Plus–minus sign

The plus–minus sign or plus-or-minus sign (±) and the complementary minus-or-plus sign (∓) are symbols with broadly similar 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.
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 x = 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
which describes the two solutions to the quadratic equation ax + bx + c = 0.
Similarly, the trigonometric identity
can be interpreted as a shorthand for two equations: one with + on both sides of the equation, and one with − on both sides.
The minus–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 or x − y + z (but not x + y + z or x − y − z). The ∓ always has the opposite sign to ±.
The above expression can be rewritten as x ± (y − z) to avoid use of ∓, but cases such as the trigonometric identity are most neatly written using the "∓" sign:
which represents the two equations:
Another example is the conjugate of the perfect squares
which represents the two equations:
A related usage is found in this presentation of the formula for the Taylor series of the sine function:
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), which gives +1 when n is even, and −1 when n is odd. In older texts one occasionally finds (−), which means the same.
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 s1, s2, ... and specifying a value of +1 or −1 separately for each, or some appropriate relation, like s3 = s1 · (s2) 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. 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.
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.
The plus–minus sign resembles the Chinese characters 土 (Radical 32) and 士 (Radical 33), whereas the minus–plus sign resembles 干 (Radical 51).