What do you mean numbers are imaginary?

Typical Complex Number Lectures begin with the definition. While there's nothing inherently wrong with that, it doesn't help us to understand what complex numbers even are, let alone why they show up everywhere in Physics and Engineering. How can something that technically doesn't even exist, the IMAGINARY Unit, be so pivotal in describing the REAL world? Most Professors will explain, as with any text book, that when you multiply by i, you rotate 90 degrees in the Argand Plane. But why? What does the square root of -1 have to do with rotation, and how do scalar numbers posses 2 dimensions? Well as it turns out, complex numbers were always built on the concept of rotation. So at their core, they encompass amplitude and phase as polar objects! So let's start there, taking a look at what it means for a number to have phase, and how that leads us naturally to "discovering" the Square Root of -1, how that unlocks a deeper understanding of complex numbers, and why it appears everywhere. ~Credits~ Background Music:    • Grant's Etude      • Grant's New Etude   References & Further Reading: Riley, K. F., Hobson, M. P., & Bence, S. J. (2023). Mathematical methods for physics and engineering. Cambridge University Press. #educational #maths #science #physics #programming #education #mathematics #educationalvideo