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Review of Complex Numbers
Introduction to Complex Numbers
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Complex numbers could be represented by the form
x
y

 y
x 
Where x and y are real numbers
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Complex numbers are denoted: N = {x}+j{y}, where x
is considered the REAL part and Y is considered the
IMAGINARY part
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If x = 0, N is considered an IMAGINARY NUMBER
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If y = 0, N is considered a REAL number
Properties of Complex Numbers
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The sum of two complex numbers is a complex
number:
(x1 + jy1) + (x2 + jy2) = (x1 + x2) + j(y1 + y2);
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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:
(x1 + jy1)(x2 + jy2) = x1(x2 + jy2) + (jy1)(x2 + iy2)
= x1x2 + x1(jy2) + (jy1)x2 + (jy1)(jy2)
= x1x2 + ix1y2 + iy1x2 + i2y1y2
= (x1x2 + {−1}y1y2) + i(x1y2 + y1x2)
= (x1x2 − y1y2) + i(x1y2 + y1x2)
Calculating with complex Numbers
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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) − 14 + 7i
= −4 + 13i .
Calculating with complex Numbers
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Applying Cramer’s rule:
Solve for w!
Calculating with complex Numbers
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Class exercise: solve the system:
(1 + i)z + (2 − i)w = −3i
(1 + 2i)z + (3 + i)w = 2 + 2i.
Calculating with complex Numbers
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Example 2: solve the system: z2 = 1 + i.
Let z = x + iy.
>> (x + iy)2 = x2 − y2 + 2xyi = 1 + i,
>> x2 − y2 = 1 and 2xy = 1.
>> x ≠0 and y = 1/(2x)
>>
>> 4x4 − 4x2 − 1 = 0
>>
>>
>>
>>
Calculating with complex Numbers
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Class exercise: solve the system: z2 = 1 + i√3
Cartesian and polar representation of a complex number
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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.
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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 = √(x2 + y2)
tanθ = (y/x)
θ = arctan(y/x)+ (0 or Π) (Π is added iff x is negative)
The Euler Formula
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ej θ = cos θ + j sin θ
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Z = x+ jy = r cos θ + j rsin θ = r (cos θ + j sin θ) = r ej θ
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R is the distance of the point z from the origin
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1/Z = 1/ r ej θ =( 1/ r) e-j θ
Conjugate of a complex number
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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) = x2+y2 = |z|2
Some useful identities
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1e±j Π =-1 ; e±j nΠ =-1 for n odd integer
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e±j 2nΠ =1 for n integer
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ej Π/2 = j
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E-j Π/2 = -j
Examples
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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
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Express the following numbers in polar form (also sketch
the geometric representation):
• 2+j3
• r = |z| = √(22+32) = √13
• Θ = tan-1(3/2) = 56.30
• 2+j3 = √13ej56.3º
Examples
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Represent the following numbers in the complex plane and
express them in Cartesian form:
• 2 ej Π/3
• 4 e- j 3Π/4
• Use the MATLAB function pol2cart to convert the
above numbers from polar to Cartesian form
Examples
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Represent the following numbers in the complex plane and
express them in Cartesian form:
• 2 ej Π/3
• = 2cos(Π/3) + 2jsin(Π/3)
• =2(1/2) +2 j(√3/2)
• =1+j√3
Examples
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Determine z1z2 and z1/z2 for
• z1 = 3 + j4 = 5ej 53.1º
• z2 = 2 + j3 = √13 ej 56.3º
Solve this problem in both polar and Cartesian forms
Solve this problem using MATLAB
Examples
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Determine z1z2 and z1/z2 for
• z1 = 3 + j4 = 5ej 53.1º
• z2 = 2 + j3 = √13 ej 56.3º
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Polar:
• z1z2 = (3+j4)(2+j3) = (6-12)+j(8+9) = -6+j17
• z1/z2 = (3+j4)(2-j3) /(22+32) = (18/13) – j(1/13)
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Cartesian:
• z1z2 = (5ej 53.1º )(√13 ej 56.3º )= 5√13 ej( 53.1º+ 56.3º ) =5√13 ej( 109.4º)
• z1/z2 = (5ej 53.1º )/(√13 ej 56.3º )=(5/√13) ej( 53.1º- 56.3º ) =(5/√13) ej(-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.
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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) )/(32 + 42 ω2)
• (6+4ω2)/(9+16 ω2) - j5ω/9+ω2)
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b) Express X(ω) in polar form and find its magnitude and
angle.
• X(ω) =[√(4 + ω2) ej arctan(w/2)]/ [√(9 + 16ω2) ej arctan(4w/3)]
• √((4 + ω2)/ √(9 + 16ω2)) ej (arctan(w/2)-arctan(4w/3))