Grundziffern

Grundziffern are the basic symbols used to represent numbers in a positional numeral system. In a system with base b, the Grundziffern (digits) are the symbols 0, 1, ..., b−1, which together with their position determine the value of a number. The collection of these symbols is also called the digit set or alphabet of the numeral system.

In the common decimal system, which has base 10, the Grundziffern are the ten digits 0 through 9. A number is written as a string of these digits, and its value is obtained by summing each digit multiplied by the base raised to the power of its position: N = a_k b^k + a_{k-1} b^{k-1} + ... + a_0 b^0, where 0 ≤ a_i < b. For decimal numbers, b = 10.

Other numeral systems use different base and corresponding digit sets. Binary uses 0 and 1, octal uses 0–7, and hexadecimal uses 0–9 and letters A–F to extend the digit set beyond 9. In general, a base b system can use any finite set of symbols to represent the digits, with the standard constraint 0 ≤ digit < b. The exact symbols chosen are conventional, but the underlying rule—the digits together with place values—remains the same.

Grundziffern are distinct from the value they represent; a number can be written with or without leading zeros, which do not change its value. They are a fundamental concept in number representation, base conversion, and computer science, where different bases are routinely used for encoding and processing data.