Additive inverse

The additive inverse, or opposite, of a number n is the number which, when added to n, yields zero. The additive inverse of n is denoted -n.

For example:

• The additive inverse of 7 is -7, because 7 + (-7) = 0;
• The additive inverse of -0.3 is 0.3, because -0.3 + 0.3 = 0.

Thus by the last example, -(-0.3) = 0.3.

The additive inverse of a number is its inverse element under the binary operation of addition. It can be calculated using multiplication by -1; that is, -n = -1 × n.

Types of numbers with additive inverses include:

• Integers;
• Rational numbers;
• Real numbers;
• Complex numbers.

Types of numbers without additive inverses (of the same type) include:

• Natural numbers;
• Cardinal numbers;
• Ordinal numbers.

But note that we can construct the integers out of the natural numbers by formally including additive inverses. Thus we can say that natural numbers do have additive inverses, but because these additive inverses are not themselves natural numbers, the set of natural numbers is not closed under taking additive inverses.

See also: Multiplicative inverse