Nyquist Sampling Rate

To reconstruct a signal from samples you must sample at more than twice its highest frequency. Below that the high frequencies do not simply vanish, which would be tolerable. They fold down and reappear as false low-frequency tones that are mathematically indistinguishable from real ones, and no amount of processing afterwards can separate them. That is aliasing, and it is the reason wagon wheels spin backwards in old films.

Human hearing reaches about 20 kHz, so audio needs at least 40 kHz. CDs use 44.1 kHz, and the extra 4.1 kHz is not sloppiness. It is the transition band an anti-aliasing filter needs in order to fall from passing 20 kHz to blocking 22.05 kHz, because a filter with an infinitely steep edge does not exist. The specific figure 44,100 came from a much more mundane constraint: early digital audio was recorded onto video tape, and 44,100 is what you get from three samples per line on 245 usable lines at 60 fields per second.

The theorem is stated as strictly greater than twice, and the strictness matters more than it looks. Sampling a 1 kHz sine at exactly 2 kHz can land every sample on a zero crossing and record silence. The classic mistake, though, is filtering after the converter instead of before it. Once aliased, the false tone is inside your passband and it is a real part of your data forever. The anti-alias filter has to be analogue, and it has to come first.