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1 Complex algebra and the complex plane

Topic 1 NotesJeremy Orloff1 Complex algebra and the Complex planeWe will start with a review of the basic algebra and geometry of Complex numbers. Mostlikely you have encountered this previously in or MotivationThe equationx2= 1 has no real solutions, yet we know that this equation arises naturallyand we want to use its roots. So we make up a new symbol for the roots and call it acomplex symbols iwill stand for the solutions to the equationx2= 1. We willcall these new numbers Complex numbers.

Proof. jei j= jcos( ) + isin( )j= q cos2( ) + sin2( ) = 1: In words, this says that ei is always on the unit circle { this is useful to remember! Likewise, if z= rei then jzj= r. You can calculate this, but it should be clear from the de nitions: jzjis the distance from zto the origin, which is exactly the same de nition as for r.

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