complex_numbers

# complex_numbers - Basics of Complex Numbers(I 1 General i-1...

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Basics of Complex Numbers (I) 1. General i - 1, so i 2 = - 1, i 3 = - i , i 4 = 1 and then it starts over again ( i 5 = i , i 6 = - 1,. . . ). Any complex number z can be written as the sum of a real part and an imaginary part: z = [Re z ] + i [Im z ] , where the numbers or variables in the []’s are real . So z = x + y i with x and y real is in this form but w = 1 / ( a + b i ) is not (see ”Rationalizing” below). Thus, Im z = y , but Re w 6 = 1 /a . Complex Conjugate: The complex conjugate of z , which is written as z * , is found by changing the sign of every i in z : z * = [Re z ] - i [Im z ] so if z = 1 a + b i , then z * = 1 a - b i . Note: There may be “hidden” i ’s in the variables; if a is a complex number, then z * = 1 / ( a * - b i ). Magnitude: The magnitude squared of a complex number z is: zz * ≡ | z | 2 = [Re z ] 2 - ( i ) 2 [Im z ] 2 = [Re z ] 2 + [Im z ] 2 0 , where the last equality shows that the magnitude is positive (except when z = 0). Basic rule: if you need to make something real, multiply by its complex conjugate.

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complex_numbers - Basics of Complex Numbers(I 1 General i-1...

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