|In Unicode||U+00B1 ± PLUS-MINUS SIGN (HTML |
|See also||U+2213 ∓ MINUS-OR-PLUS SIGN (HTML |
The plus–minus sign (also, plus or minus sign),is a mathematical symbol with multiple meanings.
- In mathematics, it generally indicates a choice of exactly two possible values, one of which is the negation of the other.
- In experimental sciences, the sign commonly indicates the confidence interval or error in a measurement, often the standard deviation or standard error. The sign may also represent an inclusive range of values that a reading might have.
- In engineering the sign indicates the tolerance, which is the range of values that are considered to be acceptable, safe, or which comply with some standard, or with a contract.
- In botany it is used in morphological descriptions to notate "more or less".
- In chemistry the sign is used to indicate a racemic mixture.
- In chess, the sign indicates a clear advantage for the white player; the complementary minus or plus sign, , indicates the same advantage for the black player.
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 William Oughtred's Clavis Mathematicae (1631).
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
describing the two solutions to the quadratic equation ax2 + bx + c = 0.
Similarly, the trigonometric identity
can be interpreted as a shorthand for two equations: one withon 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), 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:is generally used in conjunction with the sign, in such expressions as
which represents the two equations:
Another example 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 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 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).for an approximation is most commonly encountered in presenting the numerical value of a quantity together with its
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
The symbols chess notation to denote an advantage for white and black respectively. However, the more common chess notation would be only and . If a difference is made, the symbols and denote a larger advantage than and .and are used in
- In Unicode: U+00B1 ± PLUS-MINUS SIGN
- In ISO 8859-1, -7, -8, -9, -13, -15, and -16, the plus–minus symbol is code 0xB1hex. This location was copied to Unicode.
- The symbol also has a HTML entity representations of
- The rarer minus–plus sign is not generally found in legacy encodings, but is available in Unicode as U+2213 ∓ MINUS-OR-PLUS SIGN so can be used in HTML using
- In TeX 'plus-or-minus' and 'minus-or-plus' symbols are denoted
- Although these characters may also be produced using underlining or overlining + symbol ( + or + ), this is deprecated because the formatting may stripped at a later date, changing the meaning. It also makes the meaning less accessible to blind users with screen readers.
- 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:
|Look up 士, 土, or 干 in Wiktionary, the free dictionary.|
- 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, PMID 7124681.
- Engineering tolerance
- Eade, James (2005), Chess For Dummies (2nd ed.), John Wiley & Sons, p. 272, ISBN 9780471774334.
- Cajori, Florian (1928), A History of Mathematical Notations, Volumes 1-2, Dover, p. 245, ISBN 9780486677668.