Equals sign

# Equals sign

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The equality sign, equals sign, or "=" is a mathematical symbol used to indicate equality. It was invented in 1557 by Robert Recorde
Robert Recorde
Robert Recorde was a Welsh physician and mathematician. He introduced the "equals" sign in 1557.-Biography:A member of a respectable family of Tenby, Wales, he entered the University of Oxford about 1525, and was elected a fellow of All Souls College in 1531...

. The equals sign is placed between the things stated to have the same value, as in an equation
Equation
An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...

. It is the Unicode
Unicode
Unicode is a computing industry standard for the consistent encoding, representation and handling of text expressed in most of the world's writing systems...

and ASCII
ASCII
The American Standard Code for Information Interchange is a character-encoding scheme based on the ordering of the English alphabet. ASCII codes represent text in computers, communications equipment, and other devices that use text...

character 003D (in hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...

).

## History

The etymology of the word equal is from the Latin word aequalis, meaning "uniform, identical, or equal," from aequus "level, even, just."

The "=" symbol that is now universally accepted by mathematics for equality was first recorded by Welsh mathematician Robert Recorde
Robert Recorde
Robert Recorde was a Welsh physician and mathematician. He introduced the "equals" sign in 1557.-Biography:A member of a respectable family of Tenby, Wales, he entered the University of Oxford about 1525, and was elected a fellow of All Souls College in 1531...

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 Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...

gemini):

...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.

...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 two things can be more equal.

According to Scotland's St Andrews University Maths History 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, the equals sign is used for definitions (ex: let x = 2) as well as conditions and denoted equalities (ex: if x = 2 then x/2 = 1).

The equals sign is also used as a grammatical tone letter
Tone letter
Tone letters are letters that represent the tones of a language, most commonly in languages with contour tones.-Chao tone letters :A series of iconic tone letters based on a musical staff was invented by Yuen Ren Chao and adopted into the International Phonetic Alphabet.Combinations of these tone...

in the orthographies of Budu
Budu language
Budu is a Bantu language spoken by the Budu people in the Wamba Territory in the Orientale Province of the Democratic Republic of the Congo. Its orthography uses the special character ɨ, ʉ, ɛ and ɔ, as well as colon ꞉ and short equal sign ꞊ for tones....

in the Congo-Kinshasa, in Krumen
Krumen language
Krumen is a dialect continuum spoken by the Krumen people of Liberia and Ivory Coast . It is a branch of the Grebo languages, a subfamily of the Kru languages and ultimately of the Niger–Congo languages. It had 48,300 speakers as of 1993...

, Mwan
Mwan language
Mwan is a Mande language of Ivory Coast....

and Dan
Dan language
Dan is a Mande language spoken primarily in Côte d'Ivoire and Liberia . There is also a population of about 800 speakers in Guinea. Dan is a tonal language, with three main tones and two glide/contour tones....

in the Ivory Coast. The Unicode character used for the tone letter (U+A78A) is different from the mathematical symbol (U+003D).

### In computer programming

Traditional computer programming languages and their descendants (e.g. the Algol family, Pascal
Pascal (programming language)
Pascal is an influential imperative and procedural programming language, designed in 1968/9 and published in 1970 by Niklaus Wirth as a small and efficient language intended to encourage good programming practices using structured programming and data structuring.A derivative known as Object Pascal...

-People:* St. Ada, a 7th-century French abbess* Ada, Countess of Atholl * Ada, Countess of Holland* Ada de Warenne, Countess of Northumbria and Huntingdon...

, Eiffel
Eiffel (programming language)
Eiffel is an ISO-standardized, object-oriented programming language designed by Bertrand Meyer and Eiffel Software. The design of the language is closely connected with the Eiffel programming method...

, etc) use the equals sign as a boolean
Boolean
Boolean may refer to:* Boolean algebra, a logical calculus of truth values or set membership* Boolean algebra , a set with operations resembling logical ones* Boolean data type, a certain datatype in computer science...

equality-operator as well as for defining constants, types, and parameters. This is consistent with mathematical usage and marks an intention for such languages to be as similar as possible to mathematical notation. A colon
Colon (punctuation)
The colon is a punctuation mark consisting of two equally sized dots centered on the same vertical line.-Usage:A colon informs the reader that what follows the mark proves, explains, or lists elements of what preceded the mark....

-equals (`:=`) is typically used to denote assignment.

In C-like
C (programming language)
C is a general-purpose computer programming language developed between 1969 and 1973 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system....

languages, it is instead used as an assignment operator
Assignment (computer science)
In computer programming, an assignment statement sets or re-sets the value stored in the storage location denoted by a variable name. In most imperative computer programming languages, assignment statements are one of the basic statements...

. C-like languages therefore use a double equals sign () as an equality operator.

Some languages (e.g. BASIC
BASIC
BASIC is a family of general-purpose, high-level programming languages whose design philosophy emphasizes ease of use - the name is an acronym from Beginner's All-purpose Symbolic Instruction Code....

) use a single equals sign to denote both functions, sometimes with an optional `let` keyword to resolve any ambiguities.

#### Usage of several equal-signs

In PHP
PHP
PHP is a general-purpose server-side scripting language originally designed for web development to produce dynamic web pages. For this purpose, PHP code is embedded into the HTML source document and interpreted by a web server with a PHP processor module, which generates the web page document...

, the triple equals sign () denotes identicalness, meaning that not only do the two values evaluate to the same boolean value, they are also of the same data type. For instance, the expression `0 false` is true, but `0 `

### ` false` is not, because the number 0 is an integer value (whereas false is a boolean). 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 expression `0 `

` false` is true, but `0 `

## ` 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 `` false` is false. The operator is flexible and may be defined arbitrarily by any given type. For example a value of type `Range` is a range of integers, such as `1800..1899`. `(1800..1899) `

` 1844` is false, since the types are different (Range vs. Integer); however `(1800..1899) `

### ` 1844` is true, since Range defines

to mean "inclusion in the range". Note that under these semantics,

### is non-commutative; e.g. `1844 `

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

.

### Approximately equal

Symbols used to denote items that are approximately equal
Approximation
An approximation is a representation of something that is not exact, but still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws.Approximations may be used because...

are "wavy
Wave
In physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass...

" equals signs.
• (Unicode 2248),
• (Unicode 2243), a combination of ≈ and =, also used to indicate asymptotically equal to
• (Unicode 2245), another combination of ≈ and =, which is also sometimes used to indicate isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

or congruence
Congruence relation
In abstract algebra, a congruence relation is an equivalence relation on an algebraic structure that is compatible with the structure...

• ~ (Unicode 007E), which is also sometimes used to indicate proportionality
Proportionality (mathematics)
In mathematics, two variable quantities are proportional if one of them is always the product of the other and a constant quantity, called the coefficient of proportionality or proportionality constant. In other words, are proportional if the ratio \tfrac yx is constant. We also say that one...

• (Unicode 2252), in Japanese
Japanese language
is a language spoken by over 130 million people in Japan and in Japanese emigrant communities. It is a member of the Japonic language family, which has a number of proposed relationships with other languages, none of which has gained wide acceptance among historical linguists .Japanese is an...

, this is commonly used.

### Not equal

The symbol used to denote inequation
Inequation
In mathematics, an inequation is a statement that two objects or expressions are not the same, or do not represent the same value. This relation is written with a crossed-out equal sign as inx \neq y....

— when items are not equal — is a slashed
Slash (punctuation)
The slash is a sign used as a punctuation mark and for various other purposes. It is now often called a forward slash , and many other alternative names.-History:...

equals sign "≠" (Unicode 2260).

Most programming languages, limiting themselves to the ASCII
ASCII
The American Standard Code for Information Interchange is a character-encoding scheme based on the ordering of the English alphabet. ASCII codes represent text in computers, communications equipment, and other devices that use text...

character set, use `~=`, `!=`, `/=`, `=/=`, or `<>` to represent their boolean
Boolean logic
Boolean algebra is a logical calculus of truth values, developed by George Boole in the 1840s. It resembles the algebra of real numbers, but with the numeric operations of multiplication xy, addition x + y, and negation −x replaced by the respective logical operations of...

inequality operator.

### Identity

The triple bar
Triple bar
The triple bar, ≡, is a symbol used in formal logic. It has the appearance of a "=" sign with a third line.Logically, it has a similar meaning to the if and only if coupler ⇔...

symbol "≡" (U+2261) is often used to indicate an identity
Identity (mathematics)
In mathematics, the term identity has several different important meanings:*An identity is a relation which is tautologically true. This means that whatever the number or value may be, the answer stays the same. For example, algebraically, this occurs if an equation is satisfied for all values of...

, a definition
Definition
A definition is a passage that explains the meaning of a term , or a type of thing. The term to be defined is the definiendum. A term may have many different senses or meanings...

(which can also be represented by "≝", U+225D), or a congruence relation
Congruence relation
In abstract algebra, a congruence relation is an equivalence relation on an algebraic structure that is compatible with the structure...

in modular arithmetic
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....

. The symbol "" can be used to express that an item corresponds to another.

### Isomorphism

The symbol "" is often used to indicate isomorphic algebraic structures or congruent
Congruence (geometry)
In geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...

geometric figures.

### In logic

Equality of truth values, i.e. bi-implication or logical equivalence
Logical equivalence
In logic, statements p and q are logically equivalent if they have the same logical content.Syntactically, p and q are equivalent if each can be proved from the other...

, may be denoted by various symbols including =, ~, and ⇔.

### In a double-barreled name

A possibly unique case of the equal sign in a person's name, specifically in a double-barreled name, was by pioneer aviator Alberto Santos=Dumont
Alberto Santos-Dumont
Alberto Santos-Dumont , was a Brazilian early pioneer of aviation. The heir of a wealthy family of coffee producers, Santos Dumont dedicated himself to science studies in Paris, France, where he spent most of his adult life....

, as he is also known to not only have often used 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.

## Misconceptions

The equals sign can be used incorrectly within a mathematical argument, if used in a manner that connects steps of math in a non-standard way, rather than to show equality. For example, if one were finding the sum of the numbers 1, 2, 3, 4, and 5, one might say:
1 + 2 = 3 + 3 = 6 + 4 = 10 + 5 = 15. This is clearly 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.

• Equality (mathematics)
• Logical equality
Logical equality
Logical equality is a logical operator that corresponds to equality in Boolean algebra and to the logical biconditional in propositional calculus...

• Plus and minus signs
Plus and minus signs
The plus and minus signs are mathematical symbols used to represent the notions of positive and negative as well as the operations of addition and subtraction. Their use has been extended to many other meanings, more or less analogous...

• Double hyphen
Double hyphen
The double hyphen is a punctuation mark that consists of two parallel hyphens. It is not to be confused with two consecutive hyphens , which usually represents an em dash or en dash...

• 2 + 2 = 5