Plus-minus sign

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The plus-minus sign (±) is a mathematical symbol commonly used to indicate the precision of an approximation, or as a convenient shorthand for a quantity which has two possible values opposite in sign.

In mathematics, the sign is pronounced "plus or minus" and indicates that there are exactly two possible answers, one of which is positive and one of which is negative.

In most experimental sciences however, the sign is pronounced "give or take" and it indicates an inclusive range of values that a reading might have.

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[edit] Usage for indicating precision

The use of ± for an approximation is most commonly encountered for 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 to 5.7 + 0.2. More precisely, in scientific usage it usually comes with a probability of being within the interval, usually that of 2 standard deviations, or 95.4%.

A percentage may also be used to indicate the error margin. For example, 230 V ± 10% refers to a voltage within 10% of either side of 230 V (207 V - 253 V).

[edit] Usage as shorthand for two values of opposite signs

In mathematical equations, the use of ± may be found as shorthand, for example to present two equations in one formula. The best-known example is offered by the formula for the solutions of quadratic equations:

If ax2 + bx + c = 0, then

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

Written out in full, this states that there are two solutions to the equation, namely

x = \frac{-b + \sqrt {b^2-4ac\ }}{2a}

and

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!} + \dots \pm \frac{1}{(2n+1)!} x^{2n+1} + \dots.

This mild abuse of notation is meant to indicate that the sign 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.

[edit] Minus-plus sign

There is also another character, the minus-or-plus sign (∓), which is rarely seen. It only takes on significant meaning when used in conjunction with the "±" sign. It can be used alongside "±" in such expressions as "x ± y ∓ z", which can be interpreted as "x + y − z" or/and "x − y + z", but neither "x + y + z" nor "x − y − z". The upper "−" in "∓" is considered attached to the "+" of "±" (and the lower symbols work in the same way) even though there is no visual indication of the dependency. The original expression can be rewritten as "x ± (y − z)" 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.

[edit] Encodings

In ISO-8859-1,7,8,9,13,15 and 16, the plus-minus symbol is given by the code B1hex 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-or-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. These characters can also be seen as an underlined or overlined + symbol. ( + + ).

[edit] See also

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