# Opposite – Wikipedia, the free encyclopedia

In mathematics, the opposite (The symmetric for sum, The inverse additive) of a number

${displaystyle n,}$

is the number that, added with

${displaystyle n,}$

, gives zero. The additive inverse of

${displaystyle n,}$

is denoted

${displaystyle -n,}$

In our everyday language « opposite » would be equivalent to « opposite ».

Arithmetically, it can be calculated by multiplying by

${displaystyle -1,}$

, that is to say,

${displaystyle -n=-1times n}$

. Algebraically speaking, the opposite of an element of a group is its element symmetric with respect to the binary operation «

${displaystyle +,}$

« (when using additive notation).

For example:

• The opposite of
${displaystyle 8,}$

is

${displaystyle -8,}$

, because

${displaystyle 8+(-8)=0,}$

;

• The opposite of
${displaystyle -0,3,}$

is

${displaystyle 0,3,}$

, because

${displaystyle -0,3+0,3=0,}$

.

So, for the example above,

${displaystyle -(-0,3)=0,3,}$

.

Sets of numbers in which each element has opposites:

Sets of numbers without opposites:

Note that the integers are constructed from the natural numbers to which the opposites are formally added.