# −1

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.

 ← −2 −1 0 →
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Cardinal−1, minus one, negative one
Ordinal−1st (negative first)
Arabic١
Chinese numeral负一，负弌，负壹
Bengali
Binary (byte)
 S&M: 1000000012 2sC: 111111112
Hex (byte)
 S&M: 0x10116 2sC: 0xFF16

Negative one bears relation to Euler's identity since eiπ = −1.

In software development, −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.[1]

## 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

${\displaystyle x+(-1)\cdot x=1\cdot x+(-1)\cdot x=(1+(-1))\cdot x=0\cdot x=0}$

where we used the fact that any real x times 0 equals 0, implied by cancellation from the equation

${\displaystyle 0\cdot x=(0+0)\cdot x=0\cdot x+0\cdot x\,}$

In other words,

${\displaystyle x+(-1)\cdot x=0\,}$

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.

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

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

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

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

${\displaystyle (-1)\cdot (-1)=1}$

The above arguments hold in any ring, a concept of abstract algebra generalizing integers and real numbers.

### Square roots of −1

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.[2] 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/x, 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.

## Computer representation

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.

## Programming languages

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.