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:
   • Convolution and the Fourier Transform...  

Convolution playlist:
   • Convolution and the Fourier Transform  

How Imaginary Numbers were invented:
   • How Imaginary Numbers Were Invented  

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