 4 = 10 + 5 = 15. Structurally, this is shorthand for :( 1 + 2 = 3) + 3 = 6+ 4 = 10) + 5 = 15, but the notation is incorrect, because each part of the equality has a different value. If interpreted strictly as it says, it would imply that :3 = 6 = 10 = 15 = 15. A correct version of the argument would be :1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10, 10 + 5 = 15. This difficulty results from subtly different uses of the sign in education. In early, arithmetic-focused grades, the equal sign may be ''operational''; like the equal button on an electronic calculator, it demands the result of a calculation. Starting in algebra courses, the sign takes on a ''relational'' meaning of equality between two calculations. Confusion between the two uses of the sign sometimes persists at the university level.


Encodings

* Related: *


See also

* 2 + 2 = 5 *
Double hyphen In Latin script, the double hyphen is a punctuation mark that consists of two parallel hyphens. It was a development of the earlier , which developed from a Central European variant of the virgule slash, originally a form of scratch comma. S ...
*
Equality (mathematics) In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object. The equality b ...
*
Logical equality Logical equality is a logical operator that corresponds to equality in Boolean algebra and to the logical biconditional in propositional calculus. It gives the functional value ''true'' if both functional arguments have the same logical valu ...
*
Plus and minus signs The plus and minus signs, and , are mathematical symbols used to represent the notions of positive and negative, respectively. In addition, represents the operation of addition, which results in a sum, while represents subtraction, resul ...


Notes


References

* * Boyer, C. B.: ''A History of Mathematics'', 2nd ed. rev. by Uta C. Merzbach. New York: Wiley, 1989 (1991 pbk ed. )


External links


Earliest Uses of Symbols of RelationImage of the page of ''The Whetstone of Witte'' on which the equal sign is introduced
{{DEFAULTSORT:Equal Sign Mathematical symbols Welsh inventions 1557 introductions Definition Assignment operations Equivalence (mathematics)