Plus–minus sign

The plus–minus sign, ±, is a 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
  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
  minus-plus sign
  , ∓, indicates the same advantage for the black player.

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.^[1]

Usage

In mathematics

In

If x² = 9, one may give the solution as x = ±3. This indicates that the equation has two solutions: x = +3 and x = −3. A common use of this notation is found in the quadratic formula

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

which describes the two solutions to the quadratic equation ax²+bx+c = 0.

Similarly, the

${\displaystyle \sin(A\pm B)=\sin(A)\cos(B)\pm \cos(A)\sin(B)}$

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

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

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

${\displaystyle \cos(A\pm B)=\cos(A)\cos(B)\mp \sin(A)\sin(B)}$

which represents the two equations:

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

Another example is the conjugate of the perfect squares

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

which represents the two equations:

${\displaystyle x^{3}+y^{3}=(x+y)\left((x-y\right)^{2}+xy)}$

${\displaystyle x^{3}-y^{3}=(x-y)\left((x+y\right)^{2}-xy)}$

A related usage is found in this presentation of the formula for the Taylor series of the sine function:

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

Here, the plus-or-minus sign indicates that the term may be added or subtracted depending on whether 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 (−1)ⁿ, which gives +1 when n is even, and −1 when n is 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 as s₁, s₂, ... and specifying a value of +1 or −1 separately for each, or some appropriate relation, like ${\displaystyle s_{3}=s_{1}\cdot (s_{2})^{n}\,,}$ or similar.

In statistics

The use of ± for an approximation is most commonly encountered in presenting the numerical value of a quantity, together with its

Operations involving uncertain values should always try to preserve the uncertainty, in order to avoid propagation of error. If ${\displaystyle ~n=a\pm b\;,}$ any operation of the form ${\displaystyle ~m=f(n)~}$ must return a value of the form ${\displaystyle ~m=c\pm d~}$, where c is ${\displaystyle \,f(n)\,}$ and d is range updated using interval arithmetic.

In chess

The symbols ± and ∓ are used in

• In medicine, it may mean "with or without" in some cases.^[6][7]
• In
  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].^[8]

Encodings

• In Unicode: U+00B1 ± PLUS-MINUS SIGN
• In
  -16, the plus–minus symbol is code 0xB1_hex
  . This location was copied to Unicode.
• The symbol also has a
  HTML entity
  representations of ±, ±, and ±.
• The rarer minus–plus sign is not generally found in legacy encodings, but is available in Unicode as U+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 + 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 with screen readers.

Typing

• Windows: Alt+241 or Alt+0177 (numbers typed on the numeric keypad).
• Macintosh: ⌥ Option+⇧ Shift+= (equal sign on the non-numeric keypad).
• Unix-like systems: Compose,+,- or ⇧ Shift+Ctrl+u B1space (second works on Chromebook)
• In the Vim text editor (in Insert mode): Ctrl+k +- or Ctrl+v 177 or Ctrl+v x B1 or Ctrl+v u 00B1
• AutoCAD shortcut string: %%p

Similar characters

Look up 士, 土, or 干 in Wiktionary, the free dictionary.

The plus–minus sign resembles the

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

References