Mixedbase

Mixedbase, or mixed-radix numeral system, is a positional numeral system in which each digit uses a different base. Instead of a single base for all positions, a sequence of bases b0, b1, b2, … is used. The i-th digit di ranges from 0 to bi−1, and the value is computed as d0 + d1*b0 + d2*(b0*b1) + d3*(b0*b1*b2) + … . This structure lets different positions contribute with different weights.

A simple example uses bases 60, 60, and 24 to model seconds, minutes, and hours. A triple (s, m, h) with 0 ≤ s < 60, 0 ≤ m < 60, 0 ≤ h < 24 represents the value s + 60m + 3600h. For instance, (34, 12, 2) encodes 2 hours, 12 minutes, and 34 seconds.

The factorial number system is a well-known special case of mixed base systems, with bases 2, 3, 4, 5, and so on. In this system, the i-th digit ranges from 0 to i, and the place values are factorials (i!). Mixed-base representations are useful in various computational and mathematical contexts because they align digit capacity with the natural size of different quantities or combinatorial structures.

Applications include time and calendar calculations, permutation indexing (Cantor expansion), and mixed-radix counters in hardware and software. They provide efficient encoding and conversion when different positions naturally have different granularities, offering a flexible alternative to uniform-base systems.