# Plus-minus sign

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Plus-minus sign
±

The plus-minus sign (±) is a mathematical symbol commonly used either

The sign is normally pronounced "plus or minus". In experimental sciences, the sign commonly indicates the confidence interval or error 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 mathematics, it may indicate two possible values: one positive, and one negative.

## Precision indication

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. For example, "5.7±0.2" denotes a quantity that is specified or estimated to be within 0.2 units of 5.7; it may be anywhere in the range from 5.7 − 0.2 (i.e., 5.5) to 5.7 + 0.2 (5.9). 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 (207 V to 253 V). 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 could write 5.7+0.2
−0.1
.

## Shorthand

In mathematical equations, the use of ± may be found as shorthand, to present two equations in one formula: + or −, represented with ±.

For example, given the equation x2 = 1, one may give the solution as x = ±1, such that both x = +1 and x = −1 are valid solutions.

More generally we have the quadratic formula:

If ax2 + bx + c = 0 then

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

Written out in full, this states that there are two solutions to the equation, i.e. that

$\text{either } x = \frac{-b + \sqrt {b^2-4ac}}{2a} \text{ or } x = \frac{-b - \sqrt {b^2-4ac}}{2a}.$

Another example is found in the trigonometric identity

$\sin(x \pm y) = \sin(x) \cos(y) \pm \cos(x) \sin(y).\,$

This stands for two identities: one with "+" on both sides of the equation, and one with "−" on both sides.

A somewhat different use is found in this presentation of the formula for the Taylor series of the sine function:

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

This mild abuse of notation is intended to indicate 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 would use the expression (−1)n, which gives +1 when n is even and −1 when n is odd.

## Minus-plus sign

There is another character, the minus-plus sign (). It is generally used in conjunction with the "±" sign, in such expressions as "x ± y z", which can be interpreted as meaning "x + yz" or/and "xy + z", but not "x + y + z" nor "xyz". 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. The original expression can be rewritten as "x ± (yz)" to avoid confusion, but cases such as the trigonometric identity

$\cos(x \pm y) = \cos(x) \cos(y) \mp \sin(x) \sin(y)$

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

$\cos(x + y) = \cos(x)\cos(y) - \sin(x) \sin(y)\,$
$\cos(x - y) = \cos(x)\cos(y) + \sin(x) \sin(y)\,$

but never

$\cos(x + y) = \cos(x)\cos(y) + \sin(x) \sin(y)\,$
$\cos(x - y) = \cos(x)\cos(y) - \sin(x) \sin(y)\,$

because the signs are exclusively alternating.

## Other uses

± and are used in chess notation to denote an advantage for white and black respectively.

## Encodings

• In ISO 8859-1, -7, -8, -9, -13, -15, and -16, the plus-minus symbol is given by the code 0xB1hex 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 &plusmn;.
• On Windows systems, it may be entered by means of Alt codes, by holding the ALT key while typing the numbers 0177 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).
• 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 codepoint U+2213 and so can be used in HTML using &#x2213; or &#8723;.
• In TeX 'plus-or-minus' and 'minus-or-plus' symbols are denoted \pm and \mp, respectively.
• These characters are also seen written as a (rather untidy) underlined or overlined + symbol. ( +  or + ).

## Similar characters

The plus-minus sign resembles the Chinese character , whereas the minus-plus sign resembles .

## References

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• plus minus sign — plius minus ženklas statusas T sritis informatika apibrėžtis Ženklas ±. Kodai: 177 (ISO/IEC 8859 13, dešimtainis), U+00B1. Vartojamas skaičių intervalui nuo –n iki +n žymėti, pvz., ±0,1. atitikmenys: angl. plus minus sign …   Enciklopedinis kompiuterijos žodynas

• plus/minus sign — noun Date: 1971 the sign ± used to indicate a quantity (as 2 in “the square root of 4 is ±2”) taking on both an algebraically positive value and its negative and to indicate a plus or minus quantity (as 4 in “the population age was 30 ± 4 years”) …   New Collegiate Dictionary

• plus-minus sign — noun the symbol ±, meaning plus or minus , used to indicate the precision of an approximation (as in The result is 10 ± 0.3 , meaning the result is somewhere between 10 0.3, that is, 9.7, and 10 + 0.3, that is, 10.3), or as a convenient shorthand …   Wiktionary

• Plus-minus — or plus/minus (+/ ) may refer to:*Plus minus sign, a mathematical symbol (±) *Plus/minus, an ice hockey statistic *+/ , an American rock band …   Wikipedia

• plus/minus symbol — noun see plus/minus sign …   New Collegiate Dictionary

• minus sign — minus .sign also minus n a sign ( ) showing that a number is less than zero, or that the second of two numbers is to be ↑subtracted from the first →↑plus sign …   Dictionary of contemporary English

• Plus and minus signs — This article is about the mathematical symbols. For other uses of plus , see Plus (disambiguation). For other uses of minus , see Minus (disambiguation). The plus and minus signs (+ and −) are mathematical symbols used to represent the notions of …   Wikipedia

• minus-plus sign — noun the symbol ∓, meaning minus or plus , used alongside the plus minus sign to show that a negative value is to be taken where the positive value is indicated by the plus minus sign, and vice versa (as in (x nbsp;± nbsp;1) nbsp;/ nbsp;(x nbsp;∓ …   Wiktionary

• minus — [mī′nəs] prep. [LME < L, less, neut. sing. of minor,MINOR] 1. reduced by the subtraction of; less [four minus two] 2. Informal without; lacking [minus a toe] adj. 1. indicating or involving subtraction [a minus sign] …   English World dictionary

• Minus (disambiguation) — Minus may refer to: Minus sign, a mathematical symbol; See Plus and minus signs minus is a Web based file hosting service m nus, a Canadian record label; See Minus (record label) Mínus, an Icelandic band Minus a song by Beck from Odelay Minus a… …   Wikipedia