The Romans formed their numerals* from seven symbols: 1=1, V = 5, X = 10, L = 50, C = 100, D = 500, and M (and alternate forms ∞ and 1) = 1,000. All numerals were made by adding these symbols together or by subtracting one from another. When two symbols were next to each other, their values were added if the smaller numeral was on the right and subtracted if the smaller numeral was on the left. For example, the Romans made the numeral 4 by adding basic signs together (IIII = 4) or by subtracting the value of the numeral on the left from that on the right (IV = 4).
The Romans also had a system for expressing fractions. One horizontal stroke (—) represented 1/12. The fraction 3/12 was written “ = —.” The letter S (for the Latin word semis) was used to represent 1/2. It could stand alone or precede other numbers to form new fractions, such as the fraction 7/12, which was written as “S —” (1/2 + 1/12).
* numeral symbol used to represent a number
The basic Roman numbering system worked well for small numbers, but it proved awkward for larger numbers. The inventive Romans solved this problem. They modified the basic signs to indicate numbers above 1,000. A horizontal bar above a numeral indicated that the number was to be multiplied by 1,000—for example, X = 10,000. Bars above and on both sides of the symbol—such as |X|—meant that the value was to be multiplied by 100,000. A D that enclosed a small semicircle (D) stood for 5,000. Beginning in the 100s B.C., the Romans used these symbols to represent words that referred to numbers. For example, X stood for the Roman coin called a denarius, which was worth ten of the smallest Roman coins.
Like the Greeks, the Romans used the abacus for counting. The abacus is a device consisting of a frame and columns of beads, which represent numbers. The beads are manipulated along wire or wooden rods to perform calculations. (See also Alphabets and Writing; Calendars; Mathematics, Greek.)