Complex number polar form review

The polar form of complex numbers emphasizes their graphical attributes: $\text{absolute value}$‍ (the distance of the number from the origin in the complex plane) and $\text{angle}$‍ (the angle that the number forms with the positive Real axis). These are also called $\text{modulus}$‍ and $\text{argument}$‍.

Note that if we expand the parentheses in the polar representation, we get the number's rectangular form:

Polar form is really useful for multiplying and dividing complex numbers:

Let's evaluate $(1+\sqrt{3}i{)}^6$‍. First, we convert to polar form:

$(1+\sqrt{3}i)=2(\cos {60}^{\circ }+i\sin {60}^{\circ })$‍

Now we use the rule from above:

Let's find the solutions to the equation $z^3=27$‍. First, we define $r$‍ and $\theta$‍ to be the absolute value and angle of $z$‍. So $z^3$‍ is $r^3[\cos (3\cdot \theta )+i\sin (3\cdot \theta )]$‍.

The number $27$‍ can be written as $27[\cos (k\cdot {360}^{\circ })+i\sin (k\cdot {360}^{\circ })]$‍.

We obtain two equations from the main equation $z^3=27$‍:

The solution of the first equation is $r=3$‍. The solution of the second equation is $\theta =k\cdot {120}^{\circ }$‍, which has three distinct solutions: $0^{\circ }$‍, ${120}^{\circ }$‍, and ${240}^{\circ }$‍. These correspond to the following three solutions: