|Cardinal||−1, minus one, negative one|
|Ordinal||−1st (negative first)|
In mathematics, −1 (also known as negative one) is the additive inverse of 1, that is, the number that when added to 1 gives the additive identity element, 0. It is the negative integer greater than negative two (−2) and less than 0.
Multiplying a number by −1 is equivalent to changing the sign of the number – that is, for any x we have (−1) ⋅ x = −x. This can be proved using the distributive law and the axiom that 1 is the multiplicative identity:
Here we have used the fact that any number x times 0 equals 0, which follows by cancellation from the equation
In other words,
so (−1) ⋅ x is the additive inverse of x, i.e. (−1) ⋅ x = −x, as was to be shown.
The square of −1, i.e. −1 multiplied by −1, equals 1. As a consequence, a product of two negative numbers is positive.
For an algebraic proof of this result, start with the equation
The first equality follows from the above result, and the second follows from the definition of −1 as additive inverse of 1: it is precisely that number which when added to 1 gives 0. Now, using the distributive law, we see that
The third equality follows from the fact that 1 is a multiplicative identity. But now adding 1 to both sides of this last equation implies
Although there are no real square roots of −1, the complex number i satisfies i2 = −1, and as such can be considered as a square root of −1. The only other complex number whose square is −1 is −i because there are exactly two square roots of any non‐zero complex number, which follows from the fundamental theorem of algebra. In the algebra of quaternions – where the fundamental theorem does not apply – which contains the complex numbers, the equation x2 = −1 has infinitely many solutions.
Exponentiation of a non‐zero real number can be extended to negative integers. We make the definition that x−1 = 1/x, meaning that we define raising a number to the power −1 to have the same effect as taking its reciprocal. This definition is then extended to negative integers, preserving the exponential law xaxb = x(a + b) for real numbers a and b.
Exponentiation to negative integers can be extended to invertible elements of a ring, by defining x−1 as the multiplicative inverse of x.
A −1 that appears as a superscript of a function does not mean taking the (pointwise) reciprocal of that function, but rather the inverse function of the function. For example, sin−1(x) is a notation for the arcsine function, and in general f −1(x) denotes the inverse function of f(x),. When a subset of the codomain is specified inside the function, it instead denotes the preimage of that subset under the function.