# Plus-minus 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 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".

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

## Usage

### In mathematics

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

- [math]\displaystyle{ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}. }[/math]

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

Similarly, the trigonometric identity

- [math]\displaystyle{ \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 "+" 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:

- [math]\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. }[/math]

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

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

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

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

- [math]\displaystyle{ \cos(A \pm B) = \cos(A) \cos(B) \mp \sin(A) \sin(B) }[/math]

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

- [math]\displaystyle{ \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]

but *not*

- [math]\displaystyle{ \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]

because the signs are exclusively alternating.

Another example is

- [math]\displaystyle{ x^3 \pm 1 = (x \pm 1)\left(x^2 \mp x + 1\right) }[/math]

which represents two equations.

## Encodings

- In Unicode: U+00B1 ± PLUS-MINUS SIGN (HTML
`±`

**·**`±`

) - In ISO 8859-1, -7, -8, -9, -13, -15, and -16, the plus-minus symbol is given by the code 0xB1
_{hex}Since the first 256 code points of Unicode are identical to the contents of ISO-8859-1 this symbol is also at Unicode code point U+00B1. - The symbol also has a HTML entity representation of
`±`

. - The rarer minus-plus sign (∓) is not generally found in legacy encodings and does not have a named HTML entity but is available in Unicode with code point U+2213 and 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

- On Windows systems, it may be entered by means of Alt codes, by holding the ALT key while typing the numbers 0177 or 241 on the numeric keypad.
- On Unix-like systems, it can be entered by typing the sequence compose + -.
- On Macintosh systems, it may be entered by pressing option shift = (on the non-numeric keypad).
- On the Chromebook, it may be entered by pressing shift, ctrl and u, and then writing the unicode for plus-minus (00B1).

## Similar characters

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

## See also

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

## References

- ↑
^{1.0}^{1.1}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
- ↑
^{3.0}^{3.1}Eade, James (2005),*Chess For Dummies*(2nd ed.), John Wiley & Sons, p. 272, ISBN 9780471774334, https://books.google.com/books?id=7eZxKNQu-JoC&pg=PA272. - ↑ Cajori, Florian (1928),
*A History of Mathematical Notations, Volumes 1-2*, Dover, p. 245, ISBN 9780486677668, https://books.google.com/books?id=7juWmvQSTvwC&pg=PA245.