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−1

Inmathematics, −1 (negative oneorminus one) is the additive inverseof1, 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.

Algebraic properties[edit]

Multiplication[edit]

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.

Square of −1[edit]

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

0 = −1 ⋅ 0 = −1 ⋅ [1 + (−1)].

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

0 = −1 ⋅ [1 + (−1)] = −1 ⋅ 1 + (−1) ⋅ (−1) = −1 + (−1) ⋅ (−1).

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.^[1]: p.48

Square roots of −1[edit]

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.^[2] 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.^[3][4]

Inverse and invertible elements[edit]

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 x^ax^b = x^{(a + b)} for real numbers a and b.

A −1 superscriptinf ⁻¹(x) takes the inverse functionoff(x), where ( f(x))⁻¹ specifically denotes a pointwise reciprocal.^[a] Where fisbijective 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.

Rings[edit]

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.^[1]: p.49

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^[5]

x q(x) = 1 ⇒ deg (x) + deg (q(x)) = deg (1)

                 ⇒ 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 continuous, it is not a unit in F.

Uses[edit]

Sequences[edit]

Integer sequences commonly use −1 to represent an uncountable set, in place of "∞" as a value resulting from a given index.^[6]

As an example, the number of regular convex polytopesinn-dimensional space is,

{1, 1, −1, 5, 6, 3, 3, ...} for n = {0, 1, 2, ...} (sequence A060296 in the OEIS).

−1 can also be used as a null value, from an index that yields an empty set ∅ornon-integer where the general expression describing the sequence is not satisfied, or met.^[6]

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} (sequence A356136 in the OEIS).

A non-integer or empty element is often represented by 0 as well.

Computing[edit]

Insoftware development, −1 is a common initial value for integers and is also used to show that a variable contains no useful information.^{[citation needed]}

See also[edit]

• Balanced ternary
• Menelaus's theorem

References[edit]