Review of Complex Numbers

Introduction to Complex Numbers Complex numbers could be represented by the form Where x and y are real numbers Complex numbers are denoted: N = {x}+j{y}, where x is considered the REAL part and Y is considered the IMAGINARY part If x = 0, N is considered an IMAGINARY NUMBER If y = 0, N is considered a REAL number

Properties of Complex Numbers The sum of two complex numbers is a complex number: (x 1 + jy 1 ) + (x 2 + jy 2 ) = (x 1 + x 2 ) + j(y 1 + y 2 ); Example, Express the following complex numbers in the form x + iy, x, y real: (−3 + i)(14 − 2i) The product of two complex numbers is a complex number: (x 1 + jy 1 )(x 2 + jy 2 ) = x 1 (x 2 + jy 2 ) + (jy 1 )(x 2 + iy 2 ) = x 1 x 2 + x 1 (jy 2 ) + (jy 1 )x 2 + (jy 1 )(jy 2 ) = x 1 x 2 + ix 1 y 2 + iy 1 x 2 + i 2 y 1 y 2 = (x 1 x 2 + {−1}y 1 y 2 ) + i(x 1 y 2 + y 1 x 2 ) = (x 1 x 2 − y 1 y 2 ) + i(x 1 y 2 + y 1 x 2 )

Calculating with complex Numbers Example 1: solve the system (1 + i)z + (2 − i)w = 2 + 7i 7z + (8 − 2i)w = 4 − 9i. The determinant of the coefficient matrix is = (1 + i)(8 − 2i) − 7(2 − i) = (8 − 2i) + i(8 − 2i) − i = −4 + 13i.

Calculating with complex Numbers Applying Cramer’s rule: Solve for w!

Calculating with complex Numbers Class exercise: solve the system: (1 + i)z + (2 − i)w = −3i (1 + 2i)z + (3 + i)w = 2 + 2i.

Calculating with complex Numbers Example 2: solve the system: z 2 = 1 + i. Let z = x + iy. >> (x + iy) 2 = x 2 − y 2 + 2xyi = 1 + i, >> x 2 − y 2 = 1 and 2xy = 1. >> x ≠0 and y = 1/(2x) >> >> 4x 4 − 4x 2 − 1 = 0 >>

Calculating with complex Numbers Class exercise: solve the system: z 2 = 1 + i√3

Cartesian and polar representation of a complex number Every complex number z = x+iy can be represented by a point on the Cartesian plane known as complex plane by the ordered pair (x, y).

Cartesian and polar representation of a complex number The Cartesian coordinate pair (x, y) is also equivalent to the polar coordinate pair (r,θ), where r is the (nonnegative) length of the vector corresponding to (x, y), and θ is the angle of the vector relative to positive real line. x = r cos θ y = r sin θ Z = x + jy = r cos θ + j rsin θ = r (cos θ + j sin θ) |z| = r = √(x 2 + y 2 ) tanθ = (y/x) θ = arctan(y/x)+ (0 or Π) (Π is added iff x is negative)

The Euler Formula e j θ = cos θ + j sin θ Z = x+ jy = r cos θ + j rsin θ = r (cos θ + j sin θ) = r e j θ R is the distance of the point z from the origin 1/Z = 1/ r e j θ =( 1/ r) e- j θ

Conjugate of a complex number Let z = x + jy The complex conjugate of z is the complex number defined by z* = x − jy. Geometrically, the complex conjugate of z is obtained by reflecting z in the real axis z* = x − jy = r e- j θ z + z* = (x + jy) + (x – jy) = 2x = 2Re(z) zz* = (x + jy) (x – jy) = x 2 +y 2 = |z| 2

Some useful identities 1e ±j Π =-1 ; e ±j nΠ =-1 for n odd integer e ±j 2nΠ =1 for n integer e j Π/2 = j E -j Π/2 = -j

Examples Express the following numbers in polar form (also sketch the geometric representation): 2+j3 1 – j3 Use the MATLAB function cart2pol to convert the above numbers to polar form

Examples Express the following numbers in polar form (also sketch the geometric representation): 2+j3 r = |z| = √( ) = √13 Θ = tan -1 (3/2) = j3 = √13e j56.3º

Examples Represent the following numbers in the complex plane and express them in Cartesian form: 2 e j Π/3 4 e- j 3Π/4 Use the MATLAB function pol2cart to convert the above numbers from polar to Cartesian form

Examples Represent the following numbers in the complex plane and express them in Cartesian form: 2 e j Π/3 = 2cos(Π/3) + 2jsin(Π/3) =2(1/2) +2 j(√3/2) =1+j√3

Examples Determine z 1 z 2 and z 1 /z 2 for z1 = 3 + j4 = 5e j 53.1º z2 = 2 + j3 = √13 e j 56.3º Solve this problem in both polar and Cartesian forms Solve this problem using MATLAB

Examples Determine z 1 z 2 and z 1 /z 2 for z1 = 3 + j4 = 5e j 53.1º z2 = 2 + j3 = √13 e j 56.3º Polar: z1z2 = (3+j4)(2+j3) = (6-12)+j(8+9) = -6+j17 z1/z2 = (3+j4)(2-j3) /( ) = (18/13) – j(1/13) Cartesian: z1z2 = (5e j 53.1º )(√13 e j 56.3º )= 5√13 e j( 53.1º+ 56.3º ) =5√13 e j( 109.4º) z1/z2 = (5e j 53.1º )/(√13 e j 56.3º )=(5/√13) e j( 53.1º- 56.3º ) =(5/√13) e j(-3.2º)

Examples Consider X(ω), a complex function of a real variable ω: X(ω) = (2 + j ω)/(3 + j4 ω) a)Express X(ω) in Cartesian form, and find its real and imaginary parts. b)Express X(ω) in polar form and find its magnitude and angle.

Examples Consider X(ω), a complex function of a real variable ω: X(ω) = (2 + j ω)/(3 + j4 ω) a)Express X(ω) in Cartesian form, and find its real and imaginary parts. X(ω) = ((2 + j ω)(3 - j4 ω) )/( ω 2 ) (6+4ω 2 )/(9+16 ω 2 ) - j5ω/9+ω 2 ) b)Express X(ω) in polar form and find its magnitude and angle. X(ω) =[√(4 + ω 2 ) e j arctan(w/2) ]/ [√(9 + 16ω 2 ) e j arctan(4w/3) ] √((4 + ω 2 )/ √(9 + 16ω 2 )) e j (arctan(w/2)-arctan(4w/3))