−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.

Negative one bears relation to Euler's identity since e^πi = −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

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

0, 1, −1, i, and −i in the complex or cartesian plane

In other words,

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.

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

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.^[2] 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 x^ax^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.

Inductive dimension

The Inductive dimension of the empty set is defined to be −1.

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.

References