# Roman numerals

A number system in which each symbol represents a fixed value regardless of its position; this differs from the place-value system of Arabic numerals. The earliest form of the Roman system was, however, decimal. In this primitive version a series of I's represented any number from 1 to 9, and a new symbol was introduced for each higher power of 10: X for 10, C for 100, and M for 1,000. The symbols V, L, and D, which stand for 5, 50, and 500, are thought to have been introduced by the Etruscans. For example, VIII is 8, XL is 40, MCD is 1400, and MCDXLVIII is 1448.

A common remark is that multiplication and division using Roman numerals is so awkward as to be totally impractical. However, an article by James G. Kennedy in The American Mathematical Monthly in 1980 gives algorithms for these operations that are actually more straightforward in the Roman system than in the Arabic. In multiplication the first step is to rewrite the numbers in a simple place-value notation. Seven columns are set up, headed by the symbols M, D, C, L, X, V, and I, and tallies are marked in each column corresponding to the number of times that symbol appears in the multicand. For example, if the multiplicand is XIII (13), one tally is marked in the X column and three tallies are marked in the I column. The multiplier is written in the same way. The multiplication itself is done by forming partial products according to two simple rules. In most cases the partial product given by any one tally in the multiplier is simply the set of tallies that represents the multiplicand, shifted to the left an appropriate number of columns. If the multiplier digit is I, the multiplicand is not shifted at all; the multiplicand is shifted one place to the left for V, two places for X, three places for L, and so on. The second rule is applied only when one Etruscan character is multiplied by another. In such cases the tallies representing the multiplicand digit are written twice in the appropriately shifted column and an additional tally is written one column to the right. Once a partial product has been formed for every tally in the multiplier, the tallies in each column are accumulated and replaced by the Roman symbol at the head of the column, giving the final answer. Only a slight change in the method is needed for Roman numerals in "subtractive notation," where 10 is written as IX, and so on. If all this sounds not quite so simple, the method for multiplying Arabic numbers is just as involved if they are written in explicit form. Furthermore, Arabic operations require a multiplication table giving the 100 products of all the possible pairs of Arabic digits. No comparable table is needed with Roman numerals, where all arithmetical operations can be defined in terms of shifting rules, addition, and subtraction.

## Related category

• TYPES OF NUMBER