Booth multiplication algorithm, why it works?
Booth's algorithm works because 99 * N = 100 * N - N, but the latter is easier to calculate (thus using fewer brain resources).

In binary, multiplication by powers of two are simply shifts, and in hardware, shifts can be essentially free (routing requires no gates) though variable shifts require either multiplexers or multiple clock cycles.

Thus instead of multiplying n * 7 as
(n * 4) + (n * 2) + (n * 1)
which requires 2 additions, Booth recoding allows us to implement it as
(n * 8) - (n * 1) requiring one subtraction.

The rest of the description is just a formalisation and generalisation of this idea.