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Fractions

The base "Roman fraction" is S, indicating 1 ⁄ 2. The use of S (as in VIIS to indicate 7 1 ⁄ 2) is attested in some ancient inscriptions and also in the now rare apothecaries' system (usually in the form SS): but while Roman numerals for whole numbers are essentially decimal, S does not correspond to 5 ⁄ 10, as one might expect, but 6 ⁄ 12.

The Romans used a duodecimal rather than a decimal system for fractions, as the divisibility of twelve (12 = 2 × 3) makes it easier to handle the common fractions of 1 ⁄ 3 and 1 ⁄ 4 than does a system based on ten (10 = 2 × 5). Notation for fractions other than 1 ⁄ 2 is mainly found on surviving Roman coins, many of which had values that were duodecimal fractions of the unit as. Fractions less than 1 ⁄ 2 are indicated by a dot (·) for each uncia "twelfth", the source of the English words inch and ounce; dots are repeated for fractions up to five twelfths. Six twelfths (one half), is S for semis "half". Uncia dots were added to S for fractions from seven to eleven twelfths, just as tallies were added to V for whole numbers from six to nine. The arrangement of the dots was variable and not necessarily linear. Five dots arranged like (⁙) (as on the face of a die) are known as a quincunx, from the name of the Roman fraction/coin. The Latin words sextans and quadrans are the source of the English words sextant and quadrant.

Each fraction from 1 ⁄ 12 to 12 ⁄ 12 had a name in Roman times; these corresponded to the names of the related coins:

Other Roman fractional notations included the following: