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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 notation for a value that can be of either sign.

In mathematics, the sign is pronounced "plus or minus" and indicates that there are two possible answers: one positive, and one negative. In most experimental sciences, the sign is pronounced "give or take" and indicates an inclusive range of values that a reading might have.[citation needed]

Contents

Precision indication

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 (i.e. 5.5) to 5.7 + 0.2 (5.9). More precisely, in scientific usage it usually comes with a probability of being within the interval, usually that of 1 or 2 standard deviations (68.3% or 95.4%).

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: equation+ OR equation- represented with equation±.

A well-known example is offered by 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 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. 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.

Minus-plus sign

There is another character, the minus-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 + yz" or/and "xy + z", but neither "x + y + z" nor "xyz". 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 ± (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.

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 ±.
  • 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.
  • 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 + ).

See also

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