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Let S be the set of complex numbers of the form a + bi, where a and b are integers. We say that $$z \in S$$ is a unit if there exists a $$w \in S$$ such that

zw = 1. Find the number of units in S.

Aug 12, 2019

$$z w = 1 \Rightarrow |z||w| = 1 \Rightarrow |w| = \dfrac{1}{|z|}\\ \text{The only way this can occur is if a=\pm 1,~b=0 or a=0, b=\pm 1}\\ \text{If z=\pm 1, then w=z. If z = \pm i, then w = -z}\\ \text{Thus there are 4 units in S, \{1,-1,i,-i\}}$$