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barcode reader code in asp.net Imaginary axis jb (a + jb) = r in Software
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= 5313 A = 5 5313 Appendix A
Linear Algebra and Complex Numbers
EXAMPLE A2
Convert the number A = 4 60 to its complex form
Solution
b = 4 sin( 60 ) = 4 sin(60 ) = 2 3 Thus, A = 2 j 2 3
a = 4 cos( 60 ) = 4 cos(60 ) = 2 Addition and subtraction of complex numbers take place according to the following rules: (a1 + j b1 ) + (a2 + j b2 ) = (a1 + a2 ) + j (b1 + b2 ) (a1 + j b1 ) (a2 + j b2 ) = (a1 a2 ) + j (b1 b2 ) (A15) Multiplication of complex numbers in polar form follows the law of exponents That is, the magnitude of the product is the product of the individual magnitudes, and the angle of the product is the sum of the individual angles, as shown below (A)(B) = (Aej )(Bej ) = ABej ( + ) = AB ( + ) (A16) If the numbers are given in rectangular form and the product is desired in rectangular form, it may be more convenient to perform the multiplication directly, using the rule that j 2 = 1, as illustrated in equation A17 (a1 + j b1 )(a2 + j b2 ) = a1 a2 + j a1 b2 + j a2 b1 + j 2 b1 b2 = (a1 a2 + j 2 b1 b2 ) + j (a1 b2 + a2 b1 ) = (a1 a2 b1 b2 ) + j (a1 b2 + a2 b1 ) Division of complex numbers in polar form follows the law of exponents That is, the magnitude of the quotient is the quotient of the magnitudes, and the angle of the quotient is the difference of the angles, as shown in equation A18 A A Aej A = = = ( ) j B Be B B (A18) (A17) Division in the rectangular form can be accomplished by multiplying the numerator and denominator by the complex conjugate of the denominator Multiplying the denominator by its complex conjugate converts the denominator to a real number and simpli es division This is shown in Example A4 Powers and roots of a complex number in polar form follow the laws of exponents, as shown in equations A19 and A20 An = (Aej )n = An ej n = An n (A19)

