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The plus and minus signs, and , are
mathematical symbols A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. ...
used to represent the notions of
positive Positive is a property of positivity and may refer to: Mathematics and science * Positive formula, a logical formula not containing negation * Positive number, a number that is greater than 0 * Plus sign, the sign "+" used to indicate a posi ...
and negative, respectively. In addition, represents the operation of addition, which results in a sum, while represents subtraction, resulting in a
difference Difference, The Difference, Differences or Differently may refer to: Music * ''Difference'' (album), by Dreamtale, 2005 * ''Differently'' (album), by Cassie Davis, 2009 ** "Differently" (song), by Cassie Davis, 2009 * ''The Difference'' (al ...
. Their use has been extended to many other meanings, more or less analogous. ''Plus'' and ''minus'' are
Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
terms meaning "more" and "less", respectively.


History

Though the signs now seem as familiar as the
alphabet An alphabet is a standardized set of basic written graphemes (called letters) that represent the phonemes of certain spoken languages. Not all writing systems represent language in this way; in a syllabary, each character represents a syllab ...
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 could be written either from right to left or left to right), with the reverse sign indicating subtraction:
Nicole Oresme Nicole Oresme (; c. 1320–1325 – 11 July 1382), also known as Nicolas Oresme, Nicholas Oresme, or Nicolas d'Oresme, was a French philosopher of the later Middle Ages. He wrote influential works on economics, mathematics, physics, astrology an ...
's manuscripts from the 14th century show what may be one of the earliest uses of as a sign for plus. In early 15th century Europe, the letters "P" and "M" were generally used. The symbols (P with overline, , for (more), i.e., plus, and M with overline, , for (less), i.e., minus) appeared for the first time in Luca Pacioli's mathematics
compendium A compendium (plural: compendia or compendiums) is a comprehensive collection of information and analysis pertaining to a body of knowledge. A compendium may concisely summarize a larger work. In most cases, the body of knowledge will concern a sp ...
, , first printed and published in
Venice Venice ( ; it, Venezia ; vec, Venesia or ) is a city in northeastern Italy and the capital of the Veneto region. It is built on a group of 118 small islands that are separated by canals and linked by over 400 bridges. The isla ...
in 1494. The sign is a simplification of the la, et (comparable to the evolution of the ampersand ). The may be derived from a
tilde The tilde () or , is a grapheme with several uses. The name of the character came into English from Spanish, which in turn came from the Latin '' titulus'', meaning "title" or "superscription". Its primary use is as a diacritic (accent) i ...
written over when used to indicate subtraction; or it may come from a shorthand version of the letter itself. In his 1489 treatise,
Johannes Widmann Johannes Widmann (c. 1460 – after 1498) was a German mathematician. The + and - symbols first appeared in print in his book ''Mercantile Arithmetic'' or ''Behende und hüpsche Rechenung auff allen Kauffmanschafft'' published in Leipzig in 1489 ...
referred to the symbols and as ''minus'' and ''mer'' (Modern German ; "more"): They weren't used for addition and subtraction in the treatise, but were used to indicate surplus and deficit; usage in the modern sense is attested in a 1518 book by
Henricus Grammateus Henricus Grammateus (also known as Henricus Scriptor, Heinrich Schreyber or Heinrich Schreiber; 1495 – 1525 or 1526) was a German mathematician. He was born in Erfurt. In 1507 he started to study at the University of Vienna, where he subsequen ...
.
Robert Recorde Robert Recorde () was an Anglo-Welsh physician and mathematician. He invented the equals sign (=) and also introduced the pre-existing plus sign (+) to English speakers in 1557. Biography Born around 1512, Robert Recorde was the second and las ...
, the designer of the
equals sign The equals sign (British English, Unicode) or equal sign (American English), also known as the equality sign, is the mathematical symbol , which is used to indicate equality in some well-defined sense. In an equation, it is placed between tw ...
, introduced plus and minus to Britain in 1557 in ''
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 , is the : The ''Coßike'' practise, with the rule of ''Equation'': and the of ''Surde Nombers. ...
'': "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 In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary o ...
that indicates addition, as in . It can also serve as a
unary operator In mathematics, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operation o ...
that leaves its
operand In mathematics, an operand is the object of a mathematical operation, i.e., it is the object or quantity that is operated on. Example The following arithmetic expression shows an example of operators and operands: :3 + 6 = 9 In the above exam ...
unchanged ( means the same as ). This notation may be used when it is desired to emphasize the positiveness of a number, especially in contrast with the negative numbers ( versus ). The plus sign can also indicate many other operations, depending on the mathematical system under consideration. Many algebraic structures, such as
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s and matrix rings, have some operation which is called, or is equivalent to, addition. It is though conventional to use the plus sign to only denote
commutative operation In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
s. The symbol is also used in chemistry and
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
. For more, see .


Minus sign

The minus sign, , has three main uses in mathematics: # The subtraction operator: a
binary operator In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary o ...
to indicate the operation of subtraction, as in 5 − 3 = 2. Subtraction is the inverse of addition. # The
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
whose value for any
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
or
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
argument is the additive inverse of that argument. For example, if , then , but if , then . Similarly, . # A prefix of a numeric constant. When it is placed immediately before an unsigned numeral, the combination names a negative number, the additive inverse of the positive number that the numeral would otherwise name. In this usage, '' names a number the same way 'semicircle' names a geometric figure, with the caveat that 'semi' does not have a separate use as a function name. In many contexts, it does not matter whether the second or the third of these usages is intended: is the same number. When it is important to distinguish them, a raised minus sign is sometimes used for negative constants, as in
elementary education Primary education or elementary education is typically the first stage of formal education, coming after preschool/kindergarten and before secondary school. Primary education takes place in ''primary schools'', ''elementary schools'', or fir ...
, the programming language APL, and some early graphing calculators. All three uses can be referred to as "minus" in everyday speech, though the binary operator is sometimes read as "take away". In American English nowadays, −5 (for example) is generally referred to as "negative five" though speakers born before 1950 often refer to it as "minus five". (Temperatures tend to follow the older usage; −5° is generally called "minus five degrees".) Further, a few textbooks in the United States encourage to be read as "the opposite of " or "the additive inverse of "—to avoid giving the impression that is necessarily negative (since itself may already be negative). In mathematics and most programming languages, the rules for the order of operations mean that is equal to :
Exponentiation Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to r ...
binds more strongly than the unary minus, which binds more strongly than multiplication or division. However, in some programming languages (
Microsoft Excel Microsoft Excel is a spreadsheet developed by Microsoft for Windows, macOS, Android and iOS. It features calculation or computation capabilities, graphing tools, pivot tables, and a macro programming language called Visual Basic for App ...
in particular), unary operators bind strongest, so in those cases is 25, but is −25. Similar to the plus sign, the minus sign is also used in chemistry and
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
. For more, see below.


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 : becomes , or even as : becomes .


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 minus") is one grade lower than . In some occasions, this is extended to two plus or minus signs (e.g., being two grades higher than ). Positive and negative are sometimes abbreviated as and .


Mathematics

In mathematics the
one-sided limit In calculus, a one-sided limit refers to either one of the two limits of a function f(x) of a real variable x as x approaches a specified point either from the left or from the right. The limit as x decreases in value approaching a (x approach ...
means approaches from the right (i.e., right-sided limit), and means approaches from the left (i.e., left-sided limit). For example, as but as .


Blood

Blood types Blood is a body fluid in the circulatory system of humans and other vertebrates that delivers necessary substances such as nutrients and oxygen to the cells, and transports metabolic waste products away from those same cells. Blood in the ...
are often qualified with a plus or minus to indicate the presence or absence of the
Rh factor The Rh blood group system is a human blood group system. It contains proteins on the surface of red blood cells. After the ABO blood group system, it is the most likely to be involved in transfusion reactions. The Rh blood group system consists ...
. For example, A+ means
type A blood The ABO blood group system is used to denote the presence of one, both, or neither of the A and B antigens on erythrocytes. For human blood transfusions, it is the most important of the 43 different blood type (or group) classification system ...
with the Rh factor present, while B− means type B blood with the Rh factor absent.


Music

In music, augmented chords are symbolized with a plus sign, although this practice is not universal (as there are other methods for spelling those chords). For example, "C+" is read "C augmented chord". Sometimes the plus is written as a superscript.


Uses in computing

As well as the normal mathematical usage, plus and minus signs may be used for a number of other purposes in computing. Plus and minus signs are often used in
tree view In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are u ...
on a computer screen—to show if a folder is collapsed or not. In some programming languages,
concatenation In formal language theory and computer programming, string concatenation is the operation of joining character strings end-to-end. For example, the concatenation of "snow" and "ball" is "snowball". In certain formalisations of concatenat ...
of strings is written , and results in . In most programming languages, subtraction and negation are indicated with the ASCII hyphen-minus character, . In APL a raised minus sign (Unicode U+00AF) is used to denote a negative number, as in . While in J a negative number is denoted by an
underscore An underscore, ; also called an underline, low line, or low dash; is a line drawn under a segment of text. In proofreading, underscoring is a convention that says "set this text in italic type", traditionally used on manuscript or typescript a ...
, as in . In C and some other computer programming languages, two plus signs indicate the increment operator and two minus signs a decrement; the position of the operator before or after the variable indicates whether the new or old value is read from it. For example, if x equals 6, then increments x to 7 but sets y to 6, whereas would set both x and y to 7. By extension, is sometimes used in computing terminology to signify an improvement, as in the name of the language
C++ C++ (pronounced "C plus plus") is a high-level general-purpose programming language created by Danish computer scientist Bjarne Stroustrup as an extension of the C programming language, or "C with Classes". The language has expanded significan ...
. In
regular expression A regular expression (shortened as regex or regexp; sometimes referred to as rational expression) is a sequence of characters that specifies a search pattern in text. Usually such patterns are used by string-searching algorithms for "find" ...
s, is often used to indicate "1 or more" in a pattern to be matched. For example, means "one or more of the letter x". 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, 1 / −0 is negative infinity (-\infty) whereas 1 / 0 is positive infinity ( \infty).


Other uses

In physics, the use of plus and minus signs for different
electrical charge Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described ...
s was introduced by
Georg Christoph Lichtenberg Georg Christoph Lichtenberg (1 July 1742 – 24 February 1799) was a German physicist, satirist, and Anglophile. As a scientist, he was the first to hold a professorship explicitly dedicated to experimental physics in Germany. He is remembered for ...
. In chemistry, superscripted plus and minus signs are used to indicate an ion with a positive or negative charge of 1 (e.g., NH). If the charge is greater than 1, a number indicating the charge is written before the sign (as in SO). The minus sign is also used, in place of an en dash, for a single covalent bond between two atoms as in the
skeletal formula The skeletal formula, or line-angle formula or shorthand formula, of an organic compound is a type of molecular structural formula that serves as a shorthand representation of a molecule's bonding and some details of its molecular geometry. A ...
. In the
International Phonetic Alphabet The International Phonetic Alphabet (IPA) is an alphabetic system of phonetic notation based primarily on the Latin script. It was devised by the International Phonetic Association in the late 19th century as a standardized representation ...
, subscripted plus and minus signs are used as diacritics to indicate advanced or retracted articulations of speech sounds. The minus sign is also used as tone letter in the orthographies of Dan, Krumen,
Karaboro The Karaboro languages are spoken in Burkina Faso by approximately 65,000 people (SIL 1995/1991). They belong to the Senufo subfamily, but are separated from other Senufo languages by a small band of unrelated languages. Within Senufo they are th ...
, Mwan, Wan,
Yaouré Yaure (Yaouré, Yohowré, Youré) is a Mande language of Ivory Coast Ivory Coast, also known as Côte d'Ivoire, officially the Republic of Côte d'Ivoire, is a country on the southern coast of West Africa. Its capital is Yamoussoukro, in ...
, , Nyabwa and Godié. The Unicode character used for the tone letter (U+02D7) is different from the mathematical minus sign. The plus sign sometimes represents in the orthography of
Huichol The Huichol or Wixárika are an indigenous people of Mexico and the United States living in the Sierra Madre Occidental range in the states of Nayarit, Jalisco, Zacatecas, and Durango, as well as in the United States in the states of California ...
. In the algebraic notation used to record games of
chess Chess is a board game for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to dist ...
, the plus sign is used to denote a move that puts the opponent into
check Check or cheque, may refer to: Places * Check, Virginia Arts, entertainment, and media * ''Check'' (film), a 2021 Indian Telugu-language film * ''The Checks'' (episode), a 1996 TV episode of ''Seinfeld'' Games and sports * Check (chess), a thr ...
, while a double plus is sometimes used to denote
double check In chess and other related games, a double check is a check delivered by two pieces simultaneously. In chess notation, it is almost always represented the same way as a single check ("+"), but it is sometimes symbolized by "++" (however, "++" i ...
. Combinations of the plus and minus signs are used to evaluate a move (+/−, +/=, =/+, −/+). In linguistics, a superscript plus sometimes replaces the asterisk, which denotes unattested
linguistic reconstruction Linguistic reconstruction is the practice of establishing the features of an unattested ancestor language of one or more given languages. There are two kinds of reconstruction: * Internal reconstruction uses irregularities in a single language t ...
. In
botanical name A botanical name is a formal scientific name conforming to the ''International Code of Nomenclature for algae, fungi, and plants'' (ICN) and, if it concerns a plant cultigen, the additional cultivar or Group epithets must conform to the ''Internat ...
s, a plus sign denotes graft-chimaera. A plus sign written at the beginning of an international phone number is the " international prefix symbol" that "serves to remind the subscriber to dial the international prefix which differs from country to country and also serves to identify the number following as the international telephone number."


Character codes

The hyphen-minus sign, , is the original
ASCII ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding standard for electronic communication. ASCII codes represent text in computers, telecommunications equipment, and other devices. Because ...
version of the minus sign, which doubles as a
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. ''Son-in-law'' is an example of a hyphenated word. The hyphen is sometimes confused with dashes ( figure ...
. It is usually shorter in length than the plus sign and often at a different height to the plus-sign's cross bar. It can be used as a substitute for the true minus sign when the character set is limited to
ASCII ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding standard for electronic communication. ASCII codes represent text in computers, telecommunications equipment, and other devices. Because ...
. Most programming languages and other computer readable languages do this, since ASCII is generally available as a subset of most character encodings, while U+2212 is a Unicode feature only. Also several other software programs usable for calculations don't accept the U+2212 minus. For example, pasting =3−2 into Excel or 3−2= into the Windows calculator won't work. As the true minus is not available on most
keyboard layout A keyboard layout is any specific physical, visual or functional arrangement of the keys, legends, or key-meaning associations (respectively) of a computer keyboard, mobile phone, or other computer-controlled typographic keyboard. is the actua ...
s, typographers sometimes use the very similar en dash, U+2013, to represent the minus sign although it is "not preferred" in mathematical typesetting. Ways of producing the en dash are available on most computers; see .


Alternative minus signs

There is a
commercial minus sign The commercial minus sign is a typographical and mathematical symbol used in commercial and financial documents in some European languages, in specific contexts. In some commercial and financial documents, especially in Germany and Scandinavia ...
, , which is used in Germany and Scandinavia. The symbol is used to denote subtraction in
Scandinavia Scandinavia; Sámi languages: /. ( ) is a subregion in Northern Europe, with strong historical, cultural, and linguistic ties between its constituent peoples. In English usage, ''Scandinavia'' most commonly refers to Denmark, Norway, and Swe ...
.


Alternative plus sign

A
Jew Jews ( he, יְהוּדִים, , ) or Jewish people are an ethnoreligious group and nation originating from the Israelites Israelite origins and kingdom: "The first act in the long drama of Jewish history is the age of the Israelites""T ...
ish tradition that dates from at least the 19th century is to write ''plus'' using the symbol . This practice was adopted into
Israel Israel (; he, יִשְׂרָאֵל, ; ar, إِسْرَائِيل, ), officially the State of Israel ( he, מְדִינַת יִשְׂרָאֵל, label=none, translit=Medīnat Yīsrāʾēl; ), is a country in Western Asia. It is situated ...
i schools and is still commonplace today in elementary schools (including
secular Secularity, also the secular or secularness (from Latin ''saeculum'', "worldly" or "of a generation"), is the state of being unrelated or neutral in regards to religion. Anything that does not have an explicit reference to religion, either negativ ...
schools) but in fewer secondary schools.Christian-Jewish Dialogue: Theological Foundations By Peter von der Osten-Sacken (1986 – Fortress Press)
"In Israel the plus sign used in mathematics is represented by a horizontal stroke with a vertical hook instead of the sign otherwise used all over the world, because the latter is reminiscent of a cross." (Page 96)
It is also used occasionally in books by religious authors, but most books for adults use the international symbol . The reason for this practice is that it avoids the writing of a symbol that looks like a Christian cross.
Unicode Unicode, formally The Unicode Standard,The formal version reference is is an information technology standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems. The standard, wh ...
has this symbol at position .Unicode U+FB29 reference page
This form of the plus sign is also used on the control buttons at individual seats on board the El Al Israel Airlines aircraft.


See also

* En dash, a dash that looks similar to the subtraction symbol but is used for different purposes *
Plus–minus sign The plus–minus sign, , is a mathematical symbol with multiple meanings. *In mathematics, it generally indicates a choice of exactly two possible values, one of which is obtained through addition and the other through subtraction. *In experim ...
± * Glossary of mathematical symbols * ⊕ (disambiguation)


References and footnotes


External links

* * {{DEFAULTSORT:Plus and minus signs Elementary arithmetic Mathematical symbols Addition Subtraction de:Vorzeichen (Zahl)#Plus- und Minuszeichen