Plus–minus sign

Usage

In mathematics

In mathematical formulas, the ± symbol may be used to indicate a symbol that may be replaced by either of the plus and minus signs, + or −, allowing the formula to represent two values or two equations.

If x² = 9, one may give the solution as x = ±3. This indicates that the equation has two solutions: x = +3 and x = −3. A common use of this notation is found in the quadratic formula

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

which describes the two conjugate solutions to the quadratic equation ax² + bx + c = 0.

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

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

Here, the plus-or-minus sign indicates that the term may be added or subtracted depending on whether n is odd or even; a rule which can be deduced from the first few terms. A more rigorous presentation would multiply each term by a factor of (−1)ⁿ, which gives +1 when n is even, and −1 when n is odd. In older texts one occasionally finds (−)ⁿ, which means the same.

The minus–plus sign, ∓, is generally used in conjunction with the ± sign, and always has the opposite sign to ±. For example, x ± y ∓ z, is a shorthand for x + y − z or x − y + z (but not x + y + z or x − y − z).

The above expression could be rewritten as x ± (y − z) to avoid use of ∓, but cases such as this trigonometric identity are most neatly written using the "∓" sign:

$${\displaystyle \cos (A\pm B)=\cos (A)\cos (B)\mp \sin (A)\sin (B)}$$

which represents the two equations:

$${\displaystyle \begin{array}{cc}\cos (A+B)& =\cos (A)\cos (B)-\sin (A)\sin (B),\text{ and}\\ \cos (A-B)& =\cos (A)\cos (B)+\sin (A)\sin (B).\end{array}}$$

Another example is the sum and difference of cubes

$${\displaystyle x^3\pm y^3=(x\pm y)\left((x\mp y{)}^2\pm xy\right)}$$

which represents the two equations:

$${\displaystyle \begin{array}{cc}{x}^3+y^3& =(x+y)\left((x-y{)}^2+xy\right),\text{ and}\\ x^3-y^3& =(x-y)\left((x+y{)}^2-xy\right).\end{array}}$$

When both ± and ∓ signs appear in an equation, it is unambiguous that all such signs are correlated; the shorthand describes exactly two equations. When only ± signs appear, the standard mathematical convention is that they all take on the same value, so for example the trigonometric identity $${\displaystyle \sin (A\pm B)=\sin (A)\cos (B)\pm \cos (A)\sin (B)}$$ is also a shorthand for two equations: one with + on both sides of the equation, and one with − on both sides.

However, this may be modified by the surrounding text, which may state "where the ‘±’ signs are independent" or similar. If such a simple description is not possible, the equation must be re-written to provide clarity; e.g. by introducing variables such as s₁, s₂, ... and specifying the appropriate relation, such as s₃ = s₁ · (s₂)ⁿ.

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. For example, 5.7 ± 0.2 may be anywhere in the range from 5.5 to 5.9 inclusive. 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).

Operations involving uncertain values should always try to preserve the uncertainty, in order to avoid propagation of error. If n = a ± b, any operation of the form m = f(n) must return a value of the form m = c ± d, where c is f(a) and d is the range b updated using interval arithmetic.

In chess

The symbols ± and ∓ are used in chess annotation to denote a moderate but significant advantage for White and Black, respectively. Weaker and stronger advantages are denoted by ⩲ and ⩱ for only a slight advantage, and +– and –+ for a strong, potentially winning advantage, again for White and Black respectively.

Other meanings

• In medicine, it may mean "with or without" in some cases.
• In engineering, the sign indicates the tolerance, which is the range of values that are considered to be acceptable or safe, or which comply with some standard or with a contract.
• In chemistry, the sign is used to indicate a racemic mixture.
• In electronics, this sign may indicate a dual voltage power supply, such as ±5 volts means +5 volts and −5 volts, when used with audio circuits and operational amplifiers.
• In linguistics, it may indicate a distinctive feature, such as [±voiced].

Encodings

• In Unicode: U+00B1 ± PLUS-MINUS SIGN
• In ISO 8859-1, -7, -8, -9, -13, -15, and -16, the plus–minus symbol is code 0xB1_hex. This location was copied to Unicode.
• In HTML, the symbol also has character entity reference representations of ±, ±
• The rarer minus–plus sign is not generally found in legacy encodings, but is available in Unicode as U+2213 ∓ MINUS-OR-PLUS SIGN so can be used in HTML using ∓ or ∓.
• In TeX 'plus-or-minus' and 'minus-or-plus' symbols are denoted \pm and \mp, respectively.
• Although these characters may be approximated by underlining or overlining a + symbol ( + or + ), this is discouraged because the formatting may be stripped at a later date, changing the meaning. It also makes the meaning less accessible to blind users with screen readers.

Similar characters

The plus–minus sign resembles the Chinese characters 土 (Radical 32) and 士 (Radical 33), whereas the minus–plus sign resembles 干 (Radical 51).