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featuring the equals sign]] The equals sign or equality sign ( ) is a mathematical symbol used to indicate equality. It was invented in 1557 by . In an , the equals sign is placed between two (or more) expressions that have the same value. In and , it is .


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 that is now universally accepted in mathematics for equality was first recorded by Welsh mathematician in The Whetstone of Witte (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 See also and Gemini.):Recorde, Robert, The Whetstone of Witte … (London, England: Jhon Kyngstone, 1557), the third page of the chapter "The rule of equation, commonly called Algebers Rule."

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, or Gemowe 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, or Gemowe lines of one length, thus: =, because no 2 things, can be more equal.

According to Scotland's University of St Andrews History of Mathematics website:

The symbol '=' was not immediately popular. The symbol || was used by some and æ (or œ), from the Latin word aequalis meaning equal, was widely used into the 1700s.


Usage in mathematics and computer programming
In mathematics, the equals sign can be used as a simple statement of fact in a specific case (x = 2), or to create definitions (let x = 2), conditional statements (if x = 2, then …), or to express a universal equivalence (x&nbsp;+&nbsp;1)<sup>2</sup>&nbsp;=&nbsp;x<sup>2</sup>&nbsp;+&nbsp;2x&nbsp;+&nbsp;1.

The first important computer programming language to use the equals sign was the original version of , FORTRAN I, designed in 1954 and implemented in 1957. In Fortran, "=" serves as an assignment operator: X&nbsp;=&nbsp;2 sets the value of X 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 X. For example, the assignment X&nbsp;=&nbsp;X&nbsp;+&nbsp;2 increases the value of X by 2.

A rival programming-language usage was pioneered by the original version of , which was designed in 1958 and implemented in 1960. ALGOL included a relational operator that tested for equality, allowing constructions like if&nbsp;x&nbsp;=&nbsp;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 as C, , Python, , and their descendants. But "=" is used for equality and not assignment in the Pascal family, Ada, Eiffel, APL, and other languages.

A few languages, such as and PL/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 the statement) until FORTRAN IV was released in 1962, since when it has used the four characters ".EQ." 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 equals sign is also used in defining attribute–value pairs, in which an attribute is assigned a value.


Usage of several equals signs
In , the triple equals sign (===) denotes value and equality, meaning that not only do the two expressions evaluate to equal values, they are also of the same data type. For instance, the expression 0&nbsp;==&nbsp;false is true, but 0&nbsp;===&nbsp;false is not, because the number 0 is an integer value whereas false is a Boolean value.

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 0&nbsp;==&nbsp;false is true, but 0&nbsp;==&nbsp;undefined is false, even though both sides of the == act the same in Boolean context. For this reason it is recommended to avoid the == operator in JavaScript in favor of ===.

In Ruby, equality under == requires both operands to be of identical type, e.g. 0&nbsp;==&nbsp;false is false. The === operator is flexible and may be defined arbitrarily for any given type. For example, a value of type Range is a range of integers, such as 1800..1899. (1800..1899)&nbsp;==&nbsp;1844 is false, since the types are different (Range vs. Integer); however (1800..1899)&nbsp;===&nbsp;1844 is true, since === on Range values means "inclusion in the range". Note that under these semantics, === is non-symmetric; e.g. 1844&nbsp;===&nbsp;(1800..1899) is false, since it is interpreted to mean Integer#=== rather than Range#===.


Other uses
The equals sign is sometimes used in Japanese as a separator between names.


Spelling

Tone letter
The equals sign is also used as a grammatical in the orthographies of in the , in , and in the . 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 equals sign of European usage in a person's name, specifically in a double-barreled name, was by pioneer aviator Alberto Santos=Dumont, as he is also known not only to have often used an equals 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.


Linguistics
In linguistic interlinear glosses, an equals sign is conventionally used to mark clitic boundaries: the equals sign is placed between the and the word that the clitic is attached to.


Chemistry
In , the two parallel lines denoting a are commonly rendered using an equals sign.


Related symbols

Approximately equal
Symbols used to denote items that are include the following:
  • (+2248, \approx)
  • (U+2243, LaTeX \simeq), a combination of ≈ and =, also used to indicate asymptotic equality
  • (U+2245, LaTeX \cong), another combination of ≈ and =, which is also sometimes used to indicate or congruence
  • (U+223C), which is also sometimes used to indicate proportionality or similarity, being related by an equivalence relation, or to indicate that a is distributed according to a specific probability distribution (see also )
  • (U+223D), which is also used to indicate proportionality
  • (U+2250, LaTeX \doteq), which can also be used to represent the approach of a variable to a limit
  • (U+2252), commonly used in Japanese, Taiwanese and
  • (U+2253)


Not equal
The symbol used to denote (when items are not equal) is a slashed equals sign "≠" (U+2260; 2260,Alt+X in Microsoft Windows). In , this is done with the "\neq" command.

Most programming languages, limiting themselves to the and , use ~=, !=, /=, =/=, or <> to represent their inequality operator.


Identity
The symbol "≡" (U+2261, LaTeX \equiv) is often used to indicate an identity, a (which can also be represented by U+225D "≝" or U+2254 "≔"), or a congruence relation in modular arithmetic. The symbol "≘" can be used to express that an item corresponds to another.


Isomorphism
The symbol "≅" is often used to indicate algebraic structures or congruent geometric figures.


In logic
Equality of , i.e. or logical equivalence, may be denoted by various symbols including =, ~, and ⇔.


Other related symbols
Additional symbols in Unicode related to the equals sign include:
  • ()
  • () (see also assignment (computer science))
  • ()
  • ()
  • ()

  • ()
  • ()
  • ()
  • ()

  • ()
  • ().


Incorrect usage
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

1 + 2 = 3 + 3 = 6 + 4 = 10 + 5 = 15.
Structurally, this is shorthand for
((1 + 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 implies
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.


Encodings
Related:


See also


Notes
  • (1993). 9780486677668, Dover (reprint).
  • Boyer, C. B.: A History of Mathematics, 2nd ed. rev. by Uta C. Merzbach. New York: Wiley, 1989 (1991 pbk ed. )


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