Plus-minus sign

±

The plus-minus sign (±) is a mathematical symbol with multiple meanings.

• In mathematics, it generally indicates a choice of exactly two possible values, one of which is the negation of the other.
• In experimental sciences, the sign commonly indicates the confidence interval or error in a measurement, often the standard deviation or standard error.^[1] The sign may also represent an inclusive range of values that a reading might have.
• In engineering the sign indicates the tolerance, which is the range of values that are considered to be acceptable, safe, or which comply with some standard, or with a contract.^[2]
• In chemistry the sign is used to indicate a racemic mixture.
• In chess, the sign indicates a clear advantage for the white player; the complementary sign ∓ indicates the same advantage for the black player.^[3]

The sign is normally pronounced "plus or minus".

History

A version of the sign, including also the French word "ou" (meaning "or") was used in its mathematical meaning by Albert Girard in 1626, and the sign in its modern form was used as early as William Oughtred's Clavis Mathematicae (1631).^[4]

Usage

In mathematics

In mathematical formulas, the ± symbol may be used to indicate a symbol that may be replaced by either the + or − symbols, allowing the formula to represent two values or two equations. For example, given the equation x² = 1, one may give the solution as x = ±1. This indicates that the equation has two solutions, each of which may be obtained by replacing this equation by one of the two equations x = +1 or x = −1. Only one of these two replaced equations is true for any valid solution. A common use of this notation is found in the quadratic formula

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

describing the two solutions to the quadratic equation ax² + bx + c = 0.

Similarly, the trigonometric identity

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

can be interpreted as a shorthand for two equations: one with "+" on both sides of the equation, and one with "−" on both sides. The two copies of the ± sign in this identity must both be replaced in the same way: it is not valid to replace one of them with "+" and the other of them with "−". In contrast to the quadratic formula example, both of the equations described by this identity are simultaneously valid.

A third related usage is found in this presentation of the formula for the Taylor series of the sine function:

$\sin\left( x \right) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \pm \frac{1}{(2n+1)!} x^{2n+1} + \cdots.$

Here, the plus-or-minus sign indicates 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 of the same formula would multiply each term by a factor of (−1)ⁿ, which gives +1 when n is even and −1 when n is odd.

In statistics

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.^[1] 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 5.5 to 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.

In chess

The symbols ± and ∓ are used in chess notation to denote an advantage for white and black respectively. However, the more common chess notation would be only + and –. ^[3] If a difference is made, the symbol + and − denote a larger advantage than ± and ∓.

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 not

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

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

because the signs are exclusively alternating.

Another example is $x^3 \pm 1 = (x \pm 1)(x^2 \mp x + 1)$ which represents two equations.

Encodings

• In Unicode: U+00B1 ± PLUS-MINUS SIGN (HTML ± · ±)
• 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 ±.
• 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 may also be produced as an underlined or overlined + symbol ( +  or + ), but beware of the formatting being stripped at a later date, changing the meaning.

Typing

• On Windows systems, it may be entered by means of Alt codes, by holding the ALT key while typing the numbers 0177 or 241 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).

Similar characters

The plus-minus sign resembles the Chinese characters 士 and 土, whereas the minus-plus sign resembles 干.

See also

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

References

1. Brown, George W. (1982), "Standard Deviation, Standard Error: Which 'Standard' Should We Use?", American Journal of Diseases of Children 136 (10): 937–941, doi:10.1001/archpedi.1982.03970460067015.
2. Engineering tolerance
3. Eade, James (2005), Chess For Dummies (2nd ed.), John Wiley & Sons, p. 272, ISBN 9780471774334.
4. Cajori, Florian (1928), A History of Mathematical Notations, Volumes 1-2, Dover, p. 245, ISBN 9780486677668.