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68 chapter 4R R 2RL L 2LC C 2CFigure 4.1 Left: An RLC circuit connected to an alternating voltage source. Right: Two RLCcircuits connected in parallel to an alternating voltage. Observe that one of the parallelcircuits has double the values of R, L, and C as does the other.if they were real numbers and then separating the real and imaginary parts at theend of the calculation. We define the symbol z to represent a number with both realand imaginary parts (Figure 4.2 left):z = x + iy, Re z = x, Im z = y. (4.2)Here i def= √ −1 is the imaginary number, and the combination of real plus imaginarynumbers is called a complex number. In analogy to a vector in an imaginary2-D space, we also use polar coordinates to represent the same complex number:r = √ x 2 + y 2 , θ = tan −1 (y/x), (4.3)x = r cos θ, y = r sin θ. (4.4)The essence of the computing aspect of our problem is the programming ofthe rules of arithmetic for complex numbers. This is an interesting chore becausewhile most computer languages contain all the rules for real numbers, you musteducate them as to the rules for complex numbers (Fortran being the well-educatedexception). Indeed, since complex numbers are not primitive data types like doublesand floats, we will construct complex numbers as objects. We start with two complexnumbers, which we distinguish with subscripts:z 1 = x 1 + iy 1 , z 2 = x 2 + iy 2 . (4.5)Complex arithmetic rules derive from applying algebra to z’s Re and Im parts:Addition: z 1 + z 2 =(x 1 + x 2 )+i(y 1 + y 2 ), (4.6)−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 68