Roman and "Arabic" Numerals

The use of Roman numerals has been mathematically obsolete for more than 1100 years. Nonetheless, the Roman symbols for numbers continue to be used in a variety of ways, most of them rather stereotyped: to mark the hours on clock faces, to number pages in the prefaces of books, to express copyright dates, and to count items in a series (such as the Super Bowls of U.S. professional football).

The form of Roman numeration used today was established during the Middle Ages in Western Europe. It is derived from the systems actually used in Roman times, but with certain improvements. The basic Roman numerals are:

I = 1 V = 5 X = 10 L = 50 C = 100 D = 500 M = 1000

 

The symbols are repeated to form larger numbers, and when different symbols are combined, the larger unit precedes the smaller. Thus VIII represents 8, CLXXX is 180, and MMDCCXXV is 2725.

The Romans usually wrote IIII for 4 and XXXX for 40. The number 949 was DCCCCXXXXVIIII. To shorten the length of such numbers a "subtraction rule" appeared in later Roman times and was commonly used in medieval times. The "subtraction rule" allows the use of six compound symbols in which a smaller unit precedes the larger:

IV = 4 IX = 9 XL = 40 XC = 90 CD = 400 CM = 900

 

Using these symbols, 949 is written more compactly as CMXLIX. (Other "subtracted" symbols are not allowed. Thus 99 must written XCIX, not IC.) The use of subtracted symbols was never mandatory, so IIII and IV can be used interchangeably for 4.

Actually, the symbols D (500) and M (1000) were originally written using a vertical stroke with surrounding arcs; these arcs can only be approximated on this page by using parentheses. D appeared as I ) and M as ( I ). This system allowed powers of ten larger than 1000 to be written by increasing the number of arcs: 10 000 was written (( I )) and 100 000 was written ((( I ))). The Romans had no word for 1 000 000 and rarely considered numbers of that size or larger. In late Roman and medieval times, after D and M were adopted as the symbols for 500 and 1000, a custom arose of writing a bar over a number to multiply that number by 1000. Thus 10 000 became X with a bar over it and 100 000 became C with a bar over it. These "overbarred" symbols are almost never seen today.

In Roman times, only the capital letters were used for number symbols. Later, after lower case letters came into use, Roman numbers were often written in lower case. Thus "vi" means 6 and "cxxii" means 122. Sometimes cases were even mixed, as in "Mcxl" for 1140. Furthermore, the lower case letter "j" was sometimes used in place of "i". A common custom was to write "j" for the last in a series of one's, as in "xiij" for 13.

Roman numerals continued in use in Europe after the fall of the Roman Empire, and they remained in general use for centuries after our modern number system became available. As we see, their use in certain applications continues even today.

The modern system of numeration is based on place value, with the same symbol, such as 4, taking on different meaning (4, 40, 400, etc.) depending on its location within the representation of the number. Place value numeration was invented by Hindu mathematicians in India, probably by the sixth century and perhaps even earlier. The modern numerals 1, 2, 3, ..., are sometimes called "Arabic" numerals in the West because they were introduced to Europeans by Arab scholars. The key figure was the great Arab mathematician Mohammed ibn-Musa al-Khowarizmi, who taught at Baghdad sometime between 800 and 850. He wrote a book on the Hindu number system known today only in a later Latin translation as De numero indorum, "On the Hindu numbers." Subsequently he wrote a longer and very

influential work, Al-jabr w'al muqabalah, known in Europe as Algebra, which included all the techniques of arithmetic still taught in schools today. The author's name, Latinized as "Algorismus," is the root of the English word "algorithm".

The Hindu-Arabic numeration system was known in Europe by 1000, but at first it didn't make much of a dent in the use of Roman numerals. During the 1100's the "Arabic" numerals were a topic of great interest among European scholars, and several translations of the Algebra appeared. In 1202, Leonardo of Pisa (ca. 1180-1250) published a famous book Liber abaci explaining and popularizing the Hindu-Arabic system, the use of the zero, the horizontal fraction bar, and the various algorithms of the Algebra. (Leonardo is better known today by his patronymic Fibonacci, "son of Bonaccio.") Thereafter modern numerals and the standard operations of arithmetic were commonly used by scholars, but Roman numerals continued to be used for many purposes, including finance and bookkeeping, for many centuries to come.

Incidentally, the numerals 0123456789 are properly known as European digits. The numerals actually used in Arabic script, the true Arabic numerals, are of different forms; see Islamicity.com for a more complete discussion.

ARABIC NUMERALS:

Photo: From top - Modern Arabic (western); Early Arabic (western); Arabic Letters (used as numerals); Modern Arabic (eastern); Early Arabic (eastern); Early Devanagari (Indian); Later Devanagari

The system of numeration employed throughout the greater part of the world today was probably developed in India, but because it was the Arabs who transmitted this system to the West the numerals it uses have come to be called Arabic.

After extending Islam throughout the Middle East, the Arabs began to assimilate the cultures of the peoples they had subdued. One of the great

centers of learning was Baghdad, where Arab, Greek, Persian, Jewish, and other scholars pooled their cultural heritages and where in 771 an Indian scholar appeared, bringing with him a treatise on astronomy using the Indian numerical system.

Until that time the Egyptian, Greek, and other cultures used their own numerals in a manner similar to that of the Romans. Thus the number 323 was expressed like this:

Egyptian 999 nn III

Greek HHH ÆÆ III

Roman CCC XX III

The Egyptians actually wrote them from right to left, but they are set down above from left to right to call g attention to the similarities of the systems.

The Indian contribution was to substitute a single sign (in this case meaning "3" and meaning "2") indicating the number of signs in each cluster of similar signs. In this manner the Indians would render Roman CCC XX 111 as: 3 2 3.

This new way of writing numbers was economical but not flawless. The Roman numeral CCC II, for instance, presented a problem. If a 3 and a 2 respectively were substituted for the Roman clusters CCC and II, the written result was 32. Clearly, the number intended was not thirty-two but three hundred and two. The Arab scholars perceived that a sign representing "nothing" or "nought" was required because the place of a sign gave as much information as its

unitary value did. The place had to be shown even if the sign which showed it indicated a unitary value of "nothing." It is uncertain whether the Arabs or the Indians filled this need by inventing the zero, but in any case the problem was solved: now the new system could show neatly the difference between XXX II (32) and CCC II (302).

If the origin of this new method was Indian, it is not at all certain that the original shapes of the Arabic numerals also were Indian. In fact, it seems quite possible that the Arab scholars used their own numerals but manipulated them in the Indian way. The Indian way had the advantage of using much smaller clusters of symbols and greatly simplifying written computations. The modern forms of the individual numbers in both eastern Arabic and western Arabic, or European, appear to have evolved from letters of the Arabic alphabet.

The Semites and Greeks traditionally assigned numerical values to their letters and used them as numerals. This alphabetical system is still used by the Arabs, much as Roman numerals are used in the West for outlines and in enumerating kings, emperors, and popes. The new mathematical principle on which the Arabic numerals were based greatly simplified arithmetic. Their adoption in Europe began in the tenth century after an Arabic mathematical treatise was translated by a scholar in Spain and spread throughout the West.

Acknowledgements:These pages were incorporated from "ARAMCO and Its World: Arabia And The Middle East", Edited by Ismail I.

Nawwab, Peter C. Speers & Paul F. Hoye, Islam and Islamic History Section, published in 1980 by Arabian American Oil Company, Washington D.C.