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### ME3514-Complex Numbers_part_1_v1

Course: ME 3514, Fall 2011
School: Virginia Tech
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Word Count: 631

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Complex ME3514 Numbers: Complex Numbers Imaginary Part z a ib Real P R l Part Imaginary Unit, I i U it i 1 Graphic Representation: Complex Plane b Imaginary Axis Complex Number z Vector Representation of z a Real Axis Complex numbers also behaves as a vector in the complex plane Such vector is plane. defined by the complex number z and the origin of the complex plane. R. Burdisso 1 ME3514 Rectangular...

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Complex ME3514 Numbers: Complex Numbers Imaginary Part z a ib Real P R l Part Imaginary Unit, I i U it i 1 Graphic Representation: Complex Plane b Imaginary Axis Complex Number z Vector Representation of z a Real Axis Complex numbers also behaves as a vector in the complex plane Such vector is plane. defined by the complex number z and the origin of the complex plane. R. Burdisso 1 ME3514 Rectangular and Polar Forms: A complex numbers can be represented in: Rectangular Form b Complex Numbers Polar Form z OR a z a ib Where a and b are the Cartesian coordinates in the complex plane z z cos i sin Where z is the magnitude of z, and is the phase (or argument) of z, measured relative to the positive real axis Relationship between Rectangular and Polar Forms: a z cos b z sin R. Burdisso z a 2 b2 tan b a 2 ME3514 Euler s Euler's Formula: Euler's formula (or identity) Complex Numbers ei cos i sin leads to the phasor representation of complex numbers. This is: z z ei eix n 0 ix n! n 1 x 2 n i 1 x 2 n1 2n 1! 2 n ! n 0 n 1 n n 1 cos x i sin x R. Burdisso 3 ME3514 Complex Conjugate: Complex Numbers The complex conjugate of a complex number is defined as: z z * a ib z e i z* The complex conjugate is the mirror image on the real axis of the complex number The absolute value (or complex modulus) of a complex number can then be written as: z a 2 b2 a ib a ib zz * R. Burdisso 4 ME3514 Addition of Complex Numbers: Complex Numbers The addition of complex numbers can be seen as the same operation as vectors. z z1 z2 z2 z1 Subtraction can be seen as a particular case of addition where: z z1 z2 z2 z1 z2 R. Burdisso z z1 z2 5 ME3514 Product of Complex Numbers: product The of complex numbers is defined as: Complex Numbers z z1 z2 z2 z z1 z2 z1 z2 e i 1 2 ze i 2 z1 1 1 2 It is useful to think that the complex number z1 is an operator acting on z2. This p performs two actions: operator p 1) "Scales" the amplitude of z2 by abs(z1) 2) "Rotates" z2 counter clockwise by theta1 R. Burdisso 6 ME3514 Division of Complex Numbers: The division of complex numbers is defined as: z2 i z2 i2 1 z2 z2 e 2 z e i1 z1 z1 e z1 Complex Numbers z z2 z1 2 2 1 1 z1 As in the case of the product, it is useful to think that the complex number z1 is an operator acting on z2. This operator performs two actions: 1) "Scales" the amplitude of z2 by 1/abs(z1) 2) "Rotates" z2 clockwise by theta1 * * z2 z2 z1 z2 z1 z 2 * z1 z1 z1 z1 1 z1 1 In this sense, the inverse of a complex number is defined as: 1 1 z 1 R. Burdisso 1 1 1 z1 e i1 z1 z1 ei1 Unitary Circle z11 7 ME3514 Complex Function: Complex Numbers Consider the case where the phase becomes a function of time increasing at a constant rate : 1e i ( t ) e it 1 z (t ) t 1 Unitary Circle Plotting z(t) on the complex plane traces out a circle with a constant radius = 1. Plotting the real and imaginary components of z(t) vs time, we see that the real component is Re{z(t)} = cos t while the imaginary component is Im{z(t)} = sin t. Thus, z(t) is the complex representation of two harmonic motions of frequency . NOTE: Units of are rad/sec R. Burdisso 8 ME3514 Complex Function: Complex Numbers The product of a complex number times the complex number eit yields a complex number rotating with angular velocity . z (t ) Z ei eit Z ei (t ) z (t ) z t z R. Burdisso 9
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