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Question

Let S be the set of complex numbers of the form a + bi, where a and b are integers. We say that z S is a unit if there exists a w S such that zw = 1. Find the number of units in S.

Accepted Answer

Let z = a + bi and w = c + di. We want zw = 1. Then |zw| = |z||w| = 1, so |z|^2 |w|^2 = 1. Hence, (a^2 + b^2)(c^2 + d^2) = 1.If both a = b = 0, then z = 0, so zw = 0. Hence, a^2 + b^2 1. Similarly, we can show that c^2 + d^2 1. Then (a^2 + b^2)(c^2 + d^2) 1.But (a^2 + b^2)(c^2 + d^2) = 1, and the only way to get equality is if a^2 + b^2 = c^2 + d^2 = 1. If a^2 + b^2 = 1, then one of a, b must be 0 and the other must be 1. Thus, z can only be 1, -1, i, or -i. It is easy to check that all 4 complex numbers are units.