Plus–minus sign

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

For example, given the equation x² = 9, one may give the solution as x = ±3. 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 = +3 or x = −3. 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

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

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.

The minus–plus sign (also minus-or-plus sign), ∓ is generally used in conjunction with the ± sign, in such expressions as x ± y ∓ z, which can be interpreted as meaning x + y − z and/or 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, without a ∓, 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 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:

${\displaystyle {\begin{aligned}\cos(A+B)&=\cos(A)\cos(B)-\sin(A)\sin(B)\\\cos(A-B)&=\cos(A)\cos(B)+\sin(A)\sin(B)\end{aligned}}}$

Another example is

${\displaystyle x^{3}\pm 1=(x\pm 1)\left(x^{2}\mp x+1\right)}$

A third 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, in this case depending on whether n is odd or even, the rule can be deduced from the first few terms. 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.

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" 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(n) and d is range updated using interval arithmetic.

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 (from 207 V to 253 V inclusive). 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 may write 5.7+0.2
−0.1.

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 symbols + and − denote a larger advantage than ± and ∓.

• Windows: Alt+241 or Alt+0177 (numbers typed on the numeric keypad).
• Macintosh: ⌥ Option+⇧ Shift+= (equal sign on the non-numeric keypad).
• Unix-like systems: Compose,+,- or ⇧ Shift+Ctrl+u B1space (second works on Chromebook)
• AutoCAD shortcut string: %%d