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^{$π$ i} = −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.

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

In other words,

so (−1) · x, or −x, is the arithmetic inverse of x.

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.

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