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Complex Algebra Review Dr. V. Këpuska Complex Algebra Elements Definitions: j 1 R : Set of all Real Numbers Ι : Set of all Imaginary Numbers C : Set of all Complex Numbers If x,y R then z x jy C Cartezian form of a complex number Note: Real numbers can be thought of as complex numbers with imaginary part equal to zero. 24 May 2017 Veton Këpuska 2 Complex Algebra Elements If x 0 z jy I If y 0 z x R If z x jy then we define x Rez Real part of z y Imz Imaginary part of z 24 May 2017 Veton Këpuska 3 Euler’s Identity e j cos j sin j j e e cos j e cos j sin 2 j j j e e e cos j sin cos 2j 24 May 2017 Veton Këpuska 4 Polar Form of Complex Numbers z re j r R r 0 (- , ] radians z r Magnitude of z arg z z Angle (or argument) of z 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. 24 May 2017 Veton Këpuska 5 Polar Form of Complex Numbers Conversion between polar and rectangular (Cartesian) forms. z re j x jy r cos jsin x jy rcos jrsin x jy r x2 y2 x rcos 1 y y rsin tan x For z=0+j0; called “complex zero” one can not define arg(0+j0). Why? 24 May 2017 Veton Këpuska 6 Geometric Representation of Complex Numbers. Im z Im{z} Q2 Axis of Imaginaries Q1 Axis of Reals Re{z} Q3 Re Q4 Complex or Gaussian plane 24 May 2017 Veton Këpuska 7 Geometric Representation of Complex Numbers. Condition 1 Q1 or Q2 Arg{z} ≥ 0 Q3 or Q4 Arg{z} ≤ 0 Q1 or Q4 Re{z} ≥ 0 Q2 or Q3 Re{z} ≤ 0 24 May 2017 Condition 2 Im{z} ≥ 0 Im Q2 Im{z} Complex Number in Quadrant Im{z} ≤ 0 Axis of Imaginarie s z Q1 Axis of Reals Re{z} Q3 Re Q4 Complex or Gaussian plane Veton Këpuska 8 Example Im z1 1 z2 -2 -1 z3 24 May 2017 Re -1 z1 2 z1 1 j1 { 3 z1 4 z2 2 z 2 2 j 0 { z 2 z3 2 z3 1 j1 { 3 z 3 4 Veton Këpuska 9 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=ej (polar form) then what is z* also in polar form? z re j r cos jr sin z r cos jr sin r cos jr sin sin sin cos cos r cos jr sin re j If z=rej then z*=re-j 24 May 2017 Veton Këpuska 10 Geometric Representation of Conjugate Numbers If z=rej then z*=re-j Im z y x Re - -y Complex or Gaussian plane 24 May 2017 Veton Këpuska z* 11 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. 24 May 2017 Veton Këpuska 12 Addition/Subtraction of Complex Numbers Let z1 x1 jy1 , & z 2 x2 jy2 then z1 z 2 x1 x2 j y1 y2 Thus : Rez1 z 2 Rez1 Rez 2 Imz1 z 2 Imz1 Imz 2 24 May 2017 Veton Këpuska 13 Multiplication/Division of Complex Numbers Olso Let z1 r1e j1 r1 j1 j 2 e e j 2 z 2 r2 e r2 z1 r1e j1 & z 2 r2 e j 2 then z1 z 2 r1e j1 r2 e j 2 r1r2 e j1 e j 2 z1 z 2 r1r2 e j 1 2 Therefore : z1 r1 j 1 2 e z 2 r2 Therefore : z1 z 2 z1 z 2 z1 z1 z2 z2 z1 z 2 z1 z 2 24 May 2017 z1 z1 z 2 z2 Veton Këpuska 14 Useful Identities z ∈ C, ∈ R & n ∈ Z (integer set) 1) z z 3) Re z Rez 5) z1 z2 z1 z2 7) z1 z2 z1 z2 8) 10) zz z Rez Rez 12) z z 14) zn z n n n 16) 24 May 2017 2 z z z z 2) Im z Imz z1 z1 z2 z2 4) 6) z z Imz Imz 0 if 0 13) z z z 0 if nz nz 15) 9) 11) Veton Këpuska 15 Useful Identities Example: z = +j0 =2 then arg(2)=0 =-2 then arg(-2)= Im j z -2 24 May 2017 -1 0 1 Veton Këpuska 2 Re 16 Silly Examples and Tricks Im e j 0 cos0 j sin 0 1 j 0 1 e e e j 2 j 3 j 2 e j 2 cos j sin 0 j1 j 2 2 cos j sin 1 j 0 1 j /2 -1 3/2 0 3 3 cos j sin 0 j 1 j 2 2 cos2 j sin 2 1 j 0 1 j 0 j j e j j 2 jj e 2 e 24 May 2017 j 2 j 2 e j 1 j0 1 j1 j j 2 1 j3 j Veton Këpuska 1 Re -j j4 1 j5 j j 6 1 j7 j j8 1 j9 j j10 1 j11 j j12 1 j13 j j14 1 j15 j 17 Division Example Division of two complex numbers in rectangular form. z1 x1 jy1 , z2 x1 jy1 z1 x1 jy1 x1 jy1 x2 jy2 x1 x2 y1 y2 j x2 y1 x1 y2 z2 x2 jy2 x2 jy2 x2 jy2 x 22 y 22 z 2 z2 z1 x1 x2 y1 y2 x2 y1 x1 y2 j 2 2 2 2 z2 x2 y2 x2 y2 z Re 1 z2 24 May 2017 z Im 1 z2 Veton Këpuska 18 Roots of Unity Regard the equation: zN-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). st 3 z 1 z 1 ( 1 root ) 1 Example: nd 3 3 (2 root ) N=3; z -1=0 z =1 ⇒ z 2 ? rd z ? ( 3 root ) 3 24 May 2017 Veton Këpuska 19 Roots of Unity zN-1=0 has roots , k=0,1,..,N-1, where k e k e The roots of j j 2 N 2k N , k 0,1,..., N 1 are called Nth roots of unity. 24 May 2017 Veton Këpuska 20 Roots of Unity Verification: 2k j N N e 1 0 e j 2k 1 0 Applying Eulers Identity e j 2k cos2k j sin 2k cos2k j sin 2k 1 0 cos2k j sin 2k 1 j 0 cos2k 1 wich is true for k 0,1,..., N 1 sin 2k 0 24 May 2017 Veton Këpuska 21 Geometric Representation N 3 e 2 2 1 j 3 j 2 2 3 1 e e 2 j 3 j 4 3 k 0 Im k 1 k 2 -1 J2 24 May 2017 Veton Këpuska j1 J1 2/3 e 1 2 0 3 4/3 0 e j J0 0 1 Re -j1 22 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). |e 2. 2k N | 1, k The difference in angle between two consecutive roots is 2/N. k 1 k 3. j k 1 1 e k j 2 N 2 N Q.E.D The roots, if complex, appear in complex-conjugate pairs. For example for N=3, (J1)*=J2. In general the following property holds: JN-k=(Jk)* 2 N k 2N 2k 2k 2k 2k * j j j j j j * N k e N e N e N e j 2 e N 1e N e N k N k k * * For N 3 & k 1 31 2 1 2 * 24 May 2017 Veton Këpuska 23