Base19

Base19, or base-19, is a positional numeral system that uses 19 distinct digits and place-value notation. In this system, each digit represents a value from 0 to 18, with powers of 19 as the place values. The standard digit set is 0–9 for values 0–9 and A–I for values 10–18.

Digits and notation: A common convention is to use 0–9 for values 0–9 and letters A through I for values 10 through 18. For example, the base-19 number 1A3 consists of the digits 1, A (10), and 3, and represents 1×19^2 + 10×19 + 3 in decimal.

Arithmetic and operations: Base-19 follows the same general rules as other positional bases. Addition, subtraction, multiplication, and division use carries and borrows when sums or differences reach or exceed 19, and quotients and remainders are expressed in base-19. Procedures for arithmetic mirror those used in base-10, adapted to a 19-digit system.

Conversions: Converting between base-19 and decimal relies on standard methods. To decimal, multiply each digit by 19 raised to its position and sum. From decimal, repeatedly divide by 19; the remainders form the base-19 digits from least to most significant. For example, 1A3 in base-19 equals 554 in decimal, since 1×19^2 + 10×19 + 3 = 554. Conversely, 554 decimal equals 1A3 in base-19. As another check, 100 decimal equals 55 in base-19, because 5×19 + 5 = 100.

Applications and limitations: Base-19 is primarily of interest in theoretical discussions, mathematics education, and puzzle design. It is not widely used in practical computation or data encoding due to limited standardization and interoperability concerns. See also numeral systems and base conversion.