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Ch1-Complex Algebra Review

Ch1-Complex Algebra Review - Complex Algebra Review Dr V...

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Unformatted text preview: Complex Algebra Review Dr. V. Këpuska February 11, 2012 Veton Këpuska 2 Complex Algebra Elements Definitions: 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 February 11, 2012 Veton Këpuska 3 Complex Algebra Elements { } { } z of part Imaginary z of part Real Im Re define then we If If If ∴ ∴ ≡ ≡ + = ∈ = ⇒ = ∈ = ⇒ = z y z x jy x z x z y jy z x R I February 11, 2012 Veton Këpuska 4 Euler’s Identity j e e e e j e j e j e j j j j j j j 2 sin 2 cos sin cos sin cos sin cos θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ---- = + = ⇒- = + = + = February 11, 2012 Veton Këpuska 5 Polar Form of Complex Numbers 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 ] ,- ( r θ π π θ θ ≡ ∠ = ≡ ∈ ≥ ∈ = + z r z r re z j R February 11, 2012 Veton Këpuska 6 Polar Form of Complex Numbers Conversion between polar and rectangular (Cartesian) forms. 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 February 11, 2012 Veton Këpuska 7 Geometric Representation of Complex Numbers....
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Ch1-Complex Algebra Review - Complex Algebra Review Dr V...

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