What Is Polar Form for Complex Numbers?

In Cartesian form, you write complex numbers in the form $z=a+bi$, which corresponds to the coordinates $\left(a,b\right)$. Instead, you can express a complex number $z$ with the distance $r$ from the origin to $z$, and the angle $𝜃$ formed by $z$ and the real axis in the complex plane. This is called polar form. You can click here to learn how to convert between Cartesian form and polar form.

The distance from $z$ to the origin is called the norm of $z$ and is often denoted by $r$ or $\left|z\right|$. Other words for norm are absolute value, modulus and length. The angle formed by $z$ and the real axis is called the argument of $z$ and is often denoted by $𝜃$ or $\varphi$. You can think of the argument $𝜃$ as the direction of $z$.

The Cartesian form $z=a+bi$ and the polar form $z=\left(r,𝜃\right)$ are equivalent ways to write the same complex number $z$. The polar form $\left(r,𝜃\right)$ corresponds to one unique number $z$. However, a complex number $z$ does not have one unique polar form. The angles $𝜃$ and $𝜃+2\pi \cdot k$ for integers $k\in \mathbb{Z}$ point in the same direction. Therefore, the polar forms $\left(r,𝜃\right)$ and $\left(r,𝜃+2\pi \cdot k\right)$ correspond to the same complex number $z$ for all integers $k$. Usually, you want the argument to be in the interval $\left[0,2\pi \right)$ or the interval $\left(-\pi ,\pi \right]$, but arguments that differ by a multiple of $2\pi$ will still correspond to the same number.

It is not possible to order complex numbers in themselves, but with polar form, you can compare the norm of numbers. For instance, $z=\left(5,\frac{\pi }{3}\right)$ is greater than $w=\left(2,\frac{\pi }{6}\right)$, because the norm of $z$ is greater than the norm of $w$.