I came across a solution for fast multiplication of $2$ numbers. It involves something following:
Suppose each number has $n = 2m$ digits. Observe that we can split each number into two pieces each of $m$ digits, such that the product of the full numbers can easily be constructed from the products of the pieces, as follows. Let $w = 10^{m+1}$, and represent $A = a_0 + a_1w$ and $B = b_0 + b_1w$, where $a_i$ and $b_i$ are the pieces of each respective number.
I am having hard time wrapping my head around this. How do we express $A$ as $a_0 + a_1w$, and similarly $B$.
An example would really help.