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 we have . 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 times 0 equals 0, which follows by cancellation from the equation
:.
In other words,
:,
so is the additive inverse of , i.e. , 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
:.
The above arguments hold in any ring, a concept of abstract algebra generalizing integers and real numbers.
thumb|right|0, 1, −1, , and − in the [[complex plane|complex or Cartesian plane|upright]]
Although there are no real number square roots of −1, the complex number satisfies , and as such can be considered as a square root of −1. The only other complex number whose square is −1 is − 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
350px|thumb|The reciprocal function where for every except 0, 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:
:.
This definition is then applied to negative integers, preserving the exponential law for real numbers and .
A −1 superscript in takes the inverse function of , where specifically denotes a pointwise reciprocal. Where is bijective specifying an output codomain of every from every input domain , there will be
: and .
When a subset of the codomain is specified inside the function , 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 as the multiplicative inverse of ; in this context, these elements are considered units.
:
:
:
which is not possible, and therefore, is not a field. More specifically, because the polynomial is not constant, it is not a unit in .
See also
- Balanced ternary