In any
set of
numbers, a number
x has an
additive inverse iff there exists a number
y within that same set of numbers such that
x+
y=0 (provided that 0 is also within the given set of numbers). For example:
Let x be an
element of the
real numbers. Then y=-x is the additive inverse of x, since x+(-x)=0 (and 0 is a real number).
Let x be an element of the
natural numbers. Then there is no additive inverse of x within the naturals (
i.e. natural numbers are strictly
positive integers), and no 0. However, there is an additive inverse y=-x within the set of integers (since 0 is an integer).
Consider again a number x in the naturals and let another number n be a natural number also. Then there exists a natural number k such that 0<k
<n where x
=k (
mod n) (i.e. there exists a natural number q such that n*q=x-k). Then x-k
=n
=0 (mod n). So k is a "
modular additive inverse" of x in the natural numbers.