ndigit

Ndigit is a mathematical function that returns the number of digits required to represent a nonnegative integer n in a given base b (the radix). The base-10 case is the usual decimal digit count, but the concept generalizes to any base b ≥ 2. Formally, for n ≥ 0, ndigit_b(n) = floor(log_b(n)) + 1, with the convention ndigit_b(0) = 1. Equivalently, ndigit_b(n) = floor( ln(n) / ln(b) ) + 1 for n > 0.

Examples: ndigit_10(0) = 1, ndigit_10(7) = 1, ndigit_10(42) = 2. In base 2, ndigit_2(5) = 3 because 5 is 101 in binary.

Generalization and conventions: negative integers can be handled by applying the function to their absolute value, i.e., ndigit_b(n) = ndigit_b(|n|). Leading zeros do not affect the count, and the value increases by one when n reaches a new power of b (for example, in decimal, at 10, 100, 1000, and so on).

Computational aspects: in programming, ndigit_b(n) can be computed with logarithms or by repeated division by b. The logarithmic method runs in constant time for fixed-precision arithmetic but relies on floating-point operations; the division method runs in O(log_b n) time and uses only integer arithmetic.

Applications: digit counting is used in numeric formatting, data encoding, indexing, and algorithms that depend on the magnitude of numbers. It also provides a basis for understanding numeral systems and base conversions.

See also: number of digits, digit count, logarithm, radix.