http://www.welchlabs.com/blog Imaginary numbers are not some wild invention, they are the deep and natural result of extending our number system. Imaginary numbers are all about the discovery of numbers existing not in one dimension along the number line, but in full two dimensional space.
This week we use the complex plane to solve algebra problems. I think the idea of solving tough algebra problems visually is pretty fantastic. The problems I present in this episode, \(X^3-1=0\) and \(X^8-1=0\) give solutions that are roots of unity, numbers that equal one when raised to integer powers. These numbers have a special place in number theory, and show up in one of my favorite pieces of mathematics: The Discrete Fourier Transform (DFT):
$$X[k] = \frac{1}{N_F}\sum^{N_F-1}_{n=0}h_w[n]x[n]e^{-j2\pi k/N_F}$$
The DFT is a huge part of signal processing, allowing us to convert time series, such as audio signals, into frequency representations in the complex domain. Frequency representations of signals are crucial for all kinds of applications, such as audio filtering, image processing (e.g. Instagram filters), audio and image compression (e.g. mp3s, jpeg). The Fourier Transform also has lots of interesting overlap with our hearing systems work – which we’ll talk about a little in part 11.