 Karatsuba operand splitting

Fix an integer base $B\ge 2$. Let $x,y\ge 0$ have at most $n$ digits in positional notation, treating zero as one digit. For $n>1$, set $m=\lceil n/2\rceil$ using the ceiling function, and $q=B^m$. Euclidean division giveswhere $0\le b,d<q$ and $0\le a,c<B^{n-m}$. The floor function selects the high block; the remainder selects the low block. Both operands must use the same $q$, including when their actual digit lengths differ.

For odd $n$, the low block has $m$ digit positions and the high block has $n-m=m-1$. Leading zero padding aligns a shorter operand without changing its integer value. Padding to a power-of-two length is optional: the split itself works for every $n$. If power-of-two padding is used, the padded length is less than twice the original length, so it changes only a constant factor in the asymptotic analysis.