# Plus–minus sign

± | |
---|---|

Plus–minus sign | |

In Unicode | U+00B1 ± PLUS-MINUS SIGN (±, ±, ±) |

Related | |

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

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 obtained through addition and the other through subtraction.
^{[1]} - In experimental sciences, the sign commonly indicates the confidence interval or uncertainty bounding a range of possible errors in a measurement, often the standard deviation or standard error.
^{[2]}The sign may also represent an inclusive range of values that a reading might have. - In medicine, it means "with or without".
^{[3]}^{[4]} - 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.
- 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 minus-or-plus sign, ∓, indicates the same advantage for the black player.
^{[5]} - 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].
^{[6]}

## 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 1631, in William Oughtred's *Clavis Mathematicae*.^{[7]}

## Usage[edit]

### In mathematics[edit]

In mathematical formulas, the ± symbol may be used to indicate a symbol that may be replaced by either the plus and minus signs, + or −, allowing the formula to represent two values or two equations.^{[8]}

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

which describes 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.

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 nor 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 both of them 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, without a ∓, 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:

which represents the two equations:

Another example where the minus–plus sign appears is

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 term may be added or subtracted, in this case depending on whether n is odd or even, the rule can be deduced from the first few terms. 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 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 s_{1}, s_{2}, ... and specifying a value of +1 or −1 separately for each, or some appropriate relation, like or similar.

### 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.^{[2]} 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).

Operations involving uncertain values should always try to preserve the uncertainty—in order to avoid propagation of error. If any operation of the form must return a value of the form , where c is and d is range updated using interval arithmetic.

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).^{[citation needed]} 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 to only use + and –.^{[5]} If several different symbols are used together, then the symbols + and − denote a clearer advantage than ± and ∓. When finer evaluation is desired, three pairs of symbols are used: ⩲ and ⩱ for only a slight advantage; ± and ∓ for a significant advantage; and +– and –+ for a potentially winning advantage, in each case for white or black respectively.^{[9]}

## 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 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 also be produced using underlining or overlining + symbol (
__+__or + ), this is deprecated 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[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``space`(second works on Chromebook) - In the Vim text editor (in Insert mode):
`Ctrl`+`k``+``-`or`Ctrl`+`v``1``7``7`or`Ctrl`+`v``x``B``1`or`Ctrl`+`v``u``0``0``B``1` - AutoCAD shortcut string:
`%%p`

## Similar characters[edit]

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

## See also[edit]

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

## References[edit]

**^**Weisstein, Eric W. "Plus or Minus".*mathworld.wolfram.com*. Retrieved 2020-08-28.- ^
^{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. PMID 7124681. **^**Naess, I. A.; Christiansen, S. C.; Romundstad, P.; Cannegieter, S. C.; Rosendaal, F. R.; Hammerstrøm, J. (2007). "Incidence and mortality of venous thrombosis: a population-based study".*Journal of Thrombosis and Haemostasis*.**5**(4): 692–699. doi:10.1111/j.1538-7836.2007.02450.x. ISSN 1538-7933. PMID 17367492.**^**Heit, J. A.; Silverstein, M. D.; Mohr, D. N.; Petterson, T. M.; O'Fallon, W. M.; Melton, L. J. (1999-03-08). "Predictors of survival after deep vein thrombosis and pulmonary embolism: a population-based, cohort study".*Archives of Internal Medicine*.**159**(5): 445–453. doi:10.1001/archinte.159.5.445. ISSN 0003-9926. PMID 10074952.- ^
^{a}^{b}Eade, James (2005),*Chess For Dummies*(2nd ed.), John Wiley & Sons, p. 272, ISBN 9780471774334. **^**Hornsby, David.*Linguistics, A Complete Introduction*. p. 99. ISBN 9781444180336.**^**Cajori, Florian (1928),*A History of Mathematical Notations, Volume I: Notations in Elementary Mathematics*, Open Court, p. 245.**^**"Definition of PLUS/MINUS SIGN".*www.merriam-webster.com*. Retrieved 2020-08-28.**^**For details, see Chess annotation symbols#Positions.