| = | |
|---|---|
Equals sign | |
| In Unicode | U+003D =EQUALS SIGN (=) |
| Related | |
| See also | U+2260 ≠NOT EQUAL TO U+2248 ≈ALMOST EQUAL TO U+2261 ≡IDENTICAL TO |
Theequals sign (British English) orequal sign (American English), also known as theequality sign, is themathematical symbol=, which is used to indicateequality.[1] In anequation, it is placed between twoexpressions that have the same value, or for which one studies the conditions under which they have the same value.
InUnicode andASCII, it has thecode point U+003D.[2] It was invented in 1557 byRobert Recorde.
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⟩.[3]Diophantus's use of⟨ἴσ⟩, short forἴσος (ísos 'equals'), inArithmetica (c. 250 AD) is considered one of the first uses of an equals sign.[4]
The= symbol, now universally accepted in mathematics for equality, was first recorded by WelshmathematicianRobert Recorde inThe Whetstone of Witte (1557).[5] The original form of the symbol was much wider than the present form. In his book Recorde explains his design of the "Gemowe lines" (meaningtwin lines, from theLatingemellus)[6]
And to auoide the tediouſe repetition of theſe woordes : is equalle to : I will ſette as I doe often in woorke vſe, a paire of paralleles, orGemowe lines of one lengthe, thus: =, bicauſe noe .2. thynges, can be moare equalle.
And to avoid the tedious repetition of these words: "is equal to" I will set as I do often in work use, a pair of parallels, orduplicate lines of one [the same] length, thus: =, because no 2 things can be more equal.— Recorde, Robert (1557).The Whetstone of Witte. London: John Kyngstone.the third page of the chapter "The rule of equation, commonly called Algebers Rule."
The symbol= was not immediately popular. After its introduction by Recorde, 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 three influential works,Thomas Harriot'sArtis analyticae praxis,William Oughtred'sClavis mathematicae, andRichard Norwood'sTrigonometria.[7] Later used byJohn Wallis,Isaac Barrow, andIsaac Newton, which helped it spread throughout the rest of Europe.
There were several other competing symbols for equality, especially outside of England around the 16th and 17th centuries, and Recorde's version made no significant influence in the European continent until 1650 or 1660. In 1559, the French MonkJohannes Buteo published hisLogistica using the symbol for equality. In In 1571Wilhelm Xylander published an edition ofDiophantus'Arithmetica in which two parallel vertical lines|| were used for equality.[a] This version was adopted by several prominent writers, includingGiovanni Glorioso, CardinalMichelangelo Ricci, and many French and Dutch mathematicians in the hundred years after, includingRené Descartes in 1621.
A major competitor to Recorde's sign was Descartes' own symbol, introduced in hisLa Géométrie (1637). In fact, Descartes himself used the sign= for equality in a letter in 1640. Descartes does not give any reason for introducing his new symbol, howeverFlorian Cajori suggests it is because= was also being used for a difference operation at the time. Due to the prominence ofLa Géométrie, by 1675, Descartes' symbol gained favor over Recorde's in Europe, and the majority of writers of the seventeenth century on the continent either used Descartes' notation for equality or none at all. Around the turn of the 18th century, Recorde's notation gained favor rapidly. The dominating trend in mathematics of the time wasdifferential and integral calculus. The fact that both Newton andLeibniz used Recorde's symbol led to its general adoption.
In mathematics, the equals sign can be used as a simple statement of fact in a specific case ("x = 2"), or to create definitions ("letx = 2"), conditional statements ("ifx = 2, then ..."), or to express a universal equivalence ("(x + 1)2 =x2 + 2x + 1").
The first importantcomputer programming language to use the equals sign was the original version ofFortran, FORTRAN I, designed in 1954 and implemented in 1957. In Fortran,= serves as anassignment operator:X = 2 sets the value ofX 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 ofX. For example, the assignmentX = X + 2 increases the value ofX by 2.
A rival programming-language usage was pioneered by the original version ofALGOL, which was designed in 1958 and implemented in 1960. ALGOL included arelational operator that tested for equality, allowing constructions likeif x = 2 with essentially the same meaning of= as the conditional usage in mathematics. The equals 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 asC,Perl,Python,AWK, and their descendants. But= is used for equality and not assignment in thePascal family,Ada,Eiffel,APL, and other languages.
A few languages, such asBASIC andPL/I, have used the equals 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 thearithmetic IF statement) until FORTRAN IV was released in 1962, since when it has used the four characters.EQ. to test for equality. The languageB introduced the use of== with this meaning, which has been copied by its descendant C and most later languages where= means assignment.
Some languages additionally feature the "spaceship operator", or three-way comparison operator,<=>, to determine whether one value is less than, equal to, or greater than another.
In some programming languages,== and=== are used to check equality, so1844 == 1844 will return true.
InPHP, thetriple equals sign,===, denotes value andtype equality,[8] meaning that not only do the two expressions evaluate to equal values, but they are also of the same data type. For instance, the expression0 == false is true, but0 === false is not, because the number 0 is an integer value whereas false is a Boolean value.
JavaScript 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 expression0 == false is true, but0 == undefined 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===.[9]
In Ruby, equality under== requires both operands to be of identical type, e.g.0 == false is false. The=== operator is flexible and may be defined arbitrarily for any given type. For example, a value of typeRange is a range of integers, such as1800..1899.(1800..1899) == 1844 is false, since the types are different (Range vs. Integer); however(1800..1899) === 1844 is true, since=== onRange values means "inclusion in the range".[10] Under these semantics,=== isnon-symmetric; e.g.1844 === (1800..1899) is false, since it is interpreted to meanInteger#=== rather thanRange#===.[11]
The equals sign is also used as a grammaticaltone letter in the orthographies ofBudu in theCongo-Kinshasa, inKrumen,Mwan andDan in theIvory Coast.[12][13] The Unicode character used for the tone letter (U+A78A ꞊MODIFIER LETTER SHORT EQUALS SIGN)[14] is different from the mathematical symbol (U+003D).
A possibly unique case of the equals sign of European usage in a person's name, specifically in adouble-barreled name, was by pioneer aviatorAlberto Santos-Dumont, as he is also known not only to have often used adouble hyphen⹀ resembling an equal sign= between histwo 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.[15]
Instead of a double hyphen, the equals sign is sometimes used inJapanese as a separator between names. InOjibwe, the readily available equals sign on most keyboards is commonly used as a substitute for a double hyphen.
In linguisticinterlinear glosses, an equals sign is conventionally used to mark clitic boundaries: the equals sign is placed between theclitic and the word that the clitic is attached to.[16]
Inchemical formulas, the two parallel lines denoting adouble bond are commonly rendered using an equals sign (hence, atriple bond is commonly rendered using atriple bar).
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In recent years, the equals sign has been used tosymbolizeLGBT rights. The symbol has been used since 1995 by theHuman Rights Campaign, which lobbies formarriage equality, and subsequently by theUnited Nations Free & Equal, which promotesLGBT rights at the United Nations.[17]
InMorse code, the equals sign is encoded by the letters B (-...) and T (-) run together (-...-).[citation needed] The letters BT stand for Break Text, and are put between paragraphs, or groups of paragraphs in messages sent viaTelex,[citation needed] a standardised tele-typewriter. The sign, used to mean Break Text, is given at the end of atelegram to separate the text of the message from the signature.[citation needed]
Symbols used to denote items that areapproximately equal include the following:[18]
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".
The symbol used to denoteinequation (when items are not equal) is aslashed equal sign≠ (U+2260). InLaTeX, this is done with the "\neq" command.
Most programming languages, limiting themselves to the7-bit ASCIIcharacter set andtypeable characters, use~=,!=,/=, or<> to represent theirBooleaninequality operator.
Thetriple bar symbol≡ (U+2261, LaTeX\equiv) is often used to indicate anidentity, adefinition (which can also be represented byU+225D ≝EQUAL TO BY DEFINITION orU+2254 ≔COLON EQUALS), or acongruence relation inmodular arithmetic. Also, inchemistry, the triple bar can be used to represent atriple bond between atoms.
The symbol≅ is often used to indicateisomorphic algebraic structures orcongruent geometric figures.
Equality oftruth values (throughbi-implication orlogical equivalence), may be denoted by various symbols including=,~, and⇔.
The symbol (LaTeX \bumpeq) is used to show two directed line segments have the same length and direction,equipollence.
Additionalprecomposed symbols withcode points in Unicode for notations related to the equal sign include the following:[18]
::=)The equals 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
Structurally, this is shorthand for
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
A correct version of the argument would be
This difficulty results from subtly different uses of the sign in education. In early, arithmetic-focused grades, the equals sign may beoperational; like the equal button on an electronic calculator, it demands the result of a calculation. Starting in algebra courses, the sign takes on arelational meaning of equality between two calculations. Confusion between the two uses of the sign sometimes persists at the university level.[19]
Related symbols
