TITLE: Imaginary Numbers and DFT
QUESTION [5 upvotes]: I'm not a math guy per se, but i am trying to understand the DFT. I get to the point where imaginary numbers are used with Euler's formula. 
What I don't understand is why we need an imaginary plane or "i" to begin with since we can plot any point on a real plane  using sin and cos ?
I am using this site and I get to the Euler's formula page before I crash.
I am a Musician/Programmer trying to broaden my knowledge.
https://jackschaedler.github.io/circles-sines-signals/

REPLY [2 votes]: You don't need to use the complex Fourier series for many applications. The Wikipedia
article Discrete cosine transorm
describes some of them as follows:

DCTs are important to numerous applications in science and engineering, from lossy compression of audio (e.g. MP3) and images (e.g. JPEG) (where small high-frequency components can be discarded), to spectral methods for the numerical solution of partial differential equations

The article goes on to state that among the variants of the transform are

the discrete sine transform (DST), which is equivalent to a DFT of real and odd functions

Later on in more detail it states

The obvious distinction between a DCT and a DFT is that the former uses only cosine functions, while the latter uses both cosines and sines (in the form of complex exponentials). 

You can read the article for more technical details and pointers to references for
more information. The summary is that the complex Fourier series is elegant, but for
many real world applications, there are clear advantages to using DCT. This does not
negate the great importance of the DFT for theoretical purposes which is based on
the use of Euler's formula to model simple harmonic motion. In other words, simple
harmonic motion is closely related to circular motion using the important identity
$\, \sin(t)^2 + \cos(t)^2 = 1\,$ and that $\, \frac{d}{dt}\sin(t) = \cos(t),
\: \frac{d}{dt} \cos(t) = -\sin(t).\,$ The fact that these are naturally modeled by
complex  numbers is one of the most striking results in all of mathematics.