Geometry of Complex Multiplication & Division (CIE A Level Maths: Pure 3)

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Geometry of Complex Multiplication & Division

You now know how conjugation, addition and subtraction affect the geometry of complex numbers on an Argand diagram. Now we can look at the effects of multiplication and division.

What do multiplication and division look like on an Argand diagram?

  • Let z1 and z2 be two complex numbers
    • With moduli r1 and r2 respectively
    • And arguments θ1 and θ2 respectively
  • To plot z subscript 1 cross times z subscript 2 on an Argand diagram
    • The modulus will be r subscript 1 cross times r subscript 2
    • The argument will be theta subscript 1 plus space theta subscript 2
      • Subtract 2π from the argument if it is not in the range
    • To plot z subscript 1 over z subscript 2 on an Argand diagram
      • The modulus will be r subscript 1 over r subscript 2
      • The argument will be theta subscript 1 minus theta subscript 2
        • Add 2π to the argument if it is not in the range

8-3-2-mult-and-div-complex-diagram-1-1

What are the geometric representations of complex multiplication and division?

  • Let w be a given complex number with modulus r and argument θ
    • In exponential form w equals r straight e to the power of straight i theta end exponent
  • Let z be any complex number represented on an Argand diagram
  • Multiplying z by w results in z being:
    • Stretched from the origin by a scale factor of r
      • If r > 1 then z will move further away from the origin
      • If 0 < r < 1 then z will move closer to the origin
      • If r = 1 then z will remain the same distance from the origin
    • Rotated anti-clockwise about the origin by angle θ
      • If θ < 0 then the rotation will be clockwise
  • Dividing z by w results in z being:
    • Stretched from the origin by a scale factor of 1 over r
      • If r > 1 then z will move closer to the origin
      • If 0 < r < 1 then z will further away from the origin
      • If r = 1 then z will remain the same distance from the origin
    • Rotated clockwise about the origin by angle θ
      • If θ < 0 then the rotation will be anti-clockwise

8-3-2-mult-and-div-complex-diagram-2

Worked example

8-3-2-mult-and-div-complex-we-solution-1

Examiner Tip

  • If a complex number is given in Cartesian form, first convert it to polar form or exponential form to find the modulus and argument.

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Dan

Author: Dan

Expertise: Maths

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.