Plus–minus sign

For other uses, seePlus–minus (disambiguation).

Theplus–minus signorplus-or-minus sign,±,is a symbol with multiple meanings.

• Inmathematics,it generally indicates a choice of exactly two possible values, one of which is obtained throughadditionand the other throughsubtraction.
• Instatisticsandexperimental sciences,the sign commonly indicates theconfidence intervaloruncertaintybounding a range of possibleerrorsin a measurement, often thestandard deviationorstandard error.The sign may also represent an inclusive range of values that a reading might have.
• Inchess,the sign indicates a clear advantage for the white player; the complementaryminus-plus sign,∓indicates the same advantage for the black player.

±
Plus–minus sign
InUnicode	U+00B1±PLUS-MINUS SIGN(±, ±, ±)
Related
See also	U+2213∓MINUS-OR-PLUS SIGN(∓, ∓, ∓)

Plus or minus sign

Other meanings occur in other fields, including medicine, engineering, chemistry, electronics, linguistics, and philosophy.

History

A version of the sign, including also the French wordou( "or" ), was used in its mathematical meaning byAlbert Girardin 1626, and the sign in its modern form was used as early as 1631, inWilliam Oughtred'sClavis Mathematicae.

Usage

In mathematics

Inmathematical formulas,the±symbol may be used to indicate a symbol that may be replaced by either of theplus and minus signs,+or−,allowing the formula to represent two values or two equations.

Ifx²= 9,one may give the solution asx= ±3.This indicates that the equation has two solutions:x= +3andx= −3.A common use of this notation is found in thequadratic formula

$x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}},$

which describes the two solutions to thequadratic equationax²+bx+c= 0.

Similarly, thetrigonometric identity

$\sin(A\pm B)=\sin(A)\cos(B)\pm \cos(A)\sin(B)$

can be interpreted as a shorthand for two equations: one with+on both sides of the equation, and one with−on both sides.

Theminus–plus sign,∓,is generally used in conjunction with the±sign, in such expressions asx±y∓z,which can be interpreted as meaningx+y−zorx−y+z(butnotx+y+zorx−y−z). The∓always has the opposite sign to±.

The above expression can be rewritten asx± (y−z)to avoid use of∓,but cases such as the trigonometric identity are most neatly written using the "∓" sign:

$\cos(A\pm B)=\cos(A)\cos(B)\mp \sin(A)\sin(B)$

which represents the two equations:

${\begin{aligned}\cos(A+B)&=\cos(A)\cos(B)-\sin(A)\sin(B)\\\cos(A-B)&=\cos(A)\cos(B)+\sin(A)\sin(B)\end{aligned}}$

Another example is theconjugateof theperfect squares

$x^{3}\pm y^{3}=(x\pm y)\left((x\mp y)^{2}\pm xy\right)$

which represents the two equations:

${\begin{aligned}x^{3}+y^{3}&=(x+y)\left((x-y)^{2}+xy\right)\\x^{3}-y^{3}&=(x-y)\left((x+y)^{2}-xy\right)\end{aligned}}$

A related usage is found in this presentation of the formula for theTaylor seriesof the sine function:

$\sin \left(x\right)=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \pm {\frac {1}{(2n+1)!}}x^{2n+1}+\cdots$

Here, the plus-or-minus sign indicates that the term may be added or subtracted depending on whethernis odd or even; a rule which can be deduced from the first few terms. A more rigorous presentation would multiply each term by a factor of(−1)ⁿ,which gives +1 whennis even, and −1 whennis odd. In older texts one occasionally finds(−)ⁿ,which means the same.

When the standard presumption that the plus-or-minus signs all take on the same value of +1 or all −1 is not true, then the line of text that immediately follows the equation must contain a brief description of the actual connection, if any, most often of the form"where the ‘±’ signs are independent"or similar. If a brief, simple description is not possible, the equation must be re-written to provide clarity; e.g. by introducing variables such ass₁,s₂,... and specifying a value of +1 or −1 separately for each, or some appropriate relation, likes₃=s₁· (s₂)ⁿor similar.

In statistics

The use of±for an approximation is most commonly encountered in presenting the numerical value of a quantity, together with itstoleranceor its statisticalmargin of error.For example,5.7 ± 0.2may be anywhere in the range from 5.5 to 5.9 inclusive. In scientific usage, it sometimes refers to a probability of being within the stated interval, usually corresponding to either 1 or 2standard deviations(a probability of 68.3% or 95.4% in anormal distribution).

Operations involving uncertain values should always try to preserve the uncertainty, in order to avoidpropagation of error.Ifn=a±b,any operation of the formm=f(n)must return a value of the formm=c±d,wherecisf(a)anddis the rangebupdated usinginterval arithmetic.

In chess

The symbols±and∓are used inchess annotationto denote a moderate but significant advantage for White and Black, respectively.Weaker and stronger advantages are denoted by⩲and⩱for only a slight advantage, and+–and–+for a strong, potentially winning advantage, again for White and Black respectively.

Other meanings

• Inmedicine,it may mean "with or without" in some cases.
• Inengineering,the sign indicates thetolerance,which is the range of values that are considered to be acceptable or safe, or which comply with some standard or with a contract.
• Inchemistry,the sign is used to indicate aracemic mixture.
• Inelectronics,this sign may indicate a dual voltage power supply, such as ±5 volts means +5 volts and −5 volts, when used with audio circuits andoperational amplifiers.
• Inlinguistics,it may indicate adistinctive feature,such as [±voiced].

Encodings

• InUnicode:U+00B1±PLUS-MINUS SIGN
• InISO 8859-1,-7,-8,-9,-13,-15,and-16,the plus–minus symbol is code 0xB1_hex.This location was copied to Unicode.
• The symbol also has aHTML entityrepresentations of±,±,and±.
• The rarer minus–plus sign is not generally found in legacy encodings, but is available in Unicode asU+2213∓MINUS-OR-PLUS SIGNso can be used in HTML using∓or∓.
• InTeX'plus-or-minus' and 'minus-or-plus' symbols are denoted\pmand\mp,respectively.
• Although these characters may be approximated by underlining or overlining a+symbol (+or+), this is discouraged because the formatting may be stripped at a later date, changing the meaning. It also makes the meaning less accessible to blind users withscreen readers.

Typing

• Windows:Alt+241orAlt+0177(numbers typed on thenumeric keypad).
• Macintosh:⌥ Option+⇧ Shift+=(equal sign on the non-numeric keypad).
• Unix-likesystems:Compose,+,-or⇧ Shift+Ctrl+uB1space(second works onChromebook)
• In theVim text editor(in Insert mode):Ctrl+k+-orCtrl+v177orCtrl+vxB1orCtrl+vu00B1
• AutoCADshortcut string:%%p

Similar characters

The plus–minus sign resembles theChinese charactersThổ(Radical 32) andSĩ(Radical 33), whereas the minus–plus sign resemblesCàn(Radical 51).

See also

• ≈(approximately equal to)
• Engineering tolerance
• Plus and minus signs
• Sign (mathematics)
• Table of mathematical symbols