#Maths # The North Pole - Real Polar Coordinates Mathematicians are the true philosophers of our time, ranging from seeking out multiple ways to solve one seemingly irrelevant problem to toiling away for 80 years in pursuit of numerical perfection, the existence of these superhumans serves as a reminder that there will always be those who achieve more than us. Polar coordinates are one of those "I'd like to do things differently" mathematical power trips, where instead of the x-y plane we're all used to, an angular system using circular radii and angular displacement is used. Though this coordinate system make virtually anything much easier to use, it's a lot to learn for IB! # The South Pole - Imaginary Polar Coordinates Polar Coordinates can be used in tandem with complex numbers to show concepts such as [[Complex Numbers - Foundations (Maths)|De Moivre's Theorem]] as well as allowing us a simpler way to visualise these abstract concepts. Below is a grid of the complex plane in polar form: >[!Example]- Polar Grid >![[Pasted image 20230801155814.png|700]] Note how the polar grid still contains four quadrants. Noting that a right triangle can be constructed within the grid with the imaginary variable at the vertical and the real variable at the horizontal, we can create an expression for the complex number in polar form: $z = r(\cos \theta + i\sin \theta)$ Where: - $\theta$ is the acute angle of the right triangle that can be created against the horizontal - $r$ is the radius of the polar circle (the coordinate the complex number lies on)