Chapter 8, Problem 6RCC

(a)

To determine

To describe: The procedure to multiply two complex numbers

Answer to Problem 6RCC

Multiplication of $z_1$ and $z_2$ is $z_1{z}_2=r_1{r}_2\left[\cos \left({\theta }_1+{\theta }_2\right)+i\sin \left({\theta }_1+{\theta }_2\right)\right]$ .

Explanation of Solution

Given:

Calculation:

The procedure to multiply two complex numbers if they are given in polar form

Let two complex numbers $z_1$ and $z_2$ have the polar forms

  $z_1=r_1\left(\cos {\theta }_1+i\sin {\theta }_1\right)$

And,

  $z_2=r_2\left(\cos {\theta }_2+i\sin {\theta }_2\right)$

Then, multiplication of $z_1$ and $z_2$ is

  $z_1{z}_2=r_1{r}_2\left[\cos \left({\theta }_1+{\theta }_2\right)+i\sin \left({\theta }_1+{\theta }_2\right)\right]$

Conclusion:

Therefore, multiplication of $z_1$ and $z_2$ is $z_1{z}_2=r_1{r}_2\left[\cos \left({\theta }_1+{\theta }_2\right)+i\sin \left({\theta }_1+{\theta }_2\right)\right]$ .

(b)

To determine

To describe: The procedure to divide two complex numbers

Answer to Problem 6RCC

Division of $z_1$ and $z_2$ is $\frac{z_1}{z_2}=\frac{r_1}{r_2}\left[\cos \left({\theta }_1-{\theta }_2\right)+i\sin \left({\theta }_1-{\theta }_2\right)\right],\left(z_2\ne 0\right)$.

Explanation of Solution

Calculation:

The procedure to divide two complex numbers if they are given in polar form

Let two complex numbers $z_1$ and $z_2$ have the polar forms

  $z_1=r_1\left(\cos {\theta }_1+i\sin {\theta }_1\right)$

And,

  $z_2=r_2\left(\cos {\theta }_2+i\sin {\theta }_2\right)$

Then,

  $\frac{z_1}{z_2}=\frac{r_1}{r_2}\left[\cos \left({\theta }_1-{\theta }_2\right)+i\sin \left({\theta }_1-{\theta }_2\right)\right],\left(z_2\ne 0\right)$

Conclusion:

Therefore, division of $z_1$ and $z_2$ is $\frac{z_1}{z_2}=\frac{r_1}{r_2}\left[\cos \left({\theta }_1-{\theta }_2\right)+i\sin \left({\theta }_1-{\theta }_2\right)\right],\left(z_2\ne 0\right)$.