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

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.

How do I sketch complex multiplication and division 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 $z_1\times z_2$ on an Argand diagram

  • The modulus will be $r_1\times r_2$

  • The argument will be ${\theta }_1+{\theta }_2$

    • Subtract 2π from the argument if it is not in the range

  • To plot $\frac{z_1}{z_2}$ on an Argand diagram

    • The modulus will be $\frac{r_1}{r_2}$

    • The argument will be ${\theta }_1-{\theta }_2$

      • Add 2π to the argument if it is not in the range
        

Which geometric transformations represent complex multiplication and division?

• Let w be a given complex number with modulus r and argument θ

  • In exponential form $w=r{\mathrm{e}}^{\mathrm{i}\theta }$

• 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 $\frac{1}{r}$

    • 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