Equals sign

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Template:Short description Template:About Template:Redirect-several Template:Infobox symbol

A well-known equality featuring the equal sign

The equals sign (British English) or equal sign (American English), also known as the equality sign, is the mathematical symbol Template:Char, which is used to indicate equality.[1] In an equation, 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 and ASCII, it has the code point U+003D.[2] It was invented in 1557 by Robert Recorde.

History

The first use of an equals sign, equivalent to 14x+15=71 in modern notation. From The Whetstone of Witte (1557) by Robert Recorde.
Recorde's introduction of "="

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 Template:Angbr and Template:Angbr.[3] Diophantus's use of Template:Angbr, short for Template:Lang (Template:Tlit 'equals'), in Arithmetica (Template:Circa) is considered one of the first uses of an equals sign.[4]

The Template:Char symbol, now universally accepted in mathematics for equality, was first recorded by Welsh mathematician Robert Recorde in The 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" (meaning twin lines, from the Latin Template:Wikt-lang)[6] Template:Blockquote The symbol Template:Char 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 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 three influential works, Thomas Harriot's Artis analyticae praxis, William Oughtred's Clavis mathematicae, and Richard Norwood's Trigonometria.[7] Later used by John Wallis, Isaac Barrow, and Isaac Newton, which helped it spread throughout the rest of Europe.

Competing symbols

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 Monk Johannes Buteo published his Logistica using the symbol [ for equality. In In 1571 Wilhelm Xylander published an edition of Diophantus' Arithmetica in which two parallel vertical lines Template:Char were used for equality.Template:Efn This version was adopted by several prominent writers, including Giovanni Glorioso, Cardinal Michelangelo Ricci, and many French and Dutch mathematicians in the hundred years after, including René Descartes in 1621.

A major competitor to Recorde's sign was Descartes' own symbol, introduced in his La Géométrie (1637). In fact, Descartes himself used the sign Template:Char for equality in a letter in 1640. Descartes does not give any reason for introducing his new symbol, however Florian Cajori suggests it is because Template:Char was also being used for a difference operation at the time. Due to the prominence of La 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 was differential and integral calculus. The fact that both Newton and Leibniz used Recorde's symbol led to its general adoption.

Usage in mathematics and computer programming

In mathematics, the equal sign can be used as a simple statement of fact in a specific case ("Template:Nowrap"), or to create definitions ("Template:Nowrap"), conditional statements ("Template:Nowrap"), or to express a universal equivalence ("Template:Nowrap").

The first important computer programming language to use the equal sign was the original version of Fortran, FORTRAN I, designed in 1954 and implemented in 1957. In Fortran, Template:Char serves as an assignment operator: Template:Nowrap sets the value of Template:Code to 2. This somewhat resembles the use of Template:Char in a mathematical definition, but with different semantics: the expression following Template:Char is evaluated first, and may refer to a previous value of Template:Code. For example, the assignment Template:Nowrap increases the value of Template:Code by 2.

A rival programming-language usage was pioneered by the original version of ALGOL, which was designed in 1958 and implemented in 1960. ALGOL included a relational operator that tested for equality, allowing constructions like Template:Nowrap with essentially the same meaning of Template:Char 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, Template:Char is used for assignment in such languages as C, Perl, Python, AWK, and their descendants. But Template:Char is used for equality and not assignment in the Pascal family, Ada, Eiffel, APL, and other languages.

A few languages, such as BASIC and PL/I, have used the equal sign to mean both assignment and equality, distinguished by context. However, in most languages where Template:Char 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 Template:Char for equality use Template:Char 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 FORTRANTemplate:NbspIV was released in 1962, since when it has used the four characters Template:Code to test for equality. The language B introduced the use of Template:Char with this meaning, which has been copied by its descendant C and most later languages where Template:Char means assignment.

Some languages additionally feature the "spaceship operator", or three-way comparison operator, Template:Char, to determine whether one value is less than, equal to, or greater than another.

Several equal signs

In some programming languages, == and === are used to check equality, so 1844 == 1844 will return true.

In PHP, the triple equal sign, Template:Code, denotes value and type 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 expression Template:Nowrap is true, but Template:Nowrap is not, because the number 0 is an integer value whereas false is a Boolean value.

JavaScript has the same semantics for Template:Code, referred to as "equality without type coercion". However, in JavaScript the behavior of Template:Code cannot be described by any simple consistent rules. The expression Template:Nowrap is true, but Template:Nowrap is false, even though both sides of the Template:Code act the same in Boolean context. For this reason it is sometimes recommended to avoid the Template:Code operator in JavaScript in favor of Template:Code.[9]

In Ruby, equality under Template:Code requires both operands to be of identical type, e.g. Template:Nowrap is false. The Template:Code operator is flexible and may be defined arbitrarily for any given type. For example, a value of type Template:Code is a range of integers, such as Template:Code. Template:Nowrap is false, since the types are different (Range vs. Integer); however Template:Nowrap is true, since Template:Code on Template:Code values means "inclusion in the range".[10] Under these semantics, === is non-symmetric; e.g. 1844 === (1800..1899) is false, since it is interpreted to mean Integer#=== rather than Range#===.[11]

Other uses

Spelling

Tone letter

The equals sign is also used as a grammatical tone letter in the orthographies of Budu in the Congo-Kinshasa, in Krumen, Mwan and Dan in the Ivory Coast.[12][13] The Unicode character used for the tone letter (Template:Unichar)[14] is different from the mathematical symbol (U+003D).

Personal names

Template:Contains special characters

The signature of Santos-Dumont, showing a double hyphen that looks like an equals sign.

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 a double hyphen Template:Char resembling an equal sign Template:Char 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.[15]

Instead of a double hyphen, the equals sign is sometimes used in Japanese as a separator between names. In Ojibwe, the readily available equals sign on most keyboards is commonly used as a substitute for a double hyphen.

Linguistics

In linguistic interlinear glosses, an equal sign is conventionally used to mark clitic boundaries: the equals sign is placed between the clitic and the word that the clitic is attached to.[16]

Chemistry

In chemical formulas, the two parallel lines denoting a double bond are commonly rendered using an equals sign (hence, a triple bond is commonly rendered using a triple bar).

LGBT activism

Template:Expand section In recent years, the equal sign has been used to symbolize LGBT rights. The symbol has been used since 1995 by the Human Rights Campaign, which lobbies for marriage equality, and subsequently by the United Nations Free & Equal, which promotes LGBT rights at the United Nations.[17]

Telegrams and Telex

In Morse code, the equals sign is encoded by the letters B (-...) and T (-) run together (-...-).Template:Citation needed The letters BT stand for Break Text, and are put between paragraphs, or groups of paragraphs in messages sent via Telex,Template:Citation needed a standardised tele-typewriter. The sign, used to mean Break Text, is given at the end of a telegram to separate the text of the message from the signature.Template:Citation needed

Template:See also

Approximately equal

Template:Main

Symbols used to denote items that are approximately 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".

Not equal

The symbol used to denote inequation (when items are not equal) is a slashed equal sign Template:Char (U+2260). In LaTeX, this is done with the "\neq" command.

Most programming languages, limiting themselves to the 7-bit ASCII character set and typeable characters, use Template:Code, Template:Code, Template:Code, or Template:Code to represent their Boolean inequality operator.

Identity

The triple bar symbol Template:Char (U+2261, LaTeX \equiv) is often used to indicate an identity, a definition (which can also be represented by Template:Unichar or Template:Unichar), or a congruence relation in modular arithmetic. Also, in chemistry, the triple bar can be used to represent a triple bond between atoms.

Isomorphism

The symbol Template:Char is often used to indicate isomorphic algebraic structures or congruent geometric figures.

In logic

Equality of truth values (through bi-implication or logical equivalence), may be denoted by various symbols including Template:Char, Template:Char, and Template:Char.

In geometry

The symbol ≏ (LaTeX \bumpeq) is used to show two directed line segments have the same length and direction, equipollence.

Additional precomposed symbols with code points in Unicode for notations related to the equal sign include the following:[18]

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 + 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 equals 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.[19]

Encodings

Related symbols

See also

Notes

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References

Template:Wiktionary Template:Wiktionary

  1. ↑ Template:Cite web
  2. ↑ Template:Cite web
  3. ↑ Template:Cite encyclopedia
  4. ↑ Template:Cite book
  5. ↑ Template:Cite web
  6. ↑ See also geminus and Gemini.
  7. ↑ Template:Cite book
  8. ↑ Template:Cite web
  9. ↑ Template:Cite web
  10. ↑ Template:Cite book
  11. ↑ Template:Cite web
  12. ↑ Template:Cite book
  13. ↑ Template:Cite book
  14. ↑ Template:Cite web
  15. ↑ Template:Citation
  16. ↑ Template:Cite web
  17. ↑ "HRC Story: Our Logo." Template:Webarchive The Human Rights Campaign. HRC.org, Retrieved 4 December 2018.
  18. ↑ 18.0 18.1 Template:Cite web
  19. ↑ Template:Cite journal
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