plus–minus sign#Minus plus sign

{{Infobox symbol

|mark=±

|unicode= {{unichar|B1|Plus-minus Sign|html=}}

|see also= {{unichar|2213|Minus-or-plus sign|html=}}

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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 sciences, the {{char|±}} sign commonly indicates the confidence interval or uncertainty bounding a range of possible 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.{{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]}}.

Usage

=In mathematics=

In mathematical formulas, 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.{{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}}

If {{math|1=x² = 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

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

which describes the two solutions to the quadratic equation {{math|1=ax² + bx + c = 0.}}

Similarly, the trigonometric 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 {{char|+}} on both sides of the equation, and one with {{char|−}} on both sides.

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:

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

which represents the two equations:

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

Another example is the conjugate of the perfect squares

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

which represents the two equations:

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

A related usage is found in this presentation of the formula for the Taylor series of 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 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|(−1){{sup|n}}}}, which gives +1 when {{mvar|n}} is even, and −1 when {{mvar|n}} is odd. In older texts one occasionally finds {{math|(−){{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 −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 −1 separately for each, or some appropriate relation, like {{math|s₃ {{=}} s₁ · (s₂)ⁿ}} 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 tolerance or its statistical margin of error.{{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}} 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 standard deviations (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 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 to denote a moderate but significant advantage for White and Black, respectively.{{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}}. 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.For details, see {{section link|Chess annotation symbols#Positions}}.

=Other meanings=

In medicine, it may mean "with or without" in some cases.{{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}}{{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}}
In engineering, the sign indicates the 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 amplifiers.
In linguistics, it may indicate a distinctive feature, such as [±voiced].{{cite book |last=Hornsby |first=David |title=Linguistics, A Complete Introduction |isbn=9781444180336 |pages=99}}

Encodings

In Unicode: {{unichar|00B1|PLUS-MINUS SIGN}}
In ISO 8859-1, -7, -8, -9, -13, -15, and -16, the plus–minus symbol is code 0xB1_hex. This location was copied to Unicode.
In HTML, the symbol also has character entity reference representations of &pm;, ±
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 ∓ or ∓.
In TeX 'plus-or-minus' and 'minus-or-plus' symbols are denoted \pm and \mp, respectively.
Although these characters may be approximated by underlining or overlining a {{char|+}} symbol ( {{underline|+}} or {{Overline|+}} ), 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 readers.

=Typing=

Windows: {{keypress|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|+|-|chain=,}} or {{keypress|Shift|Ctrl|u}}{{nbsp}}{{keypress|B|1|space|chain=}} (second works on Chromebook)
In the 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

The plus–minus sign resembles the Chinese characters {{lang|zh|土}} (Radical 32) and {{lang|zh|士}} (Radical 33), whereas the minus–plus sign resembles {{lang|zh|干}} (Radical 51).