| Number: | −1 |
| Divisor: | 1 |
| Cardinal: | −1, minus one, |
| Ordinal: | −1st (negative first) |
| Lang1: | Arabic |
| Lang1 Symbol: | − |
| Lang2: | Chinese numeral |
| Lang2 Symbol: | 负一，负弌，负壹 |
| Lang3: | Bengali |
| Lang3 Symbol: | − |
| Lang4: | Binary (byte) |
| Lang5: | Hex (byte) |
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.
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, 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, 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.[1]
Although there are no real square roots of −1, the complex number satisfies, and as such can be considered as a square root of −1.[2] 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.[3] [4]
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.