Transcription of 1 Complex algebra and the complex plane
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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. 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 qua
1 p 1 4 2 = 1 p 3 2 = 1 p 3 p 1 2 = 1 p 3i 2: Think: Do you know how to solve quadratic equations by completing the square? This is how the quadratic formula is derived and is well worth knowing! 1.2 Fundamental theorem of algebra One of the reasons for using complex numbers is because allowing complex roots means every polynomial has exactly ...
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