In mathematics, −1 (negative one or minus 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.

In mathematics

Algebraic properties

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: x + (−1) ⋅ x = 1 ⋅ x + (−1) ⋅ x = (1 + (−1)) ⋅ x = 0 ⋅ x = 0 . Here we have used the fact that any number x times 0 equals 0, which follows by cancellation from the equation 0 ⋅ x = (0 + 0) ⋅ x = 0 ⋅ x + 0 ⋅ x . In other words, x + (−1) ⋅ x = 0 , so (−1) ⋅ x is the additive inverse of x, i.e. (−1) ⋅ x = −x , as was to be shown. The square of −1 (that is −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, it can be seen 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 (−1) ⋅ (−1) = 1 . The above arguments hold in any ring, a concept of abstract algebra generalizing integers and real numbers. Although there are no real square roots of −1, the complex number i satisfies , 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 has infinitely many solutions.

Inverse and invertible elements

[[File:Geogebra f(x)=1÷x 20211118.svg|350px|thumb|The reciprocal function where for every x except 0, f(x) represents its multiplicative inverse ]] Exponentiation of a non‐zero real number can be extended to negative integers, where raising a number to the power −1 has the same effect as taking its multiplicative inverse: x−1 = 1⁄x . This definition is then applied to negative integers, preserving the exponential law for real numbers a and b. A −1 superscript in f−1(x) takes the inverse function of f(x) , where ( f(x))−1 specifically denotes a pointwise reciprocal. Where f is bijective specifying an output codomain of every y ∈ Y  from every input domain x ∈ X , there will be f −1( f(x)) = x,  and  f −1( f(y)) = y . When a subset of the codomain is specified inside the function f , its inverse will yield an inverse image, or preimage, of that subset under the function. Exponentiation to negative integers can be further extended to invertible elements of a ring by defining x−1 as the multiplicative inverse of x; in this context, these elements are considered units. In a polynomial domain F  [ x ] over any field F , the polynomial x has no inverse. If it did have an inverse q(x) , then there would be ⇒ 1 + deg (q(x)) = 0 ⇒ deg (q(x)) = −1 which is not possible, and therefore, F  [ x ] is not a field. More specifically, because the polynomial is not constant, it is not a unit in F .

In other uses

Integer sequences commonly use −1 to represent an uncountable set, in place of "[[Infinity| ∞ ]]" as a value resulting from a given index. As an example, the number of regular convex polytopes in n -dimensional space is, {1, 1, −1, 5, 6, 3, 3, ...} for n = {0, 1, 2, ...} . −1 can also be used as a null value, from an index that yields an empty set ∅ or non-integer where the general expression describing the sequence is not satisfied, or met. For instance, the smallest k > 1 such that in the interval 1...k there are as many integers that have exactly twice n divisors as there are prime numbers is, {2, 27, −1, 665, −1, 57675, −1, 57230, −1} for n = {1, 2, ..., 9} . A non-integer or empty element is often represented by 0. In software development, −1 is a common initial value for integers and is also used to show that a variable contains no useful information.

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