Plus–minus sign
The plus–minus sign, , is a mathematical symbol with multiple meanings. History. 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". Usage. In mathematics. 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 , one  may give the solution as . This indicates that the equation has two solutions, each of which may be obtained by replacing this equation by one of the two equations or . 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 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 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, , is generally used in conjunction with the sign, in such expressions as , which can be interpreted as meaning or , but nor . 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 both of them 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 to avoid confusion, but cases such as the trigonometric identity are most neatly written using the "∓" sign: which represents the two equations: Another example where the minus –plus sign appears is A third 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, in this case depending on whether 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), which gives +1 when is even, and −1 when 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 , , ... and specifying a value of +1 or −1 separately for each, or some appropriate relation, like formula_7 or similar. 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, 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 formula_8 any operation of the form  formula_9 must return a value of the form formula_10, where is formula_11 and is range updated using interval arithmetic. A percentage may also be used to indicate the error margin. For example, 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 . 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 to only use and . If several different symbols are used together, then the symbols and denote a clearer advantage than and . When finer evaluation is desired, three pairs of symbols are used: and for only a slight advantage; and for a significant advantage; and and for a potentially winning advantage, in each case for white or black respectively. Similar characters. The plus–minus sign resembles the Chinese characters (Radical 32) and (Radical 33), whereas the minus–plus sign resembles (Radical 51). Plus–minus sign