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{-b \pm \sqrt{b^2-4ac}}{2a}, which describes the two
conjugate solutions to the
quadratic equation 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{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \pm \frac{1}{(2n+1)!} x^{2n+1} + \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. The
minus–plus sign, , is generally used in conjunction with the sign, and always has the opposite sign to . For example, , is a shorthand for or (but nor ). The above expression could be rewritten as to avoid use of , but cases such as this
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{align} \cos(A + B) &= \cos(A)\cos(B) - \sin(A) \sin(B),\text{ and} \\ \cos(A - B) &= \cos(A)\cos(B) + \sin(A) \sin(B). \end{align} Another example is the sum and
difference of cubes :x^3 \pm y^3 = (x \pm y)\left((x \mp y)^2 \pm xy\right) which represents the two equations: :\begin{align} x^3 + y^3 &= (x + y)\left((x - y)^2 + xy\right),\text{ and} \\ x^3 - y^3 &= (x - y)\left((x + y)^2 - xy\right). \end{align} When both and signs appear in an equation, it is unambiguous that all such signs are correlated; the shorthand describes exactly two equations. When only signs appear, the standard mathematical convention is that they all take on the same value, so for example the trigonometric identity :\sin(A \pm B) = \sin(A) \cos(B) \pm \cos(A) \sin(B) is also a shorthand for two equations: one with on both sides of the equation, and one with on both sides. However, this may be modified by the surrounding text, which may state
"where the ‘±’ signs are independent" or similar. If such a simple description is not possible, the equation must be re-written to provide clarity; e.g. by introducing variables such as , , ... and specifying the appropriate relation, such as .
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==