digitsets

A digit set, in the context of positional numeral systems, is the finite set of symbols used for the digits when representing numbers in a fixed base b. A representation of a number is formed as a finite sum of digits times powers of b, written as sum d_i b^i with d_i in D.

In standard decimal or binary systems, D is a complete residue system modulo b, typically D = {0,1,...,b−1}. Other digit sets are used to influence arithmetic properties; for example, balanced or signed-digit sets such as {−1,0,1} in base 2, or {−a, ..., a} for suitable a, provide symmetry around zero and can reduce carry propagation during arithmetic.

Canonical number systems study bases b together with digit sets D of size b so that every nonnegative integer has a unique representation, sum d_i b^i with d_i in D. Redundant digit sets, where representations are not unique, allow multiple encodings and can enable carry-free addition or faster arithmetic, as in signed-digit or non-adjacent form representations.

Applications include computer arithmetic and digital hardware design, where digit sets are chosen to minimize carries and improve speed; cryptographic algorithms may use signed-digit representations to speed modular operations; and theoretical work connects digit sets with fractal tilings and dynamical systems.

Beyond integers, digit sets extend to nonstandard bases, such as complex or non-integer bases, where D is a finite set of representatives and every number has an expansion in base b with digits from D. Examples include complex-base systems with small digit sets that yield stable representations for complex numbers.