alternatebase

Alternatebase, in mathematics and computer science, refers to a numeral system that uses a mixed radix rather than a single fixed base. In such systems, the weight of each digit is determined by a sequence of bases that can alternate or vary, rather than all being the same base. A representation consists of digits di and corresponding bases ri, with 0 ≤ di < ri. The value is computed as the sum of di times the product of all preceding bases: value = d0 + d1·r0 + d2·(r0·r1) + d3·(r0·r1·r2) + ...

A common special case is an alternating-base system where the bases alternate between two values, for example r0 = 2, r1 = 3, r2 = 2, r3 = 3, and so on. In this case the place values form the sequence 1, 2, 6, 12, 36, ... and digits have ranges d0 ∈ {0,1}, d1 ∈ {0,1,2}, d2 ∈ {0,1}, d3 ∈ {0,1,2}, etc. As an illustration, the number 23 can be represented as [d0,d1,d2,d3] = [1,2,1,1], since 23 = 1·1 + 2·2 + 1·6 + 1·12.

Factorial number system and certain time or date representations are related ideas under the broader umbrella of mixed radix systems. Alternatebase representations are not unique in length or digits and can be converted to and from ordinary fixed-base representations using successive division by the current radix and accumulation of remainders. Applications include encoding schemes and arithmetic algorithms where varying place values can optimize hardware or algorithmic efficiency.