−1

← −2 −1 0 →
−1 0 1 2 3 4 5 6 7 8 9 → List of numbers — Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	−1, minus one, negative one
Ordinal	−1st (negative first)
Arabic	−١
Chinese numeral	负一，负弌，负壹
Bengali	−১
Binary (byte)	S&M: 1000000012 2sC: 111111112
Hex (byte)	S&M: 0x10116 2sC: 0xFF16

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

$x+(-1)\cdot x=1\cdot x+(-1)\cdot x=(1+(-1))\cdot x=0 \cdot x=0$

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

$0\cdot x=(0+0)\cdot x=0\cdot x+0\cdot x \,$

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

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.

For an algebraic proof of this result, start with the equation

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

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

$0 =-1\cdot [1+(-1)]=-1\cdot1+(-1)\cdot(-1)=-1+(-1)\cdot(-1)$

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.^[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

1. Mathematical analysis and applications By Jayant V. Deshpande, ISBN 1-84265-189-7
2. "Ask Dr. Math". Math Forum. Retrieved 2012-10-14.