**ROMAN NUMERALS ARE EVERYWHERE AROUND US**

Football fans may have noticed that the 50th Super Bowl, played in February 2016, was branded with a logo that looked quite different from the Super Bowl logos that were used in prior years. While Super Bowls are usually numbered in roman numerals (a practice established at Super Bowl V in 1971), the National Football League broke with that tradition and went with arabic numerals for Super Bowl 50. In roman numerals, the letter L represents the number 50, so it would have been “Super Bowl L,” using roman numerals. The NFL explained that the primary reason for the change was the difficulty in designing an aesthetically pleasing logo for “Super Bowl L.”14 But why did they use roman numerals in the first place, more than 1500 years after the fall of the Roman Empire?

Although roman numerals survived the Roman Empire by almost a millennium and remained common in Europe throughout the Middle Ages, they were eventually almost completely replaced by the more convenient Hindu-Arabic numerals during the Renaissance in Europe. However, we all know that in certain contexts, roman numerals are still being used today. Hour marks on clock faces are very often labeled with roman numerals; in the credits of films, the year of production is frequently written in roman numerals; and roman numerals can be seen on the corner stones of public buildings, monuments, and gravestones. Monarchs are usually numbered in roman, for example, Henry VIII and Elizabeth II of the United Kingdom. Roman numerals are also used as generational suffixes for persons who share the same name within a family, such as Harry Phillips III. You will find roman numerals on the front page of the New York Times, indicating the volume number; and as lower-case letters they are sometimes used for the preliminary pages of books before the main page numbering begins. The most prominent use of roman numerals in the world of sports is for the modern Olympic Games. For instance, the 1996 Summer Olympics in Atlanta, Georgia, are officially known as the “Games of the XXVI Olympiad.” The numbering is useful both to distinguish between different events and also to know how many Olympic Games have already taken place (since the first modern Olympic Games in 1896). This was also the reason for numbering the Super Bowl events, and roman numerals were probably used to give them more of an “official character” and make the game seem more prestigious and dignified, comparable to the Olympic Games. However, unlike the Olympic Games, the Super Bowl is an annual event, and larger numbers tend to get longer and more complex to read when using roman numerals. So, in spite of the fact that the NFL returned to roman numerals for Super Bowl LI in 2017, this might not be the last word spoken on this issue.

Let's examine how roman numerals function. They are based on seven symbols, each of which represents a fixed value:

Numbers are formed by combining these symbols and adding the values. It might be interesting to note that the symbol for 5 is V, which is the top half of X—the symbol for 10. Symbols are placed from left to right in order of value, with the largest at the beginning. To convert a number into Roman numerals, we remove the largest roman values, write down the corresponding numerals, and continue with the remainder until the entire value is converted. As an example, consider 476, the year in which the regency of the last Roman emperor ended. The largest roman values we can begin with are four Cs, so we write down CCCC, representing 400, which leaves a remainder of 76. Now we can cover the 50 from the remaining 76 with L, to obtain CCCCL, representing 450. The remainder to be covered is 26, which can be written as two Xs (for 20), and VI (for 5 plus 1, or 6); so, 26 is written as XXVI. Thus, we arrive at CCCCLXXVI, which is 476 written in roman numerals! Conversion to an ordinary number is easy; we just have to add up the values of the letters, going from left to right.

The Romans did not have a place-value system, where the value of a numeral would depend on its position within the number. Since the numerals had definitive values that were simply added up, larger numbers generally needed more symbols. To shorten the notation, subtractive rules were introduced:

- I before V or X indicates one fewer; for example, IV (one less than five: 4) or IX (one less than ten: 9)

- X before L or C indicates ten fewer; for example, XL (ten less than fifty: 40) or XC (ten less than one hundred: 90)

- C before D or M indicates one hundred fewer; for example, CD (one hundred less than five hundred: 400) or CM (one hundred less than one thousand: 900)

Thus, the year 476 can also be written as CDLXXVI, which is unambiguous since “CD” must be subtractive notation, since the order of the letters is lower to higher when read from left to right. Although subtractive notation is the usual method of writing Roman numbers today, the ancient Romans seem to have preferred additive forms such as IIII (four) or VIIII (nine), especially in written documents. Subtractive notation was employed only when there wasn't enough space or when the numerals had to be engraved in stone. In fact, subtractive notation became popular in the Middle Ages, several hundred years after the fall of the Roman Empire. Already in the Roman era both notations lived in peaceful coexistence, and, sometimes, additive and subtractive forms have been used in the very same document. This inconsistency has survived to this day, on clock faces. Many clock faces using Roman numerals show IIII for four o'clock, but IX for nine o'clock (subtractive notation).

Adding roman numbers is particularly simple if the numbers are written in additive form. For example, if we want to add 37 and 24—that is, XXXVII and XXIIII—we just to have to put all numerals together, group them, and convert to larger-value numerals if possible. Thus, the sum would be XXXXXVIIIIII, which, upon combining five Xs to an L, five Is to a V, and two Vs to an X, yields LXI, which is 61. Subtraction is also not difficult; we only have to cross out common numerals. However, we may need to convert larger-value numerals to smaller-value numerals in order to cross out all numerals of the subtracted number. For example, to subtract 24 from 61—that is, LXI minus XXIIII—we would convert LXI to XXXXXVIIIIII and cross out the numerals forming the subtracted number (XXIIII), leaving us with XXXVII, which is indeed 37 (we basically apply the addition algorithm backward). The ancient Romans never developed a concept of zero or negative numbers, which might have something to do with this crossing-out procedure for subtraction. It is hard to make sense of crossing out something that is not there.

For very large numbers, the notation really got bulky. During the second century CE, the city of Rome already had more than one million inhabitants, making an additive number system consisting of only seven letters not a very convenient choice for a population census (one million would correspond to a sequence of thousand Ms). Therefore, the Romans developed several special notations to express very large numbers. We will just mention one of them, the apostrophus. The apostrophus notation is a system to represent multiplication by ten, starting with the numbers 500 and 1000. Roman numerals originally had their own symbols that were probably derived from tally marks. Gradually, these symbols were replaced by letters of the Roman alphabet that looked similar to them. The apostrophus notation evolved from the old symbol for 1000, which was written as ↀ (derived from the Greek Φ) or CIƆ. Adding a C on the left and the apostrophus, Ɔ, on the right, we obtain CCIƆƆ, which represents 10 times 1000, or 10,000. Adding another “pair of parentheses,” we get CCCIƆƆƆ, which is 10 times 10,000 or, 100,000, and so on. Splitting the symbol CIƆ into halves gives CI and IƆ, which turned into D for 500. Analogously, the symbols IƆƆ and IƆƆƆ would represent “halves,” that is 5,000 and 50,000, respectively. In not strictly mathematical contexts, the symbol CIƆ could also represent, symbolically, some very large number, too large to be counted. It is conjectured that the English mathematician John Wallis (1616–1703) introduced the infinity symbol, ∞, which he had derived from the roman symbol CIƆ.

As we said earlier, roman numerals are still everywhere around us; in almost every larger city, you will find them on building faces, cornerstones, or gravestones, especially in the older towns in Europe. Being able to “read” them can be both entertaining and useful. The fact that the roman numeral system doesn't use place values can also be an advantage. For young children, it is much easier to understand an additive numeral system than a place-value system. They will learn roman numerals very quickly and probably enjoy writing their age or their birthday in this style. Moreover, looking out for roman numbers on a museum tour or on a walk in the city and deciphering them will be fun and informative at the same time. Now that we have seen the development of representing numbers, we take that one step further to another counting aspect of our society, namely, the calendar.