Additive inverse

Additive inverse

In mathematics, the additive inverse of a number a is the number that, when added to a, yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the opposite to a positive number is negative, and the opposite to a negative number is positive. Zero is the additive inverse of itself.

The additive inverse of a is denoted by unary minus: −a (see the discussion below). For example, the additive inverse of 7 is −7, because 7 + (−7) = 0, and the additive inverse of −0.3 is 0.3, because −0.3 + 0.3 = 0 .

The additive inverse is defined as its inverse element under the binary operation of addition (see the discussion below), which allows a broad generalization to mathematical objects other than numbers. As for any inverse operation, double additive inverse has no net effect: −(−x) = x.

Video: Additive inverse of addition

Common examples

For a number and, generally, in any ring, the additive inverse can be calculated using multiplication by −1; that is, −n = −1 × n . Examples of rings of numbers are integers, rational numbers, real numbers, and complex numbers.

Relation to subtraction

Additive inverse is closely related to subtraction, which can be viewed as an addition of the opposite:

ab  =  a + (−b).

Conversely, additive inverse can be thought of as subtraction from zero:

a  =  0 − a.

Hence, unary minus sign notation can be seen as a shorthand for subtraction with "0" symbol omitted, although in a correct typography there should be no space after unary "−".

Other properties

In addition to the identities listed above, negation has the following ......

 

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