How do you compute the additive inverse (negation) in modular arithmetic?

Question

Accepted Answer

The additive inverse of $x$ mod $N$ is $y = -x \mod N$, which is the value where $x + y \equiv 0 \mod N$. Simply compute $N - x$.
Examples mod 5:

$x$
0
1
2
3
4

$-x \mod 5$
0
4
3
2
1

For negative numbers: Add multiples of N until positive.

$-55 \mod 9$: Add $7 	imes 9 = 63$ → $(-55 + 63) = 8$ → answer is 8

The additive inverse always exists for every element in $\mathbb{Z}_N$.
Go deeper:

Modular arithmetic (Wikipedia) — additive and multiplicative structure of ℤ_N.