View source for Plus–minus sign

{{Short description|Symbol combining both + and - signs}}
{{Other uses|Plus–minus (disambiguation)}}
{{Infobox symbol
|mark=±
|unicode=  {{unichar|B1|Plus-minus Sign|html=}}
|see also= {{unichar|2213|Minus-or-plus sign|html=}}
}}

The '''plus–minus sign''' or '''plus-or-minus sign''' ({{char|±}}) and the complementary '''minus-or-plus sign''' ({{char|∓}}) are symbols with broadly similar multiple meanings.
*In [[mathematics]], the {{char|±}} sign generally indicates a choice of exactly two possible values, one of which is obtained through [[addition]] and the other through [[subtraction]].
*In [[statistics]] and [[experimental science]]s, the {{char|±}} sign commonly indicates the [[confidence interval]] or [[measurement uncertainty|uncertainty]] bounding a range of possible [[errors and residuals in statistics|errors]] in a measurement, often the [[standard deviation]] or [[standard error]]. The sign may also represent an inclusive range of values that a reading might have.
*In [[chess]], the {{char|±}} sign indicates a clear advantage for the white player; the complementary minus-plus sign ({{char|∓}}) indicates a clear advantage for the black player.
Other meanings occur in other fields, including medicine, engineering, chemistry, electronics, linguistics, and philosophy.

==History==
A version of the sign, including also the French word ''ou'' ("or"), was used in its mathematical meaning by [[Albert Girard]] in 1626, and the sign in its modern form was used as early as 1631, in [[William Oughtred]]'s ''Clavis Mathematicae''.<ref>{{citation |title=A History of Mathematical Notations, Volume I: Notations in Elementary Mathematics |title-link=A History of Mathematical Notations |last=Cajori |first=Florian |author-link=Florian Cajori |publisher=Open Court |year=1928 |page=[http://archive.org/details/historyofmathema031756mbp/page/n263 245]}}.</ref>

==Usage==

===In mathematics===
<!-- [[± shorthand]] redirects here -->
In [[mathematical formula]]s, the {{char|±}} symbol may be used to indicate a symbol that may be replaced by either of the [[plus and minus signs]], {{char|+}} or {{char|−}}, allowing the formula to represent two values or two equations.<ref>{{cite web |title=Definition of PLUS/MINUS SIGN |website=merriam-webster.com |language=en |url=http://www.merriam-webster.com/dictionary/plus%2Fminus+sign |access-date=2020-08-28}}</ref>

If {{math|1=''x''<sup>2</sup> = 9}}, one may give the solution as {{math|1=''x'' = ±3}}. This indicates that the equation has two solutions: {{math|1= ''x'' = +3}} and {{math|1=''x'' = −3}}. A common use of this notation is found in the [[quadratic formula]]
:<math>x = \frac{-b \pm \sqrt{b^2-4ac}}{2a},</math>
which describes the two solutions to the [[quadratic equation]] {{math|1=''ax''<sup>2</sup> + ''bx'' + ''c'' = 0.}}

Similarly, the [[trigonometric identity]]
:<math>\sin(A \pm B) = \sin(A) \cos(B) \pm \cos(A) \sin(B)</math>
can be interpreted as a shorthand for two equations: one with {{char|+}} on both sides of the equation, and one with {{char|−}} on both sides.

{{anchor|Minus plus sign}}
The '''minus–plus sign''', {{char|∓}}, is generally used in conjunction with the {{char|±}} sign, in such expressions as {{math|''x'' ± ''y'' ∓ ''z''}}, which can be interpreted as meaning {{math|''x'' + ''y'' − ''z''}} or {{math|''x'' − ''y'' + ''z''}} (but {{em|not}} {{math|''x'' + ''y'' + ''z''}}  or {{math|''x'' − ''y'' − ''z''}}). The {{char|∓}} always has the opposite sign  to  {{char|±}}.

The above expression can be rewritten as {{math|''x''  ± (''y'' − ''z'')}} to avoid use of {{char|∓}}, but cases such as the trigonometric identity are most neatly written using the "∓" sign:
:<math>\cos(A \pm B) = \cos(A) \cos(B) \mp \sin(A) \sin(B) </math>
which represents the two equations:
:<math>\begin{align}
\cos(A + B) &= \cos(A)\cos(B) - \sin(A) \sin(B) \\
\cos(A - B) &= \cos(A)\cos(B) + \sin(A) \sin(B) \end{align}</math>

Another example is the [[conjugate (square roots)|conjugate]] of the [[square number|perfect squares]]
:<math>x^3 \pm y^3 = (x \pm y)\left((x \mp y)^2 \pm xy\right)</math>
which represents the two equations:
:<math>\begin{align}
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{align}</math>

A related usage is found in this presentation of the formula for the [[Taylor series]] of the sine function:
:<math>\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 </math>
Here, the plus-or-minus sign indicates that the term may be added or subtracted depending on whether {{mvar|n}} is 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 {{math|(&minus;1){{sup|''n''}}}}, which gives +1 when {{mvar|n}} is even, and &minus;1 when {{mvar|n}} is odd. In older texts one occasionally finds {{math|(&minus;){{sup|''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 &minus;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 as {{math|''s''{{sub|1}}}}, {{math|''s''{{sub|2}}}}, ... and specifying a value of +1 or &minus;1 separately for each, or some appropriate relation, like {{math|''s''<sub>3</sub> {{=}} ''s''<sub>1</sub> · (''s''<sub>2</sub>)<sup>''n''</sup>}} or similar.

===In statistics===
The use of {{char|±}} for an approximation is most commonly encountered in presenting the numerical value of a quantity, together with its [[engineering tolerance|tolerance]] or its statistical [[margin of error]].<ref name="stderror">{{cite journal |last=Brown |first=George W. |year=1982 |title=Standard deviation, standard error: Which 'standard' should we use? |journal=American Journal of Diseases of Children |volume=136 |issue=10 |pages=937–941 |pmid=7124681 |doi=10.1001/archpedi.1982.03970460067015}}</ref> For example, {{nowrap|5.7 ± 0.2}} may 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 2&nbsp;[[standard deviation]]s (a probability of 68.3% or 95.4% in a [[normal distribution]]).

Operations involving uncertain values should always try to preserve the uncertainty, in order to avoid [[propagation of uncertainty|propagation of error]]. If {{math|''n'' {{=}} ''a'' ± ''b''}}, any operation of the form {{math|''m'' {{=}} ''f''(''n'')}} must return a value of the form {{math|''m'' {{=}} ''c'' ± ''d''}}, where {{mvar|c}} is {{math|''f''(''a'')}} and {{mvar|d}} is the range {{mvar|b}} updated using [[interval arithmetic]].

===In chess===
The symbols {{char|±}} and {{char|∓}} are used in [[Chess annotation symbols|chess annotation]] to denote a moderate but significant advantage for White and Black, respectively.<ref name="chess">{{citation |title=Chess For Dummies |last=Eade |first=James |edition=2nd |publisher=John Wiley & Sons |year=2005 |isbn=9780471774334 |page=272 |url=https://books.google.com/books?id=7eZxKNQu-JoC&pg=PA272}}.</ref> Weaker and stronger advantages are denoted by {{char|⩲}} and {{char|⩱}} for only a slight advantage, and {{char|+–}} and {{char|–+}} for a strong, potentially winning advantage, again for White and Black respectively.<ref>For details, see {{section link|Chess annotation symbols#Positions}}.</ref>

===Other meanings===
*In [[medicine]], it may mean "with or without" in some cases.<ref>{{cite journal |last1=Naess |first1=I. A. |last2=Christiansen |first2=S. C. |last3=Romundstad |first3=P. |last4=Cannegieter |first4=S. C. |last5=Rosendaal |first5=F. R. |last6=Hammerstrøm |first6=J. |date=2007 |title=Incidence and mortality of venous thrombosis: a population-based study |journal=Journal of Thrombosis and Haemostasis |volume=5 |issue=4 |pages=692–699 |doi=10.1111/j.1538-7836.2007.02450.x |issn=1538-7933 |pmid=17367492 |s2cid=23648224}}</ref><ref>{{cite journal |last1=Heit |first1=J. A. |last2=Silverstein |first2=M. D. |last3=Mohr |first3=D. N. |last4=Petterson |first4=T. M. |last5=O'Fallon |first5=W. M. |last6=Melton |first6=L. J. |date=1999-03-08 |title=Predictors of survival after deep vein thrombosis and pulmonary embolism: a population-based, cohort study |journal=Archives of Internal Medicine |volume=159 |issue=5 |pages=445–453 |doi=10.1001/archinte.159.5.445 |issn=0003-9926 |pmid=10074952}}</ref>
*In [[engineering]], the sign indicates the [[engineering tolerance|tolerance]], which is the range of values that are considered to be acceptable or safe, or which comply with some standard or with a contract.
*In [[chemistry]], the sign is used to indicate a [[racemic mixture]].
*In [[electronics]], this sign may indicate a dual voltage power supply, such as ±5 volts means +5 volts and −5 volts, when used with audio circuits and [[operational amplifier]]s.
*In [[linguistics]], it may indicate a [[distinctive feature]], such as [±voiced].<ref>{{cite book |last=Hornsby |first=David |title=Linguistics, A Complete Introduction |isbn=9781444180336 |pages=99}}</ref>

==Encodings==
*In [[Unicode]]: {{unichar|00B1|PLUS-MINUS SIGN}}
*In [[ISO 8859-1]], [[ISO 8859-7|-7]], [[ISO 8859-8|-8]], [[ISO 8859-9|-9]], [[ISO 8859-13|-13]], [[ISO 8859-15|-15]], and [[ISO 8859-16|-16]], the plus–minus symbol is code 0xB1<sub>[[hexadecimal|hex]]</sub>. This location was copied to Unicode.
*In [[HTML]], the symbol also has [[character entity reference]] representations of <code>&amp;pm;</code>, <code>&amp;plusmn;</code>
*The rarer minus–plus sign is not generally found in legacy encodings, but is available in Unicode as {{unichar|2213|MINUS-OR-PLUS SIGN}} so can be used in HTML using <code>&amp;#x2213;</code> or <code>&amp;#8723;</code>.
*In [[TeX]] 'plus-or-minus' and 'minus-or-plus' symbols are denoted <code>\pm</code> and <code>\mp</code>, respectively.
*Although these characters may be approximated by underlining or overlining a {{char|+}} symbol (&nbsp;{{underline|+}}&nbsp; or {{Overline|+}}&nbsp;), 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 with [[screen reader]]s.

===Typing===
*[[Microsoft Windows|Windows]]: {{keypress|[[Alt code|Alt]]|2}}{{keypress|4|1|chain=}} or {{keypress|Alt|0}}{{keypress|1|7|7|chain=}} (numbers typed on the [[numeric keypad]]).
*Macintosh: {{keypress|Option|Shift|{{=}}}} (equal sign on the non-numeric keypad).
*[[Unix-like]] systems: {{keypress|[[Compose key|Compose]]|+|-|chain=,}} or {{keypress|Shift|Ctrl|u}}{{nbsp}}{{keypress|B|1|space|chain=}} (second works on [[Chromebook]])
* In the [[Vim (text editor)|Vim text editor]] (in Insert mode): {{keypress|Ctrl|k}} {{keypress|+|-|chain=}} or {{keypress|Ctrl|v}} {{keypress|1|7|7|chain=}} or {{keypress|Ctrl|v}} {{keypress|x}} {{keypress|B|1|chain=}} or {{keypress|Ctrl|v}} {{keypress|u}} {{keypress|0|0|B|1|chain=}}
* [[AutoCAD]] shortcut string: {{code|%%p}}

==Similar characters==
{{Wiktionary|士|土|干}}
The plus–minus sign resembles the [[Chinese character]]s {{lang|zh|土}} ([[Radical 32]]) and {{lang|zh|士}} ([[Radical 33]]), whereas the minus–plus sign resembles {{lang|zh|干}} ([[Radical 51]]).

==See also==
*[[≈]] (approximately equal to)
*[[Engineering tolerance]]
*[[Plus and minus signs]]
*[[Sign (mathematics)]]
*[[Table of mathematical symbols]]

==References==
{{Reflist|25em}}

{{Navbox punctuation}}

[[Category:Addition]]
[[Category:Elementary arithmetic]]
[[Category:Mathematical symbols]]
[[Category:Sign (mathematics)]]
[[Category:Subtraction]]