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.
*In mathematics, the sign generally indicates a choice of exactly two possible values, one of which is obtained through addition and the other through subtraction.
*In statistics and experimental sciences, the sign commonly indicates the confidence interval or uncertainty bounding a range of possible errors 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 chess, the sign indicates a clear advantage for the white player; the complementary minus-plus sign () indicates a clear advantage for the black player.
Other meanings occur in other fields, including medicine, engineering, chemistry, electronics, linguistics, and philosophy.

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 of the plus and minus signs, or , allowing the formula to represent two values or two equations.

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
:$x = \frac,$
which describes the two solutions to the quadratic equation

Similarly, the trigonometric identity
:$\sin(A \pm B) = \sin(A) \cos(B) \pm \cos(A) \sin(B)$
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 , 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:
:$\cos(A \pm B) = \cos(A) \cos(B) \mp \sin(A) \sin(B)$
which represents the two equations:
:$\begin \cos(A + B) &= \cos(A)\cos(B) - \sin(A) \sin(B) \\ \cos(A - B) &= \cos(A)\cos(B) + \sin(A) \sin(B) \end$

Another example is the conjugate of the perfect squares
:$x^3 \pm y^3 = (x \pm y)\left((x \mp y)^2 \pm xy\right)$
which represents the two equations:
:$\begin x^3 + y^3 &= (x + y)\left((x - y)^2 + xy\right) \\ x^3 - y^3 &= (x - y)\left((x + y)^2 - xy\right) \end$

A related usage is found in this presentation of the formula for the Taylor series of the sine function:
:$\sin\left( x \right) = x - \frac + \frac - \frac + \cdots \pm \frac x^ + \cdots$
Here, the plus-or-minus sign indicates that the term may be added or subtracted depending on whether 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 , 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 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 , any operation of the form must return a value of the form , where is and is the range updated using interval arithmetic.

In chess

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.

Other meanings

*In medicine, it may mean "with or without" in some cases.
*In engineering, the sign indicates the tolerance, which is the range of values that are considered to be acceptable or safe, or which comply with some standard or with a contract.
*In chemistry, the sign is used to indicate a racemic mixture.
*In electronics, this sign may indicate a dual voltage power supply, such as ±5 volts means +5 volts and −5 volts, when used with audio circuits and operational amplifiers.
*In linguistics, it may indicate a distinctive feature, such as �voiced

Encodings

*In Unicode:
*In ISO 8859-1, -7, -8, -9, -13, -15, and -16, the plus–minus symbol is code 0xB1 _hex. This location was copied to Unicode.
*In HTML, the symbol also has character entity reference representations of ±, ±
*The rarer minus–plus sign is not generally found in legacy encodings, but is available in Unicode as so can be used in HTML using ∓ or ∓.
*In TeX '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 with screen readers.

Typing

*Windows: or (numbers typed on the numeric keypad).
*Macintosh: (equal sign on the non-numeric keypad).
*Unix-like systems: or (second works on Chromebook)
* In the Vim text editor (in Insert mode): or or or
* AutoCAD shortcut string:

Similar characters

The plus–minus sign resembles the Chinese characters ( Radical 32) and ( Radical 33), whereas the minus–plus sign resembles ( Radical 51).

See also

* ≈ (approximately equal to)
*
*Plus and minus signs
*Sign (mathematics)
* Table of mathematical symbols

References

{{Navbox punctuation

Addition
Elementary arithmetic
Mathematical symbols
Sign (mathematics)
Subtraction