﻿ Units: Roman and "Arabic" Numerals

## 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).

Roman Numeration
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 as used today 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" was sometimes used in 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 be 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.

Hindu-Arabic Numeration
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 notation was used long ago in Babylonian cuneiform numerals, but our modern decimal place value system 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 merchants. The key figure was the great Persian mathematician Muhammed ibn-Musa al-Khwarizmi, 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 as Fibonacci, "son of the Bonacci," a nickname not applied to him until the 19th century). 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 more properly known as European digits. The numerals actually used in Arabic script, the true Arabic numerals, are of different forms; see Islamicity.org for a more complete discussion.

Reference