How did the Romans write fractions?

With their letter system for numerals, how did the Romans write fractions? Presumably not all their measurements came out as even numbers.

Illustration by Glen Mcbeth

Our evidence about how the Romans wrote numbers is fragmentary, and a general answer to the question of how they wrote fractions is impossible. The treatise entitled Distributio – written in AD 146 by the jurist L Volusius Maecianus for Marcus Aurelius – provides good evidence about fractions in the delimited but significant realm of money.

We know that the Romans tended to express fractions as parts of something specific: not a half or a third in abstract, but half of an as (a relatively small money denomination), a third of a certain volume unit, and so on.

Related to this is the fact that not all fractions were recognised: for instance, Maecianus tells us there is no “one-tenth of an as”, for instance, perhaps because it would not have corresponded to anything in the real world of Roman money. A part of something would have a specific name – for instance, a semis would be half an as, which in its turn was a sixteenth of a unit of currency called denarius. Names sometimes reflected the part’s mathematical relationship to another unit of measurement. Thus, semis is literally semi (half) an as.

Finally, Maecianus tells us, all named parts were denoted by a sign, be it a letter or another symbol. Thus, a semis was written as an S, while a bes (eight-twelfths of an as) as S=. It was this letter or symbol that would then be used in the numeral system, for calculations, to write out prices, and on abaci. Given that most calculations in antiquity were not carried out in writing, but on counting boards of various types, the system of notation did not matter as much.

Was the Roman system more cumbersome than ours? Perhaps. It certainly was, at least in this instance, more concrete – Maecianus thinks that Marcus Aurelius should know about the minutiae of currency denominations, as part of being a good ruler. “Don’t leave fractions to the accountants,” is possibly as good a piece of advice now as it was then.

Answered by: Dr Serafina Cuomo, author of Ancient Mathematics (Routledge, 2001).

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