# Plus–minus sign

± | |
---|---|

Plus–minus sign | |

In Unicode | U+00B1 ± PLUS-MINUS SIGN (HTML `±` · `±` ) |

Related | |

See also | U+2213 ∓ MINUS-OR-PLUS SIGN (HTML `∓` ) |

The **plus–minus sign** (±) is a mathematical symbol with multiple meanings.

- In mathematics, it generally indicates a choice of exactly two possible values, one of which is the negation of the other.
- In experimental sciences, the sign commonly indicates the confidence interval or error in a measurement, often the standard deviation or standard error.
^{[1]}The sign may also represent an inclusive range of values that a reading might have. - In engineering the sign indicates the tolerance, which is the range of values that are considered to be acceptable, safe, or which comply with some standard, or with a contract.
^{[2]} - In botany it is used in morphological descriptions to notate "more or less".
- In chemistry the sign is used to indicate a racemic mixture.
- In chess, the sign indicates a clear advantage for the white player; the complementary sign ∓ indicates the same advantage for the black player.
^{[3]}

The sign is normally pronounced "plus or minus" or "plus–minus".^{[citation needed]}

## Contents

## History[edit]

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 William Oughtred's *Clavis Mathematicae* (1631).^{[4]}

## Usage[edit]

### In mathematics[edit]

In mathematical formulas, the ± symbol may be used to indicate a symbol that may be replaced by either the + or − symbols, allowing the formula to represent two values or two equations.

For example, given the equation *x*^{2} = 9, one may give the solution as *x* = ±3. This indicates that the equation has two solutions, each of which may be obtained by replacing this equation by one of the two equations *x* = +3 or *x* = −3. Only one of these two replaced equations is true for any valid solution. A common use of this notation is found in the quadratic formula

describing the two solutions to the quadratic equation *ax*^{2} + *bx* + *c* = 0.

Similarly, the trigonometric identity

can be interpreted as a shorthand for two equations: one with "+" on both sides of the equation, and one with "−" on both sides. The two copies of the ± sign in this identity must both be replaced in the same way: it is not valid to replace one of them with "+" and the other of them with "−". In contrast to the quadratic formula example, both of the equations described by this identity are simultaneously valid.

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

Here, the plus-or-minus sign indicates that the signs of the terms alternate, where (starting the count at 0) the terms with an even index *n* are added while those with an odd index are subtracted. A more rigorous presentation of the same formula would multiply each term by a factor of (−1)^{n}, which gives +1 when *n* is even and −1 when *n* is odd.

### In statistics[edit]

The use of ⟨±⟩ 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.^{[1]}
For example, "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).

A percentage may also be used to indicate the error margin. For example, 230 ± 10% V refers to a voltage within 10% of either side of 230 V (from 207 V to 253 V inclusive). Separate values for the upper and lower bounds may also be used. For example, to indicate that a value is most likely 5.7 but may be as high as 5.9 or as low as 5.6, one may write 5.7+0.2

−0.1.

### In chess[edit]

The symbols ± and ∓ are used in chess notation to denote an advantage for white and black respectively. However, the more common chess notation would be only + and –.^{[3]} If a difference is made, the symbols + and − denote a larger advantage than ± and ∓.

## Minus–plus sign[edit]

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*" and/or "*x* − *y* + *z*", but *not* "*x* + *y* + *z*" or "*x* − *y* − *z*". The upper "−" in "∓" is considered to be associated to the "+" of "±" (and similarly for the two lower symbols) even though there is no visual indication of the dependency. (However, the "±" sign is generally preferred over the "∓" sign, so if they both appear in an equation it is safe to assume that they are linked. On the other hand, if there are two instances of the "±" sign in an expression, it is impossible to tell from notation alone whether the intended interpretation is as two or four distinct expressions.) The original expression can be rewritten as "*x* ± (*y* − *z*)" to avoid confusion, but cases such as the trigonometric identity

are most neatly written using the "∓" sign. The trigonometric equation above thus represents the two equations:

but *not*

because the signs are exclusively alternating.

Another example is

which represents two equations.

## Encodings[edit]

- In Unicode: U+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. - The symbol also has a HTML entity representation of
`±`

, or you can use`±`

. - 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. - These characters may also be produced as an underlined or overlined + symbol (
__+__or + ), but beware of the formatting being stripped at a later date, changing the meaning.

### Typing[edit]

- Windows:
`Alt`+`2``4``1`or`Alt`+`0``1``7``7`(numbers typed on the numeric keypad). - Macintosh:
`⌥ Option`+`⇧ Shift`+`=`(equal sign on the non-numeric keypad). - Unix-like systems:
`Compose`,`+`,`-`or`⇧ Shift`+`Ctrl`+`u``B``1`(second works on Chromebook)

## Similar characters[edit]

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

The plus–minus sign resembles the Chinese characters 士 and 土, whereas the minus–plus sign resembles 干.

## See also[edit]

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

## References[edit]

- ^
^{a}^{b}Brown, George W. (1982), "Standard Deviation, Standard Error: Which 'Standard' Should We Use?",*American Journal of Diseases of Children*,**136**(10): 937–941, doi:10.1001/archpedi.1982.03970460067015. **^**Engineering tolerance- ^
^{a}^{b}Eade, James (2005),*Chess For Dummies*(2nd ed.), John Wiley & Sons, p. 272, ISBN 9780471774334. **^**Cajori, Florian (1928),*A History of Mathematical Notations, Volumes 1-2*, Dover, p. 245, ISBN 9780486677668.