ⓘ −1


ⓘ −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 and less than 0.

Negative one bears relation to Eulers identity since e i π = −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

x + − 1 ⋅ x = 1 ⋅ x + − 1 ⋅ x = 1 + − 1) ⋅ x = 0 ⋅ x = 0 {\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

0 ⋅ x = 0 + 0 ⋅ x = 0 ⋅ x + 0 ⋅ x {\displaystyle 0\cdot x=0+0\cdot x=0\cdot x+0\cdot x\,}

In other words,

x + − 1 ⋅ x = 0 {\displaystyle x+-1\cdot x=0\,}

so −1 x, or − x, is the arithmetic inverse of x.


1.1. Algebraic properties 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

0 = − 1 ⋅ 0 = − 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 ⋅ − 1 = 1 {\displaystyle -1\cdot -1=1}

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


1.2. Algebraic properties Square roots of −1

Although there are no real square roots of -1, the complex number i satisfies i 2 = −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 x 2 = −1 has infinitely many solutions.


2. 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 x a x b = 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.


3. Computer representation

Most computer systems represent negative integers using twos complement. In such systems, −1 is represented using a bit pattern of all ones. For example, an 8-bit signed integer using twos 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 2 n − 1, the largest possible value that n bits can hold. For example, the 8-bit string "11111111" above represents 2 8 − 1 = 255.


4. 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.