• i² = -1
• The general form for all complex numbers is a + bi
• From this, we say:
  • Re(a + bi) = a, and
  • Im(a + bi) = -a
  • NOTE: Re – Real, and Im – Imaginary
• The complex number z and its conjugate z⁺ are
  • z = a + bi, and
  • z⁺ = a – bi
• Arithmetic Operations of Complex Numbers:
  • Addition and Subtraction: Add and subtract real, and imaginary parts with each other
  • Multiplication: Carry out algebraic expansion, if i² is present, convert it to -1
  • Division: Rationalize the denominator by multiplying conjugate pairs
  • Equivalence: Equate coefficients
• For quadratics:
  • Use the quadratic formula:
    • b² – 4ac is a negative value
    • pull out a negative and replace it with i²
    • simplify to its general form
  • Use sum of two squares
    

    Examples

    1. Solve z² + 4z + 13 = 0
       • Convert to completed square form: (z + 2)² + 9 = 0
       • Utilize i² as -1 to make it a difference of two squares: (z + 2)² – 9i² = 0
       • Proceed with the general difference of two squares method:
         • (z + 2 + 3i) (z + 2 – 3i) = 0
         • ∴ z = -2 + 3i, and -2 – 3i

 

• For Square Roots:
  

  Examples

  1. Find the square roots of 4 + 3i
     • We can say that √(4 + 3i) = a + bi
     • Square both sides: a² + b² + 2abi = 4 + 3i
     • Equate the real and imaginary parts: a – b = 4, 2ab = 3
     • Solve the equations simultaneously:
       • a = ^3√2/₄ , b = ^√2/₂
     • ∴ √(4 + 3i) = ^3√2/₄ + i^√2/₂ or –^3√2/₄ – i^√2/₂

Argand Diagram

• For the complex number z = a + bi:
  • Its magnitude is defined as |z| = √(a² + b²)
  • Its argument is defined as arg z = tan^{-1 b}/_a
  • Simply plot the imaginary (y-axis) against the real (x-axis)
    
  • Arguments: Always -π < θ < π
    
  • The position of z⁺ is a reflection in the x-axis of z

Locus

• |z – w| = r
• The locus of a point z such that |z – w| = r, is a circle with its center at w and with radius r.
  
• arg(z – w) = θ
• The locus of a point z such that arg(z – w) = θ is a ray from w, making an angle θ with positive real axis
  
• The locus of a point z such that |z – w| = |z – v| is the perpendicular bisector of the line joining w and v
  

Examples

1. On a sketch of an Argand diagram, shade the region whose points represent the complex numbers z which satisfy the inequality |z – 3i| ≤ 2. Find the greatest value of arg z for points in this region.
   • The part shaded blue is the answer
   • To find the greatest value of arg z within this region, we must use the tangent at a point on the circle which has the greatest value of θ from the horizontal (red line)
     
   • The triangle magnified:
     
   • sin α = ²/₃
   • α = 0.730
   • θ = α + ^π/₂
   • θ = 0.730 + π/2 = 2.30
2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities |z – 2 + 2i|, arg z ≤ -π¹/₄, and Re z ≥ 1
   • 
   1. Calculate the greatest possible value of Re z for points lying in the shaded region
      • The greatest value for the real part of z would be the one which is furthest right on the Re axis but within the limits of the shaded area. Graphically:
        
      • Now using circle and Pythagoras theorems, we can find the value of x:
        • x = 2 × cos ^π/₄
        • x = √2
        • ∴ Greatest value of Re z = 2 + √2

Polar Form

• For a complex number z withe magnitude R and argument θ:
  • z = R(cos θ + i sin θ) = Re^iθ
  • ∴ cos θ + i sin θ = e^iθ
• Polar Form to General Form:
  

  Examples

  1. Convert z = 4e^π/₄i to the general form
     • R = 4, arg z = ^π/₄
     • ∴ z = 4(cos ^π/₄ + i sin ^π/₄)
     • z = 4 (^√2/₂ + ^√2/₂i)
     • z = 2√2 + (2√2)i

 

• General Form to Polar Form:
  

  Examples

  1. Convert z = 2√2 + (2√2)i to polar form
     • z = 2√2 + (2√2)i
     • R = √((2√2)² + (2√2)²) = 4
     • θ = tan^{-1 2√2}/_2√2 = ^π/₄
     • ∴4(cos ^π/₄ + i sin ^π/₄) = 4e^π/₄i

Multiplication and Division in Polar Form

• To find the product of two complex numbers in polar form:
  • Multiply their magnitudes
  • Add their arguments
  • z₁z₂ = |z₁||z₂|(arg z₁ + arg z₂)
    

    Examples

    1. Find z₁z₂ in polar form given, z₁ = 2(cos ^π/₄ + i sin ^π/₄), z₂ = 4(cos ^π/₈ + i sin ^π/₈)
       • z₁z₂ = (2 × 4)(cos ^π/₄ + ^π/₈) + i sin (^π/₄ + ^π/₈)
       • ∴ z₁z₂ = 8(cos ^3π/₈ + i sin ^3π/₈)

 

• To find the quotient of two complex numbers in polar form:
  • Divide their arguments
  • Subtract their magnitudes
  • ^z₁/_z2 = ^|z₁|/_|z2| (arg z₁ – arg z₂)
    

    Examples

    1. Find ^z₁/_z2 in polar form given, z₁ = 2(cos ^π/₄ + i sin ^π/₄), z₂ = 4(cos ^π/₈ + i sin ^π/₈)
       • ^z₁/_z2 = (²/₄)(cos ^π/₄ – ^π/₈) + i sin (^π/₄ – ^π/₈)
       • ∴ ^z₁/_z2 = ¹/₂(cos ^π/₈ + i sin ^π/₈)

De Moivre’s Theorem

• zⁿ = Rⁿ(cos nθ + i sin nθ) = Rⁿe^inθ