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−1 (number)

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In mathematics, −1 is the additive inverse of 1. 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.

Negative one has some similar properties as positive one. But some are different.[1]

Negative one is related to Euler's identity. This is because the identity states [math]e^{i \pi} = -1.\![/math]

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

Algebraic properties

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

[math]x+(-1)\cdot x=1\cdot x+(-1)\cdot x=(1+(-1))\cdot x=0 \cdot x=0[/math]

where we used the fact that 0 multiplied by any real number x equals 0, shown by cancellation from the equation

[math]0\cdot x=(0+0)\cdot x=0\cdot x+0\cdot x \, [/math]
0, 1, −1, i, and −i in the complex or cartesian plane

In other words,

[math]x+(-1)\cdot x=0 \, [/math]

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

[math]0 =-1\cdot 0 =-1\cdot [1+(-1)][/math]

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

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

The second equality follows from the fact that 1 is a multiplicative identity, that is : [math]x\cdot 1=x \, [/math]. But now adding 1 to both sides of this last equation means

[math](-1) \cdot (-1) = 1[/math]

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.[2]

Exponentiation to negative integers

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 xaxb = x(a + b) for a,b non-zero real numbers is true even if a or b is negative.

Computer representation

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.