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Complex Algebra Review

Complex Algebra Review - Complex Algebra Review Dr V Kpuska...

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Complex Algebra Review Dr. V. K ë puska
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2/11/12 Veton K ë puska 2 Complex Algebra Elements u Definitions: u Note: Real numbers can be thought of as complex numbers with imaginary part equal to zero. C R C Ι R then If Numbers Complex all of Set : Numbers Imaginary all of Set : Numbers Real all of Set : 1 number complex a of form Cartezian + = - jy x z x,y j
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2/11/12 Veton K ë puska 3 Complex Algebra Elements   { } { } z of part Imaginary z of part Real Im Re define then we If 0 If 0 If + = = = = = z y z x jy x z x z y jy z x R I
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2/11/12 Veton K ë puska 4 Euler’s Identity   j e e e e j e j e j e j j j j j j j 2 cos 2 cos sin cos sin cos sin cos θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ - - - - = + = - = + = + =
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2/11/12 Veton K ë puska 5 Polar Form of Complex Numbers u Magnitude of a complex number z is a generalization of the absolute value function/operator  for real numbers. It is buy definition always non-negative. ( 29 z of argument) (or Angle z arg z of Magnitude radians ] , - ( 0 r θ π π θ θ = = + z r z r re z j R
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2/11/12 Veton K ë puska 6 Polar Form of Complex Numbers u Conversion between polar and rectangular  (Cartesian) forms. u For z=0+j0; called “complex zero” one can not define arg(0+j0).  Why? ( 29 ( 29 [ ] ( 29 ( 29 ( 29 ( 29 = + = = = + = + + = + + = = - x y y x r r y r x jy x jr r jy x j r jy x re z j 1 2 2 tan sin cos sin cos sin cos θ θ θ θ θ θ θ θ
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2/11/12 Veton K ë puska 7 Geometric Representation of Complex  Numbers. Q1 Q2 Q3 Q4 Im Re z Re{z} Im{z} |z | q Complex or  Gaussian plane Axis of  Reals Axis of  Imaginaries
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2/11/12 Veton K ë puska 8 Geometric Representation of Complex  Numbers.
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