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Complex Numbers

  • 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
    • ∴ cos θ + i sin θ = e
  • Polar Form to General Form:
    Examples
    1. Convert z = 4eπ/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) = 4eπ/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/+ π/8) + i sin (π/+ π/8)
        • ∴ z1z2 = 8(cos /8 + i sin /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) = Rneinθ
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