| Mark: | ± |
The plus–minus sign or plus-or-minus sign,, is a symbol with multiple meanings.
Other meanings occur in other fields, including medicine, engineering, chemistry, electronics, linguistics, and philosophy.
A version of the sign, including also the French word ou ("or"), was used in its mathematical meaning by Albert Girard in 1626, and the sign in its modern form was used as early as 1631, in William Oughtred's Clavis Mathematicae.[1]
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.[2]
If, one may give the solution as . This indicates that the equation has two solutions: and . A common use of this notation is found in the quadratic formula
Similarly, the trigonometric identity
\sin(A\pmB)=\sin(A)\cos(B)\pm\cos(A)\sin(B)
The minus–plus sign,, is generally used in conjunction with the sign, in such expressions as, which can be interpreted as meaning or (but or). The always has the opposite sign to .
The above expression can be rewritten as to avoid use of, but cases such as the trigonometric identity are most neatly written using the "∓" sign:
\cos(A\pmB)=\cos(A)\cos(B)\mp\sin(A)\sin(B)
\begin{align} \cos(A+B)&=\cos(A)\cos(B)-\sin(A)\sin(B)\\ \cos(A-B)&=\cos(A)\cos(B)+\sin(A)\sin(B)\end{align}
Another example is the conjugate of the perfect squares
x3\pmy3=(x\pmy)\left((x\mpy)2\pmxy\right)
\begin{align} x3+y3&=(x+y)\left((x-y)2+xy\right)\\ x3-y3&=(x-y)\left((x+y)2-xy\right)\end{align}
A related usage is found in this presentation of the formula for the Taylor series of the sine function:
\sin\left(x\right)=x-
| x3 | |
| 3! |
+
| x5 | |
| 5! |
-
| x7 | |
| 7! |
+ … \pm
| 1 | |
| (2n+1)! |
x2n+1+ …
When the standard presumption that the plus-or-minus signs all take on the same value of +1 or all -1 is not true, then the line of text that immediately follows the equation must contain a brief description of the actual connection, if any, most often of the form "where the ‘±’ signs are independent" or similar. If a brief, simple description is not possible, the equation must be re-written to provide clarity; e.g. by introducing variables such as,, ... and specifying a value of +1 or -1 separately for each, or some appropriate relation, like or similar.
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.[3] For example, 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, any operation of the form must return a value of the form, where is and is the range updated using interval arithmetic.
The symbols and are used in chess annotation to denote a moderate but significant advantage for White and Black, respectively.[4] 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.[5]
±, ±, and ±.∓ or ∓.\pm and \mp, respectively.or (numbers typed on the numeric keypad).
The plus–minus sign resembles the Chinese characters Chinese: 土 (Radical 32) and Chinese: 士 (Radical 33), whereas the minus–plus sign resembles Chinese: 干 (Radical 51).