The plus-minus sign (±) is a mathematical symbol with multiple meanings.
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 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.
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 signs of the terms alternate, where (starting the count at 0) the terms with an even index n are added while those with an odd index are subtracted. 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 ⟨±⟩ 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).
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 .
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 only + and -. If a difference is made, the symbols + and - denote a larger advantage than ± and ?.
The minus-plus sign (?) is generally used in conjunction with the "±" sign, in such expressions as "x ± y ? z", which can be interpreted as meaning "x + y - z" and/or "x - y + z", but not "x + y + z" or "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, 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. The trigonometric equation above thus represents the two equations:
because the signs are exclusively alternating.
Another example is
which represents two equations.