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.

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