I am trying to understand the real DFT and the DFT and why the distinction exists.
From what I know so far the DFT uses $e^{i2\pi kn/N}$ for basis vectors and gives the representation $$x[n]=\sum_{k=0}^{N-1}X[k]e^{i2\pi kn/N}$$ The sum is written from $k=0$ to $N-1$ for historical reasons I think instead of writing it in a way analogous to the Fourier series with the sum going from $k=-N/2$ to $N/2-1$: $$x[n]=\sum_{k=-N/2}^{N/2-1}X[k]e^{i2\pi kn/N}$$ This relying on a peculiar anomoly of the DFT where high frequencies are the same as negative frequencies: $e^{i2\pi kn/N}=e^{i2\pi (k-N)n/N}$.
Continuing the analogy with Fourier Series the real DFT gives the representation $$x[n]=\sum_{k=0}^{N/2}\left(X_R[k]\cos\left(\frac{2\pi kn}{N}\right)-X_I[k]\sin\left(\frac{2\pi kn}{N}\right)\right)$$ This can be viewed as pairing $e^{i2\pi kn/N}$ with $e^{-i2\pi kn/N}$ in the DFT representation where the sum ranges from $k=-N/2$ to $N/2-1$. This is very much like the pairing $c_n e^{in\theta}+c_{-n}e^{-in\theta}=a_n \cos n\theta + b_n \sin n\theta$ which connects the two representations of a Fourier Series:$$\sum_{-\infty}^\infty c_n e^{in\theta}= \frac{a_0}{2} + \sum_1^{\infty}(a_n \cos n\theta + b_n \sin n\theta)$$
My question then is why is the DFT so much more prevalent than the real DFT? One would expect that since the real DFT is using real valued sines and cosines as the basis and is thus representing the geometric picture better that people would like it more. I can see why the DFT and the continuous Fourier Transform would be preferred in a theoretical sense as the algebra of exponentials is simpler. But ignoring the simpler algebra, from a practical computational applied viewpoint why would the DFT be more useful? Why would representing your signal with complex exponentials be more useful in various physics, speech, image, etc. applications than decomposing your signal into sines and cosines. Also if there is anything subtle I'm missing in my above exposition I would like to know: I'm puzzled that the DFT is seen as more connected to the continuous Fourier Transform than to the Fourier Series.