totalsiffror

Totalsiffror is a term used in Swedish mathematics and computer science that can refer to two related notions involving digit counts. In one sense, totalsiffror denotes the number of digits in a single positive integer. In another sense, it denotes the total number of digits required to write all integers in a given range, typically from 1 up to some n.

For a positive integer n, the number of digits in its decimal representation is digits(n) = floor(log10(n)) + 1. Equivalently, this is the length of n when written in base-10 without leading zeros. For example, digits(7) = 1, digits(42) = 2, and digits(1234) = 4.

The range interpretation, often called the total number of digits from 1 to n, is denoted D(n) = sum_{k=1}^n digits(k). A commonly used closed-form expression for D(n) uses m = floor(log10(n)) + 1, the number of digits in n. Then D(n) = sum_{i=1}^{m-1} 9·10^{i-1}·i + (n − 10^{m-1} + 1)·m. This accounts for all full blocks of 1-digit numbers (1–9), 2-digit numbers (10–99), and so on, plus the partial block ending at n.

Examples help illustrate: D(13) = 13 (1–9 contribute 9 digits, 10–13 contribute 4 digits). D(99) = 189. D(999) = 2889.

Totalsiffror appears in problems about counting digits, numeral systems, and algorithm analysis where the cost is expressed in the number of digits processed. Related topics include digit sums, base-10 representation, and logarithmic growth.