−1

−1
Template:Numbers (0s)
Cardinal	−1, minus one, negative one
Arabic	−١
Chinese numeral	负一，负弌，负壹
Bengali	ô১
Binary (byte)	S&M: 2sC: 1000000012 111111112
Hex (byte)	S&M: 2sC: 0x1018 0xFF8

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 has some similar but slightly different properties to positive one.

Negative one bears relation to Euler's identity since $e^{i\pi }=-1.\!$

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

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 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

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

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 x^ax^b = 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.

Computer representation

There are a variety of ways that −1 (and negative numbers in general) can be represented in computer systems, the most common being as two's complement of their positive form. Since this representation could also represent a positive integer in standard binary representation, a programmer must be careful not to confuse the two. Negative one in two's complement could be mistaken for the positive integer 2ⁿ − 1, where n is the number of digits in the representation (that is, the number of bits in the data type). For example, 11111111₂ (binary) and FF₁₆ (hex) each represents −1 in two's complement, but represents 255 in standard numeric representation.

See also

• Signed zero