Multiplication rule

In this video I prove to you the following result for the multiplication of two complex numbers when given in modulus-argument form:

Rule to remember:

  • \text{If } z_1 = r_1 \left( \cos \theta _1 + i \sin \theta _1 \right) \text{  and  } z_2 = r_2 \left( \cos \theta _2 + i \sin \theta _2 \right) \\ \\ \text{then} \\ \\ \\ \left| z_1 z_2 \right| = \left| z_1 \right| \left| z_2 \right| = r_1 r_2 \\ \\ \\ \text{and} \\ \\ \arg \left( z_1 z_2 \right) = \arg \left( z_1 \right) + \arg \left( z_2 \right) = \theta _1  + \theta _2
Multiplication Rule for the Mod-Arg of two Complex Numbers : ExamSolutions Maths Tutorials - youtube Video

Division rule

In this video I prove to you the following result for the division of two complex numbers when given in modulus-argument form :

Rule to remember:

  •  \text{If } z_1 = r_1 \left( \cos {\theta _1} + i \sin {\theta _1} \right) \text{  and  } z_2 = r_2 \left( \cos {\theta _2} + i \sin {\theta _2} \right) \\ \\ \text{then} \\ \\ \left| \dfrac{z_1}{z_2} \right| = \left| \dfrac{z_1}{z_2} \right| = \dfrac{r_1}{r_2} \\ \\ \text{and} \\  \\ \arg \left( \dfrac{z_1}{z_2} \right) = \arg \left( z_1 \right) - \arg \left( z_2 \right) = {\theta _1} - {\theta _2}
Division Rule for the Mod-Arg of two Complex Numbers : ExamSolutions Maths Tutorials - youtube Video

Mixed Examples

Example:

\begin{array}{l} {\text{If }{z_1} = 6\left( {\cos \dfrac{\pi }{{12}} + i\sin \dfrac{\pi }{{12}}} \right){\text{,    }}{z_2} = 2\left( {\cos \dfrac{{2\pi }}{3} + i\sin \dfrac{{2\pi }}{3}} \right){\text{   and    }}{z_3} = 3\left( {\cos \dfrac{{5\pi }}{{12}} - i\sin \dfrac{{5\pi }}{{12}}} \right)} \\ \\ {\text{Find the following giving your answers in the form } a+ib:} \end{array}

  1. {z_1}{z_2}
  2. \dfrac{{{z_1}}}{{{z_3}}}
  3. \dfrac{{{z_1}{z_3}}}{{{z_2}}}
Multiplication & Division of Complex Numbers in Mod-Arg form Examples : ExamSolutions - youtube Video