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Chapter 1 The Basics 1.1 The Field of Complex Numbers The two dimensional R-vector space R2 of ordered pairs z =(x,y) of real numbers with multiplication (x1,y1)(x2,y2):=(x1x2−y1y2,x1y2+x2y1) isacommutativefield denotedbyC.Weidentify arealnumber x with the complex number (x,0).Via this identification C becomes a field extension of R with the unit Designed for the undergraduate student with a calculus background but no prior experience with complex variables, this text discusses theory of the most relevant mathematical topics in a student-friendly manner. A 4-credit course can include topics from Chapter 5 on nonlinear systems. 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