![Complex numbers © Tshwane University of Technology: EL Voges 19/02/25 Page 65
Phasor: The representation of a wave by a rotating arm in an Argand plane, called a phasor plane
Polar form: A complex number in the form ( ) ( )
cos sin cis
z r r j r r
θ θ θ θ
= ∠ = + =
o Also called the modulus-argument or trigonometric form
o θ MAY be in radians or degrees!
o ( 180 ;180 ]
θ ∈ − ° °
Powers of j: 2 3 4 1
1; ; 1; j
j j j j j
=
− =
− ==
−
Principal argument: ( 180 ;180 ]
θ ∈ − ° °
o Calculators give the principal argument when converting from rectangular to polar coordinates
Real part: If z x jy
= + , then Re( )
z x
=
Rectangular form: A complex number in the form x jy
+ where ,
x y ∈ and 1
j= −](https://image.slidesharecdn.com/theme4notescomplexnumbers1-230813181336-24ae15d9/85/Theme-4-Notes-Complex-Numbers-1-pdf-3-320.jpg)
Theme 4 Notes Complex Numbers (1).pdf
- 1. Complex numbers © Tshwane University of Technology: EL Voges 19/02/25 Page 63 COMPLEX NUMBERS Summary of the theory Washngton pp 334 - 361 THEORY Argand diagram: A representation of a complex number in a complex plane o The horizontal axis represents the real part of the complex numbers, while the vertical axis represents the imaginary part of the complex number o When we multiply a complex number by 1 j= − , its representation in an Argand diagram rotates anti-clock wise through 900 o A complex number and its conjugate is symmetrical about the real axis o A complex number is often represented by a directed line segment (an arrow), similar to a vector Argument/amplitude/phase angle: If z r θ = ∠ , then the argument, denoted by arg( ) z , is given by arg( ) z θ = Cartesian form: Rectangular form Complex conjugate o Rectangular form: If z x jy = + , then the complex conjugate of z, denoted by z , is given by z x jy = − o Polar form: If (cos sin ) z r r j θ θ θ = ∠ = + , then ( ) (cos sin ) z r r j θ θ θ = ∠ − = −
- 2. Complex numbers © Tshwane University of Technology: EL Voges 19/02/25 Page 64 o Exponential form: If j z re θ = , then j z re θ − = Complex number: Any number z x jy = + where , x y ∈ and 1 j= − o Pure real numbers and pure imaginary numbers are sub sets of the set of complex numbers Complex plane: The x-y plane where the x-axis represents the real numbers and the y-axis represents the imaginary numbers Conversions: ( ) 2 2 1 cos ; sin ; ; tan y x x r y r r x y θ θ θ − = = = + = De Moivre's theorems: If 1 1 1 z r θ = ∠ and 2 2 2 z r θ = ∠ , then o 1 2 1 2 1 2 ( ) z z rr θ θ = ∠ + o 1 1 2 2 1 2 ( ) z r z r θ θ = ∠ − o 1 1 1 ( ) ( ) n n z r nθ = ∠ o ( ) ( ) 1 1 360 2 1 1 1 , 0,1,2, ,( 1) k k n n n n n z r r k n θ θ π + ° + = ∠ = ∠ = − Equality: Two complex numbers 1 z a jb = + and are equal if and only if a c = and b d = Euler's formula: cos sin j e j θ θ θ ± = ± Exponential form: A complex number in the form j z re θ = o θ MUST be in radians! o De Moivre: 1/ 1/ ( 2 )/ n n j k n z r e θ π + = o Logs: ln( ) ln( ) ln j z re r j θ θ = = + Fact: Remember, , a b ab a b = ∀ ∈ but if , a b ab a b ≠ ∈ Imaginary number: Any number aj where a∈ and 1 j= − Imaginary part: If z x jy = + , then Im( ) z y = Imaginary unit: 1 i j = = − Modulus/magnitude/absolute value: If z r θ = ∠ , then the modulus, denoted by z , is given by z r =
- 3. Complex numbers © Tshwane University of Technology: EL Voges 19/02/25 Page 65 Phasor: The representation of a wave by a rotating arm in an Argand plane, called a phasor plane Polar form: A complex number in the form ( ) ( ) cos sin cis z r r j r r θ θ θ θ = ∠ = + = o Also called the modulus-argument or trigonometric form o θ MAY be in radians or degrees! o ( 180 ;180 ] θ ∈ − ° ° Powers of j: 2 3 4 1 1; ; 1; j j j j j j = − = − == − Principal argument: ( 180 ;180 ] θ ∈ − ° ° o Calculators give the principal argument when converting from rectangular to polar coordinates Real part: If z x jy = + , then Re( ) z x = Rectangular form: A complex number in the form x jy + where , x y ∈ and 1 j= −
- 4. Complex numbers © Tshwane University of Technology: EL Voges 19/02/25 Page 66 CALCULATIONS By hand: Operation Rectangular Polar Exponential 1 2 , z z ∈; n∈ 1 1 1 z x jy = + 2 2 2 z x jy = + 1 1 1 z r θ = ∠ 2 2 2 z r θ = ∠ θ in either radians or degrees 1 1 1 j z re θ = 2 2 2 j z r e θ = θ in radians Conjugate 1 1 1 z x jy z x jy = + ⇔ = − 1 1 1 1 1 1 ( ) z r z r θ θ = ∠ ⇔ = ∠ − 1 1 1 1 1 1 j j z re z re θ θ − = ⇔ = Addition/subtraction 1 2 1 2 1 2 ( ) ( ) z z x x j y y ± = ± + ± Covert to rectangular Covert to rectangular Multiplication 1 2 1 1 2 2 ( )( ) z z x jy x jy =+ + ( ) 1 2 1 2 1 2 z z rr θ θ = ∠ + ( ) 1 2 1 2 1 2 j z z rr e θ θ + = Division 1 1 1 2 2 2 2 2 2 2 z x jy x jy z x jy x jy + − + − = × ( ) 1 1 2 2 1 2 z r z r θ θ = ∠ − ( ) 1 2 1 1 2 2 j z r z r e θ θ − = Powers 1 1 1 ( ) ( ) n n z x jy = + or convert to polar ( ) ( ) ( ) 1 1 1 n n z r nθ = ∠ ( ) ( ) ( ) 1 1 1 n n j n z r e θ = Roots Convert to polar ( ) ( ) ( ) ( ) ( ) 1 1 1/ 1/ 360 1 1 1 1/ 2 1 n n k n n n k n z z r r θ θ π + ° + = = ∠ = ∠ ( ) ( ) ( ) 1 1/ 1/ 2 / 1 1 n n j k n z r e θ π + = Logs Convert to exponential Convert to exponential ( ) ( ) 1 1 1 ln ln z r jθ = + Practice using your calculator!
- 5. Complex numbers © Tshwane University of Technology: EL Voges 19/02/25 Page 67 APPLICATIONS Electrical o Reactance – effective resistance of a part in a circuit o Impedance – total effective resistance to the flow of a current o For a circuit with a resistor with resistance R, a capacitor with reactance XC and an inductor with reactance XL the impedance is ( ) L C Z R j X X = + + Magnitude: 2 2 ( ) L C z R X X = + − Phase angle: ( ) 1 tan L C X X R θ − − = SUPPLEMENTARY EXERCISE 4 (JJ Verlinde) 1. Simplify the complex numbers below. Do the operations in the rectangular form and leave your answer in the simplest rectangular form. 1.1 35 j 1.2 41 j 1.3 74 j 1.4 (2 )(3 4) j j − + 1.5 (1 3)(1 3) j j − + 1.6 ( 2)( 2) j j + − (j+2) 1.7 3 2 1 − − j j 1.8 j + −1 2 1.9 j j − 1 2 2. Convert the following complex numbers to the polar form, with the argument in radians, accurately to 2 decimal places where necessary. 2.1 1 j + 2.2 3 j − 2.3 2 2 j − + 2.4 4 3 j − − 2.5 5 j − 2.6 8 −
- 6. Complex numbers © Tshwane University of Technology: EL Voges 19/02/25 Page 68 3. Convert the following complex numbers to the rectangular form: 3.1 3 3 4 π 3.2 2 π 3.3 2120 3.4 2 240 − 3.5 4 0 3.6 5 2 π − 4. Simplify the following complex numbers. Do not convert to another form. Leave your answers in the simplest polar form. 4.1 4 2 3 4 3 4 2 π π π 4.2 6 2 8 π − 4.3 ( )( ) 4 30 2150 3 60 5. Simplify the following complex numbers by using De Moivre’s theorem. Leave your answer in the polar form with the angles in radians. 5.1 4 ) 1 ( 4 j − 5.2 2 2 j − + 5.3 4 2 ( 1 ) j j − + 5.4 8 ) 1 ( j − 6. Use De Moivre’s theorem to solve for z and leave your answer in the polar form with the angles in radians. 6.1 2 4 z j = 6.2 8 3 − = z 6.3 3 2 2 z j = − 6.4 3 1 2 j z + − = 6.5 16 4 − = z 7. Write the following complex numbers in the simplest rectangular (Cartesian) form, accurately to 3 decimal places. 7.1 2 3 π j e − 7.2 2 ln3 j e π − 7.3 3 2 j e π 8. Write the following complex numbers in the exponential form. 8.1 1 j + 8.2 3 j − 8.3 2 2 j − + 8.4 4 3 j − − 8.5 5 j − 8.6 8 − 9. Solve for x and y, both real, if: 9.1 ) 3 5 ( ) 3 2 ( 2 j y j xj j + − − = 9.2 ) 2 3 )( 2 ( ) 1 ( j j x y j xj − + = + −
- 7. Complex numbers © Tshwane University of Technology: EL Voges 19/02/25 Page 69 ANSWERS 4 If your answers are different from those given here, first do some further calculations – your answers may still be the same! If they are not, please discuss it with your lecturer. 1.1 -j 1.2 j 1.3 -1 1.4 10 5 j + 1.5 4 1.6 10 5 j − − 1.7 1 1 2 2 j − + 1.8 1 j − − 1.9 1 j − + 2.1 2 1.410.79 4 π ≈ 2.2 2 2 0.52 6 π − ≈ − 2.3 3 8 2.83 2.36 4 π ≈ 2.4 5 3.79 5 2.50 = − 2.5 5 5 1.57 2 π − ≈ − 2.6 8 8 3.14 π ≈ 3.1 3 3 2 2 j − + 3.2 -2 3.3 1 3 j − + 3.4 1 3 j − + 3.5 4 3.6 5 j 4.1 5 12 6 π 4.2 1 3 8 4 π − 4.3 2 180 3 − 5.1 π 5.2 4 3 8 8 π 5.3 1 2 2 π − 5.4 16 0 6.1 2 4 z π = or 5 2 4 z π = 6.2 2 3 z π = or 2 z π = or 5 2 3 z π = 6.3 6 8 12 z π = − or 6 7 8 12 z π = or 6 5 8 4 z π = 6.4 2 3 z π = or 4 2 3 z π = 6.5 2 4 z π = or 3 2 4 z π = or 5 2 4 z π = or 7 2 4 z π = 7.1 3 je − 7.2 3 j − 7.3 1 3 j + 8.1 4 2 j e π 8.2 6 2 j e π − 8.3 3 4 8 j e π 8.4 2.50 5 j e− 8.5 2 5 j e π − 8.6 8 j e π 9.1 ( ; ) (10;6) x y = 9.2 ( ; ) (1;9) x y =