It is well-known that the place-value system introduced into Europe from India via the Arabs played an invaluable role in modern arithmetic.

But, if you read the "Sand Reckoner" by Archimedes, what he lays out are the rudiments of a place-value system. He essentially describes the modern notation, but not rules for addition, subtraction, multiplication and division using this notation.

Another tidbit: if you see the recent movie about Shannon "The Bit Keeper", he shows the journalist a mechanical device (he designed?) which can do calculations using Roman numerals.