Why the base changes the fraction
A fraction has a terminating expansion when its denominator can be built from the factors of the number base. In decimal, the base is 10, which factors into 2 and 5. That is why halves, fifths, and tenths end neatly, while thirds repeat forever: no number of tenths can make exactly one-third.
Base twelve has different building blocks: 2 and 3. A twelfth is small enough to divide into three equal groups, so one-third is four twelfths. Written in base twelve, that is 0.4₁₂. The subscript marks the base; without it, “0.4” is normally read as decimal, or four-tenths.
More common fractions fit
The same factor match helps with other fractions. In base twelve, one-half is 0.6₁₂, one-quarter is 0.3₁₂, and one-sixth is 0.2₁₂. Each ends after one place because its denominator divides twelve.
This does not make every fraction simpler. A fraction whose denominator includes a prime factor other than 2 or 3 will still repeat in base twelve. Decimal has the advantage for fractions involving fifths; base twelve has it for thirds and several related divisions.
The practical difference is where clean divisions fall. When dividing a quantity into thirds or quarters, a base-twelve notation can show the result without an endlessly repeating string of digits.
