There is a whole lot easier a way to understand what the DFT does which doesn't involve complex things. First generate two tables of signals. Each table is a set of signals having the same length and the same number of samples as the signal to be transformed; all signals are sampled at N equi-spaced points in the time interval T.

The 0th entry in the first table is just the number one repeated N times. The 1st entry in the table is a sampled sinusoid with a period exactly equal to T. The 2nd is a sampled sinusoid with a period of one half T (twice the frequency of the 1st.) The third is a sampled sinusoid with a period one 3rd of T (three times the frequency of the 1st.) The fourth has a period one 4th of 1st (four times the frequency of the first.) You get the point. The last entry in the table is a sampled sinusoid with a period of just two samples, i.e. the signal alternates between +1 and -1 throughout its duration. Scratch your head a bit and you'll see that this yields exactly N/2+1 entries.

The second table is the same as the first except that sampled cosinusiods (the quaduratures) are used instead of sinusoids.

Now it's time to do the transform. Multiply the signal to be transformed sample by sample with the 0th entry of the first table and sum the result (the signal processing name for this vector inner product is correlation.) Record that number as the 0th result. Do the same with the 1st signal in the table and record that as the 1st result. Do this until all N/2+1 results are recorded. For mathimagical reasons we call this vector of numbers the "real" part.

Repeat this for the second table (the sampled cosinusoids) and call this vector of numbers the "imaginary" part.

That's it. Those two vectors, which we combine into a N/2+1 by 2 column matrix, are the discrete fourier transform of the signal.

It gets all weird when this is mathimaticalized with complex algebra and the convenient identities and concise notation it puts on the table but that's just artifice and in the end the above is all there is to it. Nature never ever evidenced a truly complex or imaginary signal until Bohr, et. al. found things deep down that couldn't do without them.

Formalizing at it that way, though led Cooley and Tukey to visualize the matrix decomposition that led them to the fast logarithmic time transform that revolutionized signal processing. I think I read that Gauss actually beat them to it but when I was getting educated that part of Gauss's work remained in obscurity and those guys were the gods that had just brought it into the computational realm.