−1
Template:Short description Template:About Template:Infobox number 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 Template:Mvar we have Template:Math. 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 Template:Mvar times 0 equals 0, which follows by cancellation from the equation
In other words,
so Template:Math is the additive inverse of Template:Mvar, i.e. Template:Math, 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.<ref name="MultIdRng">Template:Cite book</ref>Template:Rp
Although there are no real square roots of −1, the complex number Template:Mvar satisfies Template:Math, and as such can be considered as a square root of −1.<ref name="imaginary">Template:Cite book</ref> The only other complex number whose square is −1 is −Template:Mvar 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 Template:Math has infinitely many solutions.<ref>Template:Cite book</ref><ref>Template:Cite book</ref>
Inverse and invertible elements
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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 Template:Math for real numbers Template:Mvar and Template:Mvar.
A −1 superscript in Template:Math takes the inverse function of Template:Math, where Template:Math specifically denotes a pointwise reciprocal.Template:Efn Where Template:Math is bijective specifying an output codomain of every Template:Math from every input domain Template:Math, there will be
When a subset of the codomain is specified inside the function Template:Math, 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 Template:Math as the multiplicative inverse of Template:Mvar; in this context, these elements are considered units.<ref name="MultIdRng" />Template:Rp
In a polynomial domain Template:Math over any field Template:Math, the polynomial Template:Mvar has no inverse. If it did have an inverse Template:Math, then there would be<ref>Template:Cite book</ref>
which is not possible, and therefore, Template:Math is not a field. More specifically, because the polynomial is not constant, it is not a unit in Template:Math.