Roman numerals illustrate how numerical notations work generally. First, we don’t need to know the Latin words for numbers to use the Roman numerals because numerical notations are semasiographic, communicating information without phonetic clues to the words involved. Second, we are likely to notice that Roman numerals are organized differently than Western numbers, and noticing such differences can help uncover the organizing principles. The Romans invented a pocket calculator, an abacus sized to fit into whatever kind of pocket might be construed for a toga. This pocket abacus could handle numbers into the millions, as well as fractions. Roman numerals and the associated abacus were designed to work together smoothly so that calculations could be carried out on the abacus and then recorded easily with the numerals. The Roman numerals and abacus were eventually replaced by numerals and algorithmic calculating methods imported from Indian and Arabic sources, a process that took centuries because of reluctance to give up the familiar devices. Roman numerals are still used more than fifteen centuries after the Roman empire collapsed because of their prestige value.