Plus-minus sign

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For other uses, see plus-minus.
±

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. The plus-minus sign is frequently misused to mean "approximately".[citation needed]

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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).

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}.

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

In mathematical equations, the use of ± may be found as shorthand, to present two equations in one formula: equation+ OR equation- represented with equation±

The best-known example is offered by the formula for the solutions of quadratic equations:

If \displaystyle ax^2 + 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, namely

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

and

\displaystyle 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 meant to indicate that the sign of the terms alternate, where (starting the count at 0) the terms with an even index \displaystyle n are added while those with an odd index are subtracted. A less ambiguous presentation in this case would use the quantity ( − 1)n, which gives + 1 when n is even and − 1 when n is odd.

[edit] Minus-plus sign

There is another character, the minus-or-plus sign (∓), which is seen less often. 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 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 ±. 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 ∓
  • In TeX 'plus-or-minus' and 'minus-or-plus' symbols are encoded as \pm and \mp entities, respectively.
  • These characters are also seen written as the (highly un-semantic) underlined or overlined + symbol. ( +  or + ).

[edit] See also