dimensional analysis of the Shannon-Hartley theorem
In the equation:
$$C = B \cdot log_2 (1 + \frac{S}{N})$$
The B represents the bandwidth in Hz, and the log2(1 + S/N) represents the "information density" that you can achieve as a result of the signal to noise ratio. This expression has units of "bits/cycle", but this is rarely stated explicitly, since it's technically a dimensionless quantity. It's basically a measure of how many distinct signalling states (e.g., voltage levels) you can reliably distinguish at the receiver, given the noise level in the channel.
So, if bandwidth has units of Hz, or cycles/second, and the rest has units of bits/cycle, you end up with bits/second.
Both units have dimensions of inverse time. Hertz, formerly known as cycles per second, has the same dimensions as angular frequency, radians per second.
In other words, bits, cycles, and radians are not dimensionful quantities.