Theplus–minus sign orplus-or-minus sign (±) and the complementaryminus-or-plus sign (∓) are symbols with broadly similar multiple meanings.
Inmathematics, the± sign generally indicates a choice of exactly two possible values, one of which is obtained throughaddition and the other throughsubtraction. The∓ is typically used only in tandem with the± sign and indicates that in the case that the± is a +, the∓ would be a − (and vice-versa).
Inchess, the± sign indicates a clear advantage for the white player; the complementary minus-plus sign (∓) indicates a clear advantage for the black player.
A version of the sign, including also the French wordou ("or"), was used in its mathematical meaning byAlbert Girard in 1626, and the sign in its modern form was used as early as 1631, inWilliam Oughtred'sClavis Mathematicae.[1]
Inmathematical formulas, the± symbol may be used to indicate a symbol that may be replaced by either of theplus and minus signs,+ or−, allowing the formula to represent two values or two equations.[2]
Ifx2 = 9, one may give the solution asx = ±3. This indicates that the equation has two solutions:x = +3 andx = −3. A common use of this notation is found in thequadratic formula
which describes the two solutions to thequadratic equationax2 +bx +c = 0.
can be interpreted as a shorthand for two equations: one with+ on both sides of the equation, and one with− on both sides.
Theminus–plus sign,∓, is generally used in conjunction with the± sign, in such expressions asx ±y ∓z, which can be interpreted as meaningx +y −z orx −y +z (butnotx +y +z orx −y −z). When an interpretation is chosen,∓ always has the opposite sign to±.
The above expression can be rewritten asx ± (y −z) to avoid use of∓, but cases such as the trigonometric identity are most neatly written using the "∓" sign:
A related usage is found in this presentation of the formula for theTaylor series of the sine function:
Here, the plus-or-minus sign indicates that the term may be added or subtracted depending on whethern 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 whenn is even, and −1 whenn is odd. In older texts one occasionally finds(−)n, 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 ass1,s2, ... and specifying a value of +1 or −1 separately for each, or some appropriate relation, likes3 =s1 · (s2)n or similar.
The use of± for an approximation is most commonly encountered in presenting the numerical value of a quantity, together with itstolerance or its statisticalmargin of error.[3] 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 anormal distribution).
Operations involving uncertain values should always try to preserve the uncertainty, in order to avoidpropagation of error. Ifn =a ±b, any operation of the formm =f(n) must return a value of the formm =c ±d, wherec isf(a) andd is the rangeb updated usinginterval arithmetic.
The symbols± and∓ are used inchess 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]
Inmedicine, it may mean "with or without" in some cases.[6][7]
Inengineering, the sign indicates thetolerance, which is the range of values that are considered to be acceptable or safe, or which comply with some standard or with a contract.
Inelectronics, this sign may indicate a dual voltage power supply, such as ±5 volts means +5 volts and −5 volts, when used with audio circuits andoperational amplifiers.
The rarer minus–plus sign is not generally found in legacy encodings, but is available in Unicode asU+2213∓MINUS-OR-PLUS SIGN so can be used in HTML using∓ or∓.
InTeX '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 withscreen readers.
^Brown, George W. (1982). "Standard deviation, standard error: Which 'standard' should we use?".American Journal of Diseases of Children.136 (10):937–941.doi:10.1001/archpedi.1982.03970460067015.PMID7124681.
^Heit, J. A.; Silverstein, M. D.; Mohr, D. N.; Petterson, T. M.; O'Fallon, W. M.; Melton, L. J. (1999-03-08). "Predictors of survival after deep vein thrombosis and pulmonary embolism: a population-based, cohort study".Archives of Internal Medicine.159 (5):445–453.doi:10.1001/archinte.159.5.445.ISSN0003-9926.PMID10074952.
^Hornsby, David.Linguistics, A Complete Introduction. p. 99.ISBN9781444180336.
