The imaginary number i and the Fourier Transform

i and the Fourier Transform; what do they have to do with each other? The answer is the complex exponential. It's called complex because the "i" turns an exponential function into a spiral containing within it a cosine wave and a sine wave. By using convolution, these two functions allow the Fourier Transform to model almost any signal as a collection of sinusoids. In this video, we look at an intuitive way to understand what "i" is and what it is doing in the Fourier Transform. Other videos of interest: Convolution and the Fourier Transform: https://www.youtube.com/watch?v=9i6aDdQ9FTQ Convolution playlist: https://youtube.com/playlist?list=PLWMUMyAolbNsjohK9PJBuYP7lpcz5SBtZ How Imaginary Numbers were invented: https://www.youtube.com/watch?v=cUzklzVXJwo 0:00 - Introduction 1:15 - Ident 1:20 - Welcome 1:29 - The history of imaginary numbers 3:48 - The origin of my quest to understand imaginary numbers 4:32 - A geometric way of looking at imaginary numbers 9:37 - Looking at a spiral from different angles 10:39 - Why "i" is used in the Fourier Transform 10:44 - Answer to the last video's challenge 11:39 - How "i" enables us to take a convolution shortcut 13:05 - Reversing the Cosine and Sine Waves 15:01 - Finding the Magnitude 15:12 - Finding the Phase 15:20 - Building the Fourier Transform 15:38 - The small matter of a minus sign 16:34 - This video's challenge 17:10 - End Screen