Digit-array integer representation ($\mathrm{val}(a)={\sum }_{i=0}^{L-1}{a}_i{b}^i$)

Store an integer in a nonempty array of numeral digits, with the least significant digit first and a fixed radix $b\ge 2$. Normalized digits satisfy $0\le a_i<b$. High-order zeros do not change the value. Adding digit arrays coefficientwise or multiplying them by discrete convolution gives the correct value before normalization, because these operations add or multiply the corresponding powers of the radix. Exact temporary arithmetic, or a radix and accumulator size chosen to prevent integer overflow, is required.

When the radix is $10^k$, print the highest nonzero normalized digit without leading zeros, then every lower digit as exactly $k$ decimal characters. Internal zero blocks must be retained. With base $1000$, the low-first digits $[4,0,12]$ print as 12 followed by 000 followed by 004, giving 12000004. An all-zero representation prints a single zero, independent of its allocated length.

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.