Plus-minus sign

For other uses, see plus-minus (disambiguation).

±

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

• to indicate the precision of an approximation, or
• to indicate a value that can be of either sign.

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. It is commonly used in indicating a range of values, such as in mathematical statements.

Usage

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:

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

Another example is found in the trigonometric identity

$\sin(A \pm B) = \sin(A) \cos(B) \pm \cos(A) \sin(B).\,$

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.

Chess notation

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

Minus-plus sign

There is another symbol, 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 + y − z" or/and "x − y + z", but not "x + y + z" nor "x − y − z". 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. (However, the "±" sign is generally preferred over the "∓" sign, so if they both appear in an equation it is safe to assume that they are linked. On the other hand, if there are two instances of the "±" sign in an expression, it is impossible to tell from notation alone whether the intended interpretation is as two or four distinct expressions.) The original expression can be rewritten as "x ± (y − z)" to avoid confusion, but cases such as the trigonometric identity

$\cos(A \pm B) = \cos(A) \cos(B) \mp \sin(A) \sin(B)$

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

$\cos(A + B) = \cos(A)\cos(B) - \sin(A) \sin(B)\,$

$\cos(A - B) = \cos(A)\cos(B) + \sin(A) \sin(B)\,$

but never

$\cos(A + B) = \cos(A)\cos(B) + \sin(A) \sin(B)\,$

$\cos(A - B) = \cos(A)\cos(B) - \sin(A) \sin(B)\,$

because the signs are exclusively alternating.

Encodings

• In ISO 8859-1, -7, -8, -9, -13, -15, and -16, the plus-minus symbol is given by the code 0xB1_hex 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.
• 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 ∓ or ∓.
• 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 士, whereas the minus-plus sign resembles 干.

See also

• Plus and minus signs
• Table of mathematical symbols
• ≈ (approximately equals to)
• Engineering tolerance