± shorthand
The plus-minus sign (±) is a mathematical symbol with multiple meanings.
The sign is normally pronounced "plus or minus".
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).
In mathematical formulas, the ± symbol may be used to indicate a symbol that may be replaced by either the + or − symbols, allowing the formula to represent two values or two equations.
For example, given the equation x2 = 1, one may give the solution as x = ±1. 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 = +1 or x = −1. 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, "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).
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