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. We will also write 1 = iNote: Engineers typically usejwhile mathematicians and physicists usei. We ll follow themathematical custom in numberiis called animaginary number. This is a historical term. These areperfectly valid numbers that don t happen to lie on the real number re going tolook at the algebra , geometry and, most important for us, the exponentiation of starting a systematic exposition of Complex numbers, we ll work a simple the equationz2+z+ 1 = :We can apply the quadratic formula to getz= 1 1 42= 1 32= 1 3 12= 1 :Do you know how to solve quadratic equations by completing the square?

Properties P1-P4 should convince you that ei behaves like an exponential. 1.6.2 Complex exponentials and polar form Now let’s turn to the relation between polar coordinates and complex exponentials. Suppose z = x+ iyhas polar coordinates rand . That is, we have x= rcos( ) and y= rsin( ). Thus, we get the important relationship

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  Complex, Exponential

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