Number −1: Number

That is, if it is added to 1, the result is 0. It is the negative integer greater than negative two (−2) and less than 0.

−1 has some similar properties as positive one. But some are different.

−1 is related to Euler's identity. This is because the identity states ${\displaystyle e^{i\pi }=-1.\!}$

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

Multiplying a number by −1 is the same as changing the sign on the number. This can be proved using the distributive law and the axiom at 1 is the multiplicative identity, that is, a number multiplied by 1 is the number itself. So, 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 0 multiplied by any real number x equals 0, shown 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. So, a square of negative real numbers is positive.

To prove this with algebra, start with the equation

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

The first equality follows from the above result. The second follows from the definition of −1 as additive inverse of 1, that is, when added to 1, it 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, that is : ${\displaystyle x\cdot 1=x\,}$ . But now adding 1 to both sides of this last equation means

${\displaystyle (-1)\cdot (-1)=1}$ 

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

Square roots of −1

The complex number i satisfies i² = −1. So it is a square root of −1. The only other complex number x for which the equation x² = −1 holds is −i. In the algebra of quaternions, which has the complex plane, the equation x² = −1 has an infinity of solutions.

A non-zero real number can have a negative number as its power. We define that x⁻¹ = 1/x. This means a number raised to a power of −1 is equal to the reciprocal of that number. The exponential law x^ax^b = x^{(a + b)} for a,b non-zero real numbers is true even if a or b is negative.

There are many ways that −1 (and negative numbers in general) can be represented in computer systems. The most common is as two's complement of their positive form. In standard binary representation, this can also represent a positive integer.