In any set
, a number x
has an additive inverse iff
there exists a number y
within that same set of numbers such that x
=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=
n) (i.e. there exists a natural number q such that n*q=x-k). Then x-k=
0 (mod n). So k is a "modular
additive inverse" of x in the natural numbers.