radix

Radix, in numeral systems, is the base of a positional notation. It defines the set of digits and the weight of each position. A base-b system uses b distinct digits, typically 0 through b-1. Examples: decimal is base-10 with digits 0–9; binary is base-2 with digits 0,1; hexadecimal is base-16 with digits 0–9 and A–F.

In a base-b numeral, a number with digits d_k … d_0 represents d_k*b^k + … + d_1*b + d_0. Each digit satisfies 0 ≤ d_i < b. The radix determines how many digits are needed to represent a given quantity; the number of digits is floor(log_b(N)) + 1 for N > 0.

Radix also has important applications in computer science. Radix sort is a non-comparative sorting algorithm that processes digits in each position using the base as the radix. Radix trees, or prefix trees, store strings by their character sequences, sharing common prefixes. Base representations influence arithmetic operations, data encoding, and base conversions, which typically use division and remainder or digit extraction by powers of the radix.

In linguistics, radix (root) refers to the lexical core of a word from which affixes are derived. The term originates from the Latin radix meaning root.

See also: positional notation, base conversion, numeral systems, radix sort, radix tree, root word.