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Complex Algebra Review Dr. V. Këpuska

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11 June 2014Veton Këpuska2 Complex Algebra Elements Definitions: Note: Real numbers can be thought of as complex numbers with imaginary part equal to zero.

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11 June 2014Veton Këpuska3 Complex Algebra Elements

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11 June 2014Veton Këpuska4 Eulers Identity

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11 June 2014Veton Këpuska5 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.

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11 June 2014Veton Këpuska6 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?

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11 June 2014Veton Këpuska7 Geometric Representation of Complex Numbers. Q1Q2 Q3Q4 Im Re z Re{z} Im{z} |z| Complex or Gaussian plane Axis of Reals Axis of Imaginaries

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11 June 2014Veton Këpuska8 Geometric Representation of Complex Numbers. Q1Q2 Q3Q4 Im Re z Re{z} Im{z} |z | Complex or Gaussian plane Axis of Reals Axis of Imaginarie s Complex Number in Quadrant Condition 1Condition 2 Q1 or Q2Arg{z} 0Im{z} 0 Q3 or Q4Arg{z} 0Im{z} 0 Q1 or Q4Re{z} 0 Q2 or Q3Re{z} 0

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11 June 2014Veton Këpuska9 Example Im Re z1z z2z2 z3z3

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11 June 2014Veton Këpuska10 Conjugation of Complex Numbers Definition: If z = x+jy C then z * = x-jy is called the Complex Conjugate number of z. Example: If z=re j (polar form) then what is z * also in polar form? If z=re j then z*=re -j

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11 June 2014Veton Këpuska11 Geometric Representation of Conjugate Numbers If z=re j then z*=re -j Im Re z r Complex or Gaussian plane - r x y -y z*z*

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11 June 2014Veton Këpuska12 Complex Number Operations Extension of Operations for Real Numbers When adding/subtracting complex numbers it is most convenient to use Cartesian form. When multiplying/dividing complex numbers it is most convenient to use Polar form.

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11 June 2014Veton Këpuska13 Addition/Subtraction of Complex Numbers

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11 June 2014Veton Këpuska14 Multiplication/Division of Complex Numbers

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11 June 2014Veton Këpuska15 Useful Identities z C, R & n Z (integer set)

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11 June 2014Veton Këpuska16 Useful Identities Example: z = +j0 =2 then arg(2)=0 =-2 then arg(-2)= Im Re j -2 z 21 0

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11 June 2014Veton Këpuska17 Silly Examples and Tricks Im Re j 1 0 -j /2 3 /2

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11 June 2014Veton Këpuska18 Division Example Division of two complex numbers in rectangular form.

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11 June 2014Veton Këpuska19 Roots of Unity Regard the equation: z N -1=0, where z C & N Z + (i.e. N>0) The fundamental theorem of algebra (Gauss) states that an Nth degree algebraic equation has N roots (not necessarily distinct). Example: N=3; z 3 -1=0 z 3 =1

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11 June 2014Veton Këpuska20 Roots of Unity z N -1=0 has roots, k=0,1,..,N-1, where The roots of are called N th roots of unity.

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11 June 2014Veton Këpuska21 Roots of Unity Verification:

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11 June 2014Veton Këpuska22 J1J1 Geometric Representation Im Re 1 -j1 J2J2 0 j1 2/3 4/3 J0J0 2/3

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11 June 2014Veton Këpuska23 Important Observations 1.Magnitude of each root are equal to 1. Thus, the Nth roots of unity are located on the unit circle. (Unit circle is a circle on the complex plane with radius of 1). 2.The difference in angle between two consecutive roots is 2 /N. 3.The roots, if complex, appear in complex-conjugate pairs. For example for N=3, ( J 1 ) * = J 2. In general the following property holds: J N-k =( J k ) *

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