Worthless math insight of the day

[winterkoninkje]

The integers mod 3 are better thought of as {-1, 0, 1} rather than the traditional {0,1,2}. This makes it clear that the two non-zero elements are additive inverses of one another. And it also makes it clear that multiplication of non-zero elements just returns an "is same/different" judgment, which comes for free from our usual arithmetic: (-1)*(-1) = 1 = 1*1 and (-1)*1 = -1 = 1*(-1).