Operations with complex numbers in polar form

These operations are a quotient of complex numbers in polar form. Dividim els mòduls and restem els arguments. We divide the modules and reduce the arguments.

1. $\dfrac{56_{180^{\circ}}}{23_{77^{\circ}}}=\Big(\dfrac{56}{23}\Big)_{180^{\circ}-77^{\circ}}=\Big(\dfrac{56}{23}\Big)_{103^{\circ}}$

In this case we have left the module indicated with the fraction because the result is not an integer.

2. $\dfrac{24_{60^{\circ}}}{4_{60^{\circ}}}=\dfrac{24}{4}_{60^{\circ}-60^{\circ}}=6_{0^{\circ}}$.

1. The first we will do is the product: $ 4_{76^{\circ}}\cdot 12_{12^{\circ}} =(4\cdot 12)_{76^{\circ}+12^{\circ}}=48_{88^{\circ}} $

And now we can do the division: $160_{300^{\circ}}:(4_{76^{\circ}}\cdot 12_{12^{\circ}})=160_{300^{\circ}}:48_{88^{\circ}}={\dfrac{160}{48}}_{300^{\circ}-88^{\circ}} = {\dfrac{10}{3}}_{212^{\circ}}$

2. $\sqrt[3]{27_{130^{\circ}}}= \sqrt[3]{27}_{\frac{130^{\circ}+360^{\circ}k}{3}}= 3_{\frac{130^{\circ}+360^{\circ}k}{3}} \ $ for $k=0,1,2$.

This way, $$\displaystyle \sqrt[3]{27_{130^{\circ}}}= \left\{ \begin{array}{l} 3_{\frac{130}{3}^{\circ}} \\ 3_{\frac{490}{3}^{\circ}} \\ 3_{\frac{850}{3}^{\circ}} \end{array} \right. $$