The complex number i satisfies the equation x4 = 1 (check it: i4 = (i2)2 = (−1)2 = 1), as does its twin, the complex number −i; for that matter, so do the numbers 1 and −1 (which have dual citizenship in both the real numbers and the complex numbers). For the price of replacing addition by multiplication, we can create a model of mod n arithmetic whose elements are not the ordinary integers 0, 1, 2, …, n−1 but the complex numbers ζ0, ζ1, ζ2, …, ζn−1, where ζ is a primitive nth root of unity (that is, an nth root of unity that isn’t a 1st, 2nd, 3rd, …, or (n−1)st root of unity).