To multiply binary number A by binary number B.
We first multiply each bit in number A, by the l.s.b. in number B using the rules;
The result of this step is called a partial product. The partial product from this first step, is written below number B (with their least significant bits aligned).
We then multiply each bit of number A, by the next significant bit in number B to produce another partial product. This is written below the previous partial product, but shifted one place to the left.
We repeat this process for each bit in number B.
We then simply add the partial products together.
(It is a longer but simpler process, if we first add the first two partial products together and then add the next partial product to the result, and so on, until all the partial products have been added together.)
It can be seen from above, that for each bit in number B that is 1, the partial product is just number A shifted left, until the l.s.b. is in the same column as the multiplying bit of number B. A zero is added to the right of the partial product, for each place it has been shifted.
Obviously when the multiplying bit is 0, the partial product is zero and can therefore be omitted for the sake of clarity.