Plus-minus sign

From Wikipedia, the free encyclopedia

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.

Contents

[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 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. 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 not "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 can be found 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.

In a pinch, a crude version of these symbols can also be formed by underlining or overlining a + sign ( + + ).

[edit] See also

Retrieved from "http://en.wikipedia.org../../../p/l/u/Plus-minus_sign.html"