|Cardinal||−1, minus one, negative one|
|Ordinal||−1st (negative first)|
In mathematics, −1 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.
Negative one bears relation to Euler's identity since eiπ = −1.
Negative one has some similar but slightly different properties to positive one.
where we used the fact that any real x times 0 equals 0, implied by cancellation from the equation
In other words,
so (−1) · x, or −x, is the arithmetic inverse of x.
Square of −1
The square of −1, i.e. −1 multiplied by −1, equals 1. As a consequence, a product of two negative real numbers is positive.
For an algebraic proof of this result, start with the equation
The first equality follows from the above result. The second follows from the definition of −1 as additive inverse of 1: it is precisely that number that when added to 1 gives 0. Now, using the distributive law, we see that
The second equality follows from the fact that 1 is a multiplicative identity. But now adding 1 to both sides of this last equation implies
Square roots of −1
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. The only other complex number whose square is 1 is −i. In the algebra of quaternions, which contain the complex plane, the equation x2 = −1 has infinitely many solutions.
Exponentiation to negative integers
Exponentiation of a non-zero real number can be extended to negative integers. We make the definition that x−1 = 1/, meaning that we define raising a number to the power −1 to have the same effect as taking its reciprocal. This definition is then extended to negative integers, preserving the exponential law xaxb = x(a + b) for real numbers a and b.
Exponentiation to negative integers can be extended to invertible elements of a ring, by defining x−1 as the multiplicative inverse of x.
−1 that appears next to functions or matrices does not mean raising them to the power −1 but their inverse functions or inverse matrices. For example, f−1(x) is the inverse of f(x), or sin−1(x) is a notation of arcsine function.
Most computer systems represent negative integers using two's complement. In such systems, −1 is represented using a bit pattern of all ones. For example, an 8-bit signed integer using two's complement would represent −1 as the bitstring "11111111", or "FF" in hexadecimal (base 16). If interpreted as an unsigned integer, the same bitstring of n ones represents 2n − 1, the largest possible value that n bits can hold. For example, the 8-bit string "11111111" above represents 28 − 1 = 255.
In some programming languages, when used to index some data types (such as an array), then −1 can be used to identify the very last (or 2nd last) item, depending on whether 0 or 1 represents the first item. If the first item is indexed by 0, then −1 identifies the last item. If the first item is indexed by 1, then −1 identifies the second-to-last item.