The simplest rotation is surely a half-turn. Let's work in the complex plane with origin O, and let M be a point with complex coordinate z = x + iy. Its image M' under the rotation about O through π radians (or 180°) has complex coordinate z' = –x – iy. Thus, z' = –1 × z.
But what if we want to make only a quarter-turn? Is there a number a such that this rotation is described by a relation of the form z' = az? If so, we would have a2 = –1, since applying a quarter-turn twice amounts to making a half-turn! The only solutions are a = i and a = – i: it all depends on which way we turn.
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Another way to view rotations
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