Why calculate negative frequencies of DFT? This answer says that when calculating the DFT on a real valued signal, the (roughly) second half of the bins are the complex conjugates of the first half of the bins. Firstly I was curious if that is true?

If it is true, it seems like for frequency analysis purposes, you could forgo calculating the second half of the bins, which ought to be helpful for computation times and also for storage space if you want to store information in frequency domain since you could calculate the second half of the information on demand.

This also seems to be "more correct" to what people would expect when doing an FFT.

For instance, if using the equation:

$$X_k = \frac 1N \sum\limits_{n=0}^{N-1}x_ke^{-2i \pi kn/N}, \quad k\in[0,N), \quad k\in \mathbb Z$$

If you DFT a 4 sample cosine wave: $[1, 0, -1, 0]$, it gives the result: $[0, 0.5, 0, 0.5]$.

But, if you are only calculating the positive frequencies, it seems reasonable to modify the equation to be this:

$$X_k = \frac 2N \sum\limits_{n=0}^{N-1}x_ke^{-2i \pi kn/N}, \quad k\in[0,N/2), \quad k\in \mathbb Z$$

Which when applied to the 4 sample cosine wave gives us this result:
$[0,1]$

It seems more correct (to me anyways!) that it shows that $0\textrm{ Hz}$ (DC) has an amplitude of 0, and that $1\textrm{ Hz}$ has an amplitude of 1.  It's kind of confusing the other way, where the full amplitude of the $1\textrm{ Hz}$ wave is split between the positive and negative $1\textrm{ Hz}$ frequency.

Why is it then, that for real valued signals, we even bother calculating and reporting negative frequencies?
Mainly because its easier. The FFT is a specific algorithm to calculate the DFT. However, it only works if you calculate ALL frequencies (regardless if you want them or not). It takes in N complex values and it spits out N complex values. So the FFT can be used to evaluate the your first equation but not your second. 

In your example, that's a trivial difference; but for larger FFT sizes this makes a substantial difference. It's a lot faster to calculate the values at negative frequencies with FFT and simply throw them away then calculating only the values at positive frequencies with a different algorithm.

This being said, there are ways for making use of the redundancy:


You can pack two real signals into a single complex, do a single complex FFT and split this into odd and even parts to get the DFTs of both input signals
There is a way to do a N point real DFT using N/2 point complex FFT. It just needs one extra stage of "unwinding" the result.