 Carry normalization of a digit array

Process a digit-array integer representation from low to high positions. The integer division supplies $q=\lfloor a_i/b\rfloor$ and $0\le r<b$. Replacing the current digit by $r$ and adding $q$ to the next position preserves the represented value. Append zero storage when a positive carry leaves the allocated array. The processed prefix remains normalized and the value stays fixed.

This also works with signed temporary digits when the represented value is nonnegative. A negative carry cannot leave the final allocated position: the normalized lower digits represent a value less than that position's next radix power, which could not offset that negative carry. Positive carries eventually stop beyond the value's finite radix expansion. Thus the usual nonnegative-digit algorithm admits this useful extension without interpreting truncated division as floor division.