digits1

digits1 is a mathematical function that returns the most significant digit of a positive integer when written in base 10. For any n ≥ 1, digits1(n) is the leading decimal digit; for example, digits1(374) = 3, digits1(5) = 5, and digits1(1000) = 1. The concept can be generalized to other bases, denoted digits1_b(n), which returns the leading digit of n in base b (b ≥ 2).

Computation can be done by iterative division by 10 until a single digit remains, or by using k = floor(log10(n)) and computing floor(n / 10^k). In practice, many programming languages implement direct digit extraction routines that emulate these methods.

Properties of digits1 include its invariance under multiplication by powers of 10: digits1(n * 10^m) = digits1(n). It is determined by the interval in which n falls, with the possible outputs being the digits 1 through 9. The function is piecewise constant, changing only when n crosses a power of 10.

Distribution and related concepts: if a dataset spans multiple orders of magnitude, the leading-digit distribution often follows Benford's law, with probabilities log10(1 + 1/d) for d = 1,...,9. While this is a statistical property of real-world data, the digits1 function itself is deterministic; its study is common in analyses of digit patterns and scale invariance.

Applications and variants: digits1 is used in data analysis to classify numbers by scale, in teaching about base representation, and in simple data validation checks. It also appears in algorithms that manipulate digits and in theoretical investigations of digit distribution.

See also: Benford's law; leading digit; base representation; digit extraction.