Plus and minus signs
Encyclopedia
The plus and minus signs (+ and ) are mathematical symbols used to represent the notions of positive and negative
Sign (mathematics)
In mathematics, the word sign refers to the property of being positive or negative. Every nonzero real number is either positive or negative, and therefore has a sign. Zero itself is signless, although in some contexts it makes sense to consider a signed zero...

as well as the operations of addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....

and subtraction
Subtraction
In arithmetic, subtraction is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with...

. Their use has been extended to many other meanings, more or less analogous. Plus and minus are 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...

terms meaning "more" and "less", respectively.

## History

Though the signs now seem as familiar as the alphabet
Alphabet
An alphabet is a standard set of letters—basic written symbols or graphemes—each of which represents a phoneme in a spoken language, either as it exists now or as it was in the past. There are other systems, such as logographies, in which each character represents a word, morpheme, or semantic...

or the Hindu-Arabic numerals, they are not of great antiquity. The Egyptian hieroglyphic sign for addition, for example, resembled a pair of legs walking in the direction in which the text was written (Egyptian
Egyptian language
Egyptian is the oldest known indigenous language of Egypt and a branch of the Afroasiatic language family. Written records of the Egyptian language have been dated from about 3400 BC, making it one of the oldest recorded languages known. Egyptian was spoken until the late 17th century AD in the...

could be written either from right to left or left to right), with the reverse sign indicating subtraction:
 D54- or -D55

In Europe in the early 15th century the letters "P" and "M" were generally used.
The symbols (P with stroke
P with stroke
or "p with stroke" is a letter of the Latin alphabet, formed from P with the addition of a stroke, usually through the bowl but sometimes through the descender...

for piu, i.e., plus, and M with stroke for meno, i.e., minus) appeared for the first time in Luca Pacioli
Luca Pacioli
Fra Luca Bartolomeo de Pacioli was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and seminal contributor to the field now known as accounting...

’s mathematics compendium, Summa de arithmetica, geometria, proportioni et proportionalità, first printed and published in Venice
Venice
Venice is a city in northern Italy which is renowned for the beauty of its setting, its architecture and its artworks. It is the capital of the Veneto region...

in 1494. The + is a simplification of the Latin "et" (comparable to the ampersand
Ampersand
An ampersand is a logogram representing the conjunction word "and". The symbol is a ligature of the letters in et, Latin for "and".-Etymology:...

&). The may be derived from a tilde
Tilde
The tilde is a grapheme with several uses. The name of the character comes from Portuguese and Spanish, from the Latin titulus meaning "title" or "superscription", though the term "tilde" has evolved and now has a different meaning in linguistics....

written over m when used to indicate subtraction; or it may come from a shorthand version of the letter m itself. In his 1489 treatise Johannes Widmann
Johannes Widmann
Johannes Widmann was a German mathematician who invented the addition and the subtraction signs.Born at in Eger, Bohemia, Widmann attended the University of Leipzig in the 1480s, and published Behende und hubsche Rechenung auff allen Kauffmanschafft Johannes Widmann (c. 1460 - after 1498) was a...

referred to the symbols − and + as minus and mer (Modern German mehr; "more"): "das − ist, das ist minus, und das + ist das mer".

Henricus Grammateus
Henricus Grammateus was a German mathematician. In 1518 he published details of a new musical temperament, which is now named after him, for the harpsichord...

in 1518 makes another early use of + and − for addition and subtraction.

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 designer of the equals sign
Equals sign
The equality sign, equals sign, or "=" is a mathematical symbol used to indicate equality. It was invented in 1557 by Robert Recorde. The equals sign is placed between the things stated to have the same value, as in an equation...

, introduced plus and minus to Britain in 1557 in The Whetstone of Witte
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 Witte, whiche is the seconde parte of Arithmeteke: containing the extraction of rootes; the cossike practise, with the rule of equation; and the workes of...

: "There be other 2 signes in often use of which the first is made thus + and betokeneth more: the other is thus made – and betokeneth lesse."

## Plus sign

The plus sign is a binary operator that indicates addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....

, as in 2 + 3 = 5. It can also serve as a unary operator that leaves its operand
Operand
In mathematics, an operand is the object of a mathematical operation, a quantity on which an operation is performed.-Example :The following arithmetic expression shows an example of operators and operands:3 + 6 = 9\;...

unchanged
Identity function
In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument...

(+x means the same as x). This notation may be used when it is desired to emphasise the positiveness of a number, especially when contrasting with the negative (+5 versus −5).

The plus sign can also indicate many other operations, depending on the mathematical system under consideration. Many algebraic structure
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...

s have some operation which is called, or equivalent to, addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....

. Moreover, the symbolism has been extended to very different operations. Plus can mean:
• exclusive or (usually written ⊕): 1 + 1 = 0, 1 + 0 = 1
• logical disjunction
Logical disjunction
In logic and mathematics, a two-place logical connective or, is a logical disjunction, also known as inclusive disjunction or alternation, that results in true whenever one or more of its operands are true. E.g. in this context, "A or B" is true if A is true, or if B is true, or if both A and B are...

(usually written ∨): 1 + 1 = 1, 1 + 0 = 1

## Minus sign

The minus sign has three main uses in mathematics:
1. The subtraction operator: A binary operator to indicate the operation of subtraction
Subtraction
In arithmetic, subtraction is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with...

, as in 5 − 3 = 2. Subtraction is the inverse of addition.
2. Directly in front of a number and when it is not a subtraction operator it means a negative number. For instance −5 is negative 5.
3. A unary operator that acts as an instruction to replace the operand by its opposite. For example, if x is 3, then −x is −3, but if x is −3, then −x is 3. Similarly, −(−2) is equal to 2.

All three uses can be referred to as "minus" in everyday speech. In modern US usage, −5 (for example) is normally pronounced "negative five" rather than "minus five". "minus" may be used by speakers born before 1950, and is still popular in some contexts, but "negative" is usually taught as the only correct reading. In most other parts of the English-speaking world, "minus five" is more common. Textbooks in America encourage −x to be read as "the opposite of x" or even "the additive inverse of x" to avoid giving the impression that −x is necessarily negative.

In some contexts, different glyphs are used for these meanings; for instance in the computer language APL a raised minus sign is used in negative numbers (as in 2 − 5 gives 3), but such usage is rare.

In mathematics and most programming languages, the rules for the order of operations
Order of operations
In mathematics and computer programming, the order of operations is a rule used to clarify unambiguously which procedures should be performed first in a given mathematical expression....

mean that −52 is equal to −25. Powers bind more strongly than multiplication or division which binds more strongly than addition or subtraction. While strictly speaking, the unary minus is not subtraction, it is given the same place as subtraction. However in some programming languages and Excel
Microsoft Excel
Microsoft Excel is a proprietary commercial spreadsheet application written and distributed by Microsoft for Microsoft Windows and Mac OS X. It features calculation, graphing tools, pivot tables, and a macro programming language called Visual Basic for Applications...

in particular, unary operators bind strongest, so in these −5^2 is 25 but 0−5^2 is −25.

## Use in elementary education

Some elementary teachers use raised plus and minus signs before numbers to show they are positive or negative numbers. For example subtracting −5 from 3 might be read as positive three take away negative 5 and be shown as
3 − 5 becomes 3 + 5 = 8,

or even as
+3 − 5 becomes +3 + +5 which is +8

## Use as a qualifier

In grading systems (such as examination marks), the plus sign indicates a grade one level higher and the minus sign a grade lower. For example, B− ("B minus") is one grade lower than B. Sometimes this is extended to two plus or minus signs; for example A++ is two grades higher than A.

Positive and negative are sometimes abbreviated as +ve and −ve.

In mathematics the one-sided limit
One-sided limit
In calculus, a one-sided limit is either of the two limits of a function f of a real variable x as x approaches a specified point either from below or from above...

xa+ means x approaches a from the right, and xa means x approaches a from the left. For example, when calculating what x−1 is when x approaches 0, because x−1+∞
Extended real number line
In mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +∞ and −∞ . The projective extended real number system adds a single object, ∞ and makes no distinction between "positive" or "negative" infinity...

when x→0+ but x−1→ −∞ when x→0.

## Uses in computing

As well as the normal mathematical usage plus and minus may be used for a number of other purposes in computing.

Plus and minus signs are often used in tree view
Tree view
A tree view or an outline view is a graphical user interface element that presents a hierarchical view of information. Each item can have a number of subitems...

on a computer screen to show if a folder is collapsed or not.

In some programming languages concatenation
Concatenation
In computer programming, string concatenation is the operation of joining two character strings end-to-end. For example, the strings "snow" and "ball" may be concatenated to give "snowball"...

of string
String (computer science)
In formal languages, which are used in mathematical logic and theoretical computer science, a string is a finite sequence of symbols that are chosen from a set or alphabet....

s is written: "a" + "b" = "ab", although this usage is questioned by some for violating commutativity
Commutativity
In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...

, a property addition is expected to have.

In most programming languages, subtraction and negation are indicated with the ASCII hyphen-minus
Hyphen-minus
The hyphen-minus is a character used in digital documents and computing to represent a hyphen or a minus sign .It is present in Unicode as code point ; it is also in ASCII with the same value....

character, `-`. In APL a raised minus sign (Unicode U+00AF) is used to denote a negative number, as in ¯3) and in J
J (programming language)
The J programming language, developed in the early 1990s by Kenneth E. Iverson and Roger Hui, is a synthesis of APL and the FP and FL function-level languages created by John Backus....

a negative number is denoted by an underscore
Underscore
The underscore [ _ ] is a character that originally appeared on the typewriter and was primarily used to underline words...

, as in _5.

In C
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....

and some other computer programming languages, two plus signs indicate the increment operator and two minus signs a decrement. For example, `x++` means "increment the value of x by one" and `x--` means "decrement the value of x by one". By extension, "++" is sometimes used in computing terminology to signify an improvement, as in the name of the language C++
C++
C++ is a statically typed, free-form, multi-paradigm, compiled, general-purpose programming language. It is regarded as an intermediate-level language, as it comprises a combination of both high-level and low-level language features. It was developed by Bjarne Stroustrup starting in 1979 at Bell...

.

There is no concept of negative zero in mathematics, but in computing −0 may have a separate representation from zero. In the IEEE floating-point standard
IEEE floating-point standard
IEEE 754–1985 was an industry standard for representingfloating-pointnumbers in computers, officially adopted in 1985 and superseded in 2008 byIEEE 754-2008. During its 23 years, it was the most widely used format for...

1/−0 is negative infinity whereas 1/0 is positive infinity.

## Other uses

In chemistry, the minus sign (rather than an en dash) is used for a single covalent bond
Covalent bond
A covalent bond is a form of chemical bonding that is characterized by the sharing of pairs of electrons between atoms. The stable balance of attractive and repulsive forces between atoms when they share electrons is known as covalent bonding....

between two atoms, in skeletal formula
Skeletal formula
The skeletal formula of an organic compound is a shorthand representation of its molecular structure, developed by the organic chemist, Friedrich August Kekulé von Stradonitz. Skeletal formulae are ubiquitous in organic chemistry, because they are relatively quick and simple to draw. Carbon and...

.

Subscripted plus and minus signs are used as diacritics in the International Phonetic Alphabet
International Phonetic Alphabet
The International Phonetic Alphabet "The acronym 'IPA' strictly refers [...] to the 'International Phonetic Association'. But it is now such a common practice to use the acronym also to refer to the alphabet itself that resistance seems pedantic...

to indicate advanced or retracted articulations
Relative articulation
In descriptions of phonetics and phonology, the manner and place of articulation of a speech sound may be specified relative to some point of comparison...

of speech sounds.

The minus sign is also used as tone letter in the orthographies of 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....

, 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...

, Karaboro
Karaboro languages
The Karaboro languages are spoken in Burkina Faso by approximately 65 000 people . They belong to the Senufo subfamily, but are separated from other Senufo languages by a small band of unrelated languages. Within Senufo they are thought to be most closely related to the Senari...

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

, Wan
Wan language
Wan, or Nwa, is a Mande language of Ivory Coast. Dialects are Miamu and Kemu....

, Yaouré,
Wè language
Guéré , also called Wè , is a Kru language spoken by over 300,000 people in the Dix-Huit Montagnes and Moyen-Cavally regions of Ivory Coast.-Phonology:...

, Nyabwa
Nyabwa language
The Nyabwa language is a Kru language spoken in Ivory Coast. It is part of the Wee dialect continuum....

and Godié. The Unicode character used for the tone letter (U+02D7) is different from the mathematical minus sign.

## Character codes

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...

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...

URL
Uniform Resource Locator
In computing, a uniform resource locator or universal resource locator is a specific character string that constitutes a reference to an Internet resource....

HTML
HTML
HyperText Markup Language is the predominant markup language for web pages. HTML elements are the basic building-blocks of webpages....

(others)
Plus + U+002B `&#43;` `%2B`
Minus U+2212 `%E2%88%92` `&minus;` or `&x2212;` or `&#8722;`
Hyphen-minus - U+002D `&#45;` `%2D`

The Unicode minus sign is designed to be the same length and height as the plus and equals sign
Equals sign
The equality sign, equals sign, or "=" is a mathematical symbol used to indicate equality. It was invented in 1557 by Robert Recorde. The equals sign is placed between the things stated to have the same value, as in an equation...

s. In most fonts these are the same width as digits in order to facilitate the alignment of numbers in tables.

The hyphen-minus sign (-) is 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...

version of the minus sign, and doubles as a hyphen
Hyphen
The hyphen is a punctuation mark used to join words and to separate syllables of a single word. The use of hyphens is called hyphenation. The hyphen should not be confused with dashes , which are longer and have different uses, or with the minus sign which is also longer...

. It is usually shorter in length than the plus sign and sometimes at a different height. It can be used as a substitute for the true minus sign when the character set is limited to 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...

.

There is a commercial minus sign (⁒), which looks somewhat like an obelus
Obelus
An obelus is a symbol consisting of a short horizontal line with a dot above and below. It is mainly used to represent the mathematical operation of division. It is therefore commonly referred to as the division sign.- History :The word "obelus" comes from the Greek word for a sharpened stick,...

, at U+2052 (HTML &x2052;).

### Alternative plus sign

A Jewish tradition that dates from at least the 19th century is to write plus using a symbol like an inverted T. This practice was adopted into Israel
Israel
The State of Israel is a parliamentary republic located in the Middle East, along the eastern shore of the Mediterranean Sea...

i schools (this practice goes back to at least the 1940s) and is still commonplace today in elementary school
Elementary school
An elementary school or primary school is an institution where children receive the first stage of compulsory education known as elementary or primary education. Elementary school is the preferred term in some countries, particularly those in North America, where the terms grade school and grammar...

s (including secular schools) but in fewer secondary school
Secondary school
Secondary school is a term used to describe an educational institution where the final stage of schooling, known as secondary education and usually compulsory up to a specified age, takes place...

s. It is also used occasionally in books by religious authors, but most books for adults use the international symbol "+". The usual explanation for this practice is that it avoids the writing of a symbol "+" that looks like a Christian cross
Christian cross
The Christian cross, seen as a representation of the instrument of the crucifixion of Jesus Christ, is the best-known religious symbol of Christianity...

. 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...

has this symbol at position U+FB29 "Hebrew letter alternative plus sign" .

• Graft-chimaera
Graft-chimaera
In horticulture, a graft-chimaera may arise in grafting at the point of contact between rootstock and scion and will have properties intermediate to those of its "parents". A graft-chimaera is not a true hybrid but a mixture of cells, each with the genotype of one of its "parents": it is a chimaera...

for the meaning of + in botanical name
Botanical name
A botanical name is a formal scientific name conforming to the International Code of Botanical Nomenclature and, if it concerns a plant cultigen, the additional cultivar and/or Group epithets must conform to the International Code of Nomenclature for Cultivated Plants...

s
• List of international call prefixes that + can represent the numbers required to dial out of a country as seen in a phone number
• Plus-minus sign
Plus-minus sign
The plus-minus sign is a mathematical symbol commonly used either*to indicate the precision of an approximation, or*to indicate a value that can be of either sign....

• Table of mathematical symbols
Table of mathematical symbols
This is a listing of common symbols found within all branches of mathematics. Each symbol is listed in both HTML, which depends on appropriate fonts being installed, and in , as an image.-Symbols:-Variations:...

• En dash, a dash that looks similar to the subtraction symbol but is used for a different purpose