Basic Technical Mathematics, 11th Edition

12.6 Products, Quotients, Powers, and Roots of Complex Numbers

• Multiplication, Division, and Powers in Polar and Exponential Forms • DeMoivre’s Theorem • Roots

The operations of multiplication and division can be performed with complex numbers in polar and exponential forms, as well as rectangular form. It is convenient to use polar form for multiplication and division as well as for finding powers or roots of complex numbers.

Using exponential form and laws of exponents, we multiply two complex numbers as

$$[\hspace{0.17em}{r}_1{e}^{j{\theta }_1}]\hspace{0.17em}\hspace{0.17em}\times \hspace{0.17em}[\hspace{0.17em}{r}_2{e}^{j{\theta }_2}]\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}{r}_1{r}_2{e}^{j{\theta }_1\hspace{0.17em}+\hspace{0.17em}j{\theta }_2}\hspace{0.17em}=\hspace{0.17em}{r}_1{r}_2{e}^{j({\theta }_1\hspace{0.17em}+\hspace{0.17em}{\theta }_2)}$$

Rewriting this result in polar form, we have

$$[\hspace{0.17em}{r}_1{e}^{j{\theta }_1}]\hspace{0.17em}\hspace{0.17em}\times \hspace{0.17em}[\hspace{0.17em}{r}_2{e}^{j{\theta }_2}]\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}[\hspace{0.17em}{r}_1(\cos {\theta }_1\hspace{0.17em}+\hspace{0.17em}j\sin {\theta }_1)]\hspace{0.17em}\hspace{0.17em}\times \hspace{0.17em}[\hspace{0.17em}{r}_2(\cos {\theta }_2\hspace{0.17em}+\hspace{0.17em}j\sin {\theta }_2)]\hspace{0.17em}$$

and

$$r_1{r}_2{e}^{j({\theta }_1\hspace{0.17em}+\hspace{0.17em}{\theta }_2)}\hspace{0.17em}=\hspace{0.17em}{r}_1{r}_2[\hspace{0.17em}\cos ({\theta }_1\hspace{0.17em}+\hspace{0.17em}{\theta }_2)\hspace{0.17em}+\hspace{0.17em}j\sin ({\theta }_1\hspace{0.17em}+\hspace{0.17em}\; ...$$