When is writing complex number in polar form better then rectangular form

Question

Today I was wondering when it's better to use polar form to write complex numbers vs. rectangular form.

Let's suppose we have this complex number in rectangular form: $\sqrt{3}-i$

and we also have the same number in polar form: $2(\cos{(-\frac{\pi}{6}})+i\sin{(-\frac{\pi}{6}})),$

so both forms show this complex number, but when to use which form -- what are the advantages of polar form vs. rectangular one?

Thanks for answering and best regards.

Answer 1

wondering when it's better to use polar form to write complex numbers vs. rectangular form

In rectangular form, complex numbers are easy to add; just add their components.

In polar form, complex numbers are easy to multiply;

just multiply their magnitudes and add their arguments.

Answer 2

I would also vote to close this since the original poster doesn't seem to understand what the "polar form" of a complex number is! The polar form of $\sqrt{3}- i$ is NOT "$\frac{\pi}{6}$". For one thing, $\frac{\pi}{6}$ is a real number with imaginary part 0. Then absolute value of $\sqrt{3}- i$ is $\sqrt{(\sqrt{3})^2+ 1^2}= \sqrt{4}= 2$. The argument is $arctan\left(\frac{\sqrt{3}}{-1}\right)= arctan(-\sqrt{3})= -\frac{\pi}{6}$.

The polar form of $\sqrt{3}- i$ is $2\left(\cos\left(-\frac{\pi}{6}\right)+ i \sin\left(-\frac{\pi}{6}\right)\right)= 2\left(\cos\left(\frac{\pi}{6}\right)- i \sin\left(\frac{\pi}{6}\right)\right)$ or, in an abbreviated form, $2 cis\left(-\frac{\pi}{6}\right)$ or sometimes just "\left(2, -\fra{\pi}{6}\right)$.

In any case, the product of (a+ bi)(c+ di) is (ac- bd)+ (ad+ bc)i while the product of (r cis(t))(u cis(v)) is ru(cis(t+ v). The second is much easier. That is especially true of powers. $(a+ bi)^n$ is wicked, while $(r cis(u))^n= r^n cis(nu)$.