The plus and minus signs, and , are
mathematical symbols used to represent the notions of
positive and
negative, respectively. In addition, represents the operation of
addition, which results in a
sum
Sum most commonly means the total of two or more numbers added together; see addition.
Sum can also refer to:
Mathematics
* Sum (category theory), the generic concept of summation in mathematics
* Sum, the result of summation, the additio ...
, while represents
subtraction
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
, resulting in a
difference.
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 ...
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 s ...
or the
Hindu-Arabic numerals
Arabic numerals are the ten numerical digits: , , , , , , , , and . They are the most commonly used symbols to write decimal numbers. They are also used for writing numbers in other systems such as octal, and for writing identifiers such as c ...
, 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's
manuscripts
A manuscript (abbreviated MS for singular and MSS for plural) was, traditionally, any document written by hand – or, once practical typewriters became available, typewritten – as opposed to mechanically printed or reproduced in ...
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
Fra Luca Bartolomeo de Pacioli (sometimes ''Paccioli'' or ''Paciolo''; 1447 – 19 June 1517) was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and an early contributor to the field now known as accounting ...
'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 Regions of Italy, region. It is built on a group of 118 small islands that are separated by canals and linked by over 400 ...
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 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 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.
Robert Recorde, the designer of the
equals sign, introduced plus and minus to Britain in 1557 in ''
The Whetstone of Witte'': "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, as in . It can also serve as a
unary operator that leaves its
operand 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 number
In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed m ...
s ( versus ).
The plus sign can also indicate many other operations, depending on the mathematical system under consideration. Many
algebraic structures, such as
vector spaces and
matrix ring
In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ''U ...
s, have some operation which is called, or is equivalent to, addition. It is though conventional to use the plus sign to only denote
commutative operations.
The symbol is also used in
chemistry and
physics. For more, see .
Minus sign
The minus sign, , has three main uses in mathematics:
# The
subtraction
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
operator: a
binary operator to indicate the operation of subtraction, as in 5 − 3 = 2. Subtraction is the inverse of addition.
# The
function whose
value for any
real or
complex argument is the
additive inverse of that argument. For example, if , then , but if , then . Similarly, .
# A
prefix
A prefix is an affix which is placed before the Word stem, stem of a word. Adding it to the beginning of one word changes it into another word. For example, when the prefix ''un-'' is added to the word ''happy'', it creates the word ''unhappy'' ...
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 ...
, 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
In mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression.
For exampl ...
mean that is equal to :
Exponentiation 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 ...
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. 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 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 are often qualified with a plus or minus to indicate the presence or absence of the
Rh factor. For example, A+ means
type A blood 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
A subscript or superscript is a character (such as a number or letter) that is set slightly below or above the normal line of type, respectively. It is usually smaller than the rest of the text. Subscripts appear at or below the baseline, whil ...
.
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 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
string
String or strings may refer to:
*String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects
Arts, entertainment, and media Films
* ''Strings'' (1991 film), a Canadian anim ...
s is written , and results in .
In most programming languages, subtraction and negation are indicated with the ASCII
hyphen-minus
The hyphen-minus is the most commonly used type of hyphen, widely used in digital documents. It is the only character that looks like a minus sign or a dash in many character sets such as ASCII or on most keyboards, so it is also used as such. ...
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, as in .
In
C and some other computer programming languages, two plus signs indicate the
increment operator
Increment or incremental may refer to:
*Incrementalism, a theory (also used in politics as a synonym for gradualism)
*Increment and decrement operators, the operators ++ and -- in computer programming
*Incremental computing
*Incremental backup, wh ...
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++.
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
Signed zero is zero with an associated sign. In ordinary arithmetic, the number 0 does not have a sign, so that −0, +0 and 0 are identical. However, in computing, some number representations allow for the existence of two zeros, often denoted b ...
may have a separate representation from zero. In the
IEEE floating-point standard
The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point arithmetic established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard addressed many problems found in ...
, 1 / −0 is
negative infinity
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra on ...
(
) whereas 1 / 0 is
positive infinity
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra ...
(
).
Other uses
In physics, the use of plus and minus signs for different
electrical charges was introduced by
Georg Christoph Lichtenberg.
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.
In the
International Phonetic Alphabet
The International Phonetic Alphabet (IPA) is an alphabetic system of phonetic transcription, phonetic notation based primarily on the Latin script. It was devised by the International Phonetic Association in the late 19th century as a standa ...
,
subscript
A subscript or superscript is a character (such as a number or letter) that is set slightly below or above the normal line of type, respectively. It is usually smaller than the rest of the text. Subscripts appear at or below the baseline, whil ...
ed 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,
Mwan,
Wan,
Yaouré,
Wè,
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.
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, while a double plus is sometimes used to denote
double check. 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.
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 '' Inte ...
s, a plus sign denotes
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 between those of its "parents". A graft-chimaera is not a true hybrid but a mixture of cells, each wi ...
.
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. 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 layouts, 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, , 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
Jewish 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 schools) but in fewer
secondary school
A secondary school describes an institution that provides secondary education and also usually includes the building where this takes place. Some secondary schools provide both '' lower secondary education'' (ages 11 to 14) and ''upper secondar ...
s.
[Christian-Jewish Dialogue: Theological Foundations By Peter von der Osten-Sacken (1986 – Fortress Press)](_blank)
"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, ...
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 ±
*
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