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 ?on an Argand diagram
  • The modulus will be
  • The argument will be
    • Subtract 2π from the argument if it is not in the range
  • To plot ?on an Argand diagram
    • The modulus will be
    • The argument will be
      • Add 2π to the argument if it is not in the range

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
• 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
    • 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

轉載自savemyexams