Overview
The duodecimal system, commonly called dozenal or base‑12, is a positional numeral system that uses twelve distinct values per place instead of ten. In a base‑12 system each digit position represents a power of 12: units (12^0), twelves (12^1), one hundred forty‑fours (12^2), and so on. The system is purely an alternative way to represent integers and fractions; like any positional system it can express very large and very small numbers compactly. For a concise introductory resource see base‑12 overview.
Digits and notation
Because humans commonly use ten digit symbols (0–9), duodecimal requires two additional symbols to represent the values ten and eleven. A widely used convention is to write 10 for decimal twelve, then use letters such as A (ten) and B (eleven), so the sequence of single‑digit symbols becomes 0,1,2,3,4,5,6,7,8,9,A,B. Under that notation the decimal number 50 is written 42 in base‑12 because 4×12 + 2 = 50. Likewise 100 in base‑12 equals 144 in decimal because it stands for 1×12^2. For further details about place value and notation see notation and place value.
Divisibility and fractions
A practical appeal of base‑12 is the number 12's many integer divisors: 12 is divisible by 2, 3, 4, and 6 (its full set of positive divisors is 1,2,3,4,6,12). That property means several common rational fractions have finite representations in dozenal. For example, one half is written 0.6 (since 6/12 = 1/2) and one third is 0.4 (since 4/12 = 1/3). One quarter is 0.3 and one sixth is 0.2. By contrast, fractions whose denominators contain prime factors other than 2 or 3 (for example 5) produce repeating expansions in base‑12, so neither system makes all fractions terminate. A compact discussion with examples of terminating and repeating expansions can be found at fraction behavior.
History and cultural traces
Duodecimal counting and grouping appear in a number of traditional measures and linguistic remnants rather than as universally adopted positional systems. Common survivals include the use of dozens and grosses (12 and 144), the division of the day into two 12‑hour cycles, and the subdivision of a foot into 12 inches in some measurement systems. Historical evidence suggests various cultures used mixed bases or grouped counts in twelves for trade and timekeeping; however, large‑scale adoption of a pure base‑12 positional system has been limited. For summaries of historical uses and cultural traces see historical notes.
Advantages, drawbacks and modern interest
Advocates of dozenal arithmetic point to easier division by 2, 3 and 4 and to simpler finite representations of many everyday fractions. Opponents note that the global infrastructure of decimal accounting, measurement, education and technology makes a wholesale transition costly and impractical. In practice, many cultures combine bases (for example, grouping by twelves within a predominantly decimal society) so that some benefits of base‑12 survive without replacing decimal conventions entirely. Contemporary interest in duodecimal is mainly academic and hobbyist: enthusiasts examine notation, arithmetic pedagogy, and the comparative advantages of alternative bases.
How to convert and where it appears today
Conversion between decimal and duodecimal uses the same division and remainder methods used for other bases: repeatedly divide the decimal number by 12 and record remainders to build the base‑12 digits, or evaluate each digit by multiplying by powers of 12 to return to decimal. In everyday life dozenal thinking is still visible in expressions like "a dozen" or pricing by the dozen, in the use of gross in commerce history, and in subdivisions of time and length. Readers seeking practical conversion tools, symbol conventions and exercises will find introductory tutorials and calculators at the resources linked above.