Basewelves

Basewelves is a numerical system that uses a base-12 (duodecimal) counting system as its foundation, combined with additional symbols to represent fractions and negative numbers. It was introduced by mathematician and computer scientist Donald Knuth in the 1970s as part of his *Literate Programming* methodology, specifically in the *Web* system, which he later renamed *Literate Programming*. The term "basewelves" is a playful contraction of "base twelve" and "elves," referencing the fictional elves in J.R.R. Tolkien’s *Lord of the Rings*, who were said to use a base-20 system in some interpretations.

In basewelves, the digits 0 through 9 are used as in the decimal system, while the digits A and B represent the values 10 and 11, respectively. This allows for a more efficient representation of numbers, as 12 is a highly composite number, making it easier to divide and multiply without remainders. The system also incorporates a fractional component, where digits are separated by a semicolon (;) to denote the radix point, similar to the decimal system’s use of a decimal point. For example, the number 1A;B in basewelves represents 1×12² + 10×12¹ + 0×12⁰ + 11×12⁻¹, which equals 144 + 120 + 0.9167 (approximately).

Basewelves also includes a mechanism for representing negative numbers using a prefix of a tilde (~) or a minus sign (-). For instance, ~1A;B would denote the negative of the value represented by 1A;B. The system was designed to simplify arithmetic operations and improve readability in programming contexts, particularly for those working with large or complex calculations.

While basewelves is not widely adopted in mainstream computing, it remains a theoretical and educational tool, demonstrating how alternative numeral systems can enhance clarity and efficiency in mathematical and computational tasks. Its simplicity and elegance continue to be studied in the fields of computer science and mathematics.