# −1 (number)

{{#invoke:Hatnote|hatnote}} Template:Infobox number 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.

In computer science, −1 is a common initial value for integers and is also used to show that a variable contains no useful information.

Negative one has some similar but slightly different properties to positive one.

## Algebraic properties

Multiplying a number by −1 is equivalent to changing the sign on the number. This can be proved using the distributive law and the axiom that 1 is the multiplicative identity: for x real, we have

$x+(-1)\cdot x=1\cdot x+(-1)\cdot x=(1+(-1))\cdot x=0\cdot x=0$ where we used the fact that any real x times 0 equals 0, implied by cancellation from the equation

$0\cdot x=(0+0)\cdot x=0\cdot x+0\cdot x\,$ In other words,

$x+(-1)\cdot x=0\,$ so (−1) · x is the arithmetic inverse of x, or −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.

$0=-1\cdot 0=-1\cdot [1+(-1)]$ 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

$0=-1\cdot [1+(-1)]=-1\cdot 1+(-1)\cdot (-1)=-1+(-1)\cdot (-1)$ 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

$(-1)\cdot (-1)=1$ The above arguments hold in any ring, a concept of abstract algebra generalizing integers and real numbers.

### 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 x satisfying the equation x2 = −1 is −i. In the algebra of quaternions, containing the complex plane, the equation x2 = −1 has an infinity of 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/x, meaning that we define raising a number to the power −1 to have the same effect as taking its reciprocal. This definition then extended to negative integers preserves the exponential law xaxb = x(a + b) for a,b real numbers.

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.

## Computer representation

{{#invoke:main|main}} 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.