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# Roman Numerals: Their Origins, Impact, and Limitations

## Overview

The numeral system developed by the Romans was used by most Europeans for nearly 1800 years, far longer than the current Hindu-Arabic system has been in existence. Although the Roman numeral system provided for easy addition and subtraction, other arithmetic operations proved more difficult. Combined with the lack of an effective system for utilizing fractions and the absence of the concept of zero, the cumbersome nature of the Roman numeral system, while it served most of the needs of the Romans, hindered future mathematical advances.

## Background

The Roman numeral system for representing numbers was developed around 500 b.c. As the Romans conquered much of the world that was known to them, their numeral system spread throughout Europe, where Roman numerals remained the primary manner for representing numbers for centuries. Around a.d. 1300, Roman numerals were replaced throughout most of Europe with the more effective Hindu-Arabic system still used today.

Before examining the limitations posed by the use of Roman numerals, it is necessary to understand how Roman numerals are utilized. A numeral is any symbol used to represent a number. In the Hindu-Arabic numeral system, the numeral 3 represents the number three. When the numeral 3 is held in place by one or more zeros, the value increases by an order of magnitude, e.g., 30, 300, 3000, and so on. In the Roman numeral system, numerals are represented by various letters. The basic numerals used by the Romans are: I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1000. These numerals can be strung together, in which case they would be added together in order to represent larger numbers. For example, the number 72 would be represented as LXXII (L + X + X + I + I, or 50 + 10 + 10 + 1 + 1 in Arabic numbers).

In order to prevent numbers from becoming too long and cumbersome, the Romans also allowed for subtraction when a smaller numeral precedes a larger numeral. Therefore, the number 14 would be represented as XIV instead of XIIII. Under this system, a numeral can only precede another numeral that is equal to ten times the value of the smaller numeral or less. For example, I can only precede and, thus, be subtracted from V and X, which are equal to five and ten times the value of I, respectively. Under this rule, the number 1999 cannot be represented as MIM, because M is equal to one thousand times the value of I. The Roman representation of 1999 is MCMXCIX, or M (1000) + CM (1000-100) + XC (100-10) + IX (10-1). Most of these rules, while often used by the Romans, were not standardized until the Middle Ages. Thus, one might find 9 represented on some older documents as VIIII instead of IX.

Because the largest numeral used by the Romans was M, or 1000, it proved impractical to write extremely large numbers, such as 1,000,000, as a string of 1000 Ms. To avoid this problem the Romans wrote a bar, called a vinculum, over numerals to express that numeral as a number 1000 times its original value. Instead of writing 6000 as MMMMMM, 6000 could simply be written as V̄Ī and 1,000,000 as M̄. Using this form of notation, the Romans were capable of writing large numbers.

## Impact

The Romans adopted the symbols that they used for their numerals from a variety of sources, including their Greek counterparts. The origin of I to represent one is straightforward, derived from counting on one's hand, where one finger, which resembles I, equals one of whatever was being counted. The V came to represent five because when five items are counted on the hand, a V is formed by the space between the thumb and first finger.

Originally the Romans adopted the Greek letter X, or chi, to represent 50. Through the study of monument transcriptions historians have been able to determine that L replaced X as 50, and X came to represent 10. How X came to represent 10 is not entirely clear. One theory suggests that X was derived from one V, or five, placed on top of another, upside-down V. Thus, the two Vs formed an X. Another theory suggests that when counting to 10, Romans did so by making ten vertical marks and then crossing them out with an X in order to easily count groups of ten. This is similar to the manner in which Americans keep tallies by groups of five in which four vertical marks are crossed through with a fifth diagonal mark. Eventually the Romans adopted just X to be the numeral for 10. The symbol C came to represent 100, because it is the first letter of the Latin word for one hundred, centum. Likewise, M was adopted for 1000, because the Latin word for one thousand is mille.

Unlike the Greeks, the Romans were not concerned with pure mathematics, such as number theory, geometric proofs, and other abstract ideas. Instead, the Romans preferred utilitarian mathematics. The Romans primarily used mathematics to figure personal and government accounts, keep military records, and aid in the construction of aqueducts and buildings. The Roman numeral system allowed for simple addition and subtraction. For addition, Romans simply lined up all of the numerals from the numbers being added, and simplified. For example, in order to solve the problem 7 + 22, or VII + XXII, the numerals were first arranged in de scending order, or XXVIIII. Because VIIII, or 9, is not in acceptable form, this was changed to IX, the generally recognized manner of writing 9. The correct answer remains, XXIX, or 29. Subtraction can be done in a similar manner by crossing out similar numerals from the two different numbers.

The fact that multiplication and division were fairly difficult operations for the Romans spurred development of counting boards to aid with these operations. The counting boards, which resembled the familiar abacus, could also be used for addition and subtraction. Counting boards based on the Roman design were used throughout Europe until the Middle Ages. Even with these counting boards, multiplication and division of large numbers remained a difficult task. Therefore, Romans developed and often consulted multiplication and division tables to solve problems involving large numbers.

In addition to difficulty with the multiplication and division of numbers, several other problems severely limited the use and effectiveness of Roman numerals. One flaw of the Roman numeral system was the absence of a way to numerically express fractions. Romans were aware of fractions, but putting them to use was difficult, as they were expressed in written form. The Romans would have written three-eighths as tres octavae. The Romans usually expressed fractions in terms of the uncia. An uncia originally meant 1/12 of the Roman measure of weight (English derived the word "ounce" from uncia). Soon, however, uncia evolved to mean 1/12 of anything. Although basing the use of fractions on 1/12s, the Romans were able to express one-sixth, one-fourth, one-third, and half. While the modern numerical expression of one-fourth is ¼, the Romans would have expressed one-fourth as three unciae (3/12 = ¼). This system allowed the Romans to approximate measures, but they could not easily express exact measures.

Another flaw that limited Roman mathematics was the absence of the concept of zero. As with the previous number systems of the Sumerians, Babylonians, and Egyptians, the Romans did not have a place-value system that included the concept of zero as a placeholder for numerals. This forced the Romans to adopt the cumbersome system with numerals that represented 1, 5, 10, 50, 100, 500, and 1000, as described above. Unlike the ancient Greeks, the Romans also did not understand or explore the concept of irrational numbers. This severely limited the Romans in geometry, because much of geometry rests on an understanding of π, the ratio of the circumference of a circle to its diameter.

Although not limiting from a practical engineering standpoint, these flaws in the Roman mathematical systems limited the advancement of mathematical theory in Rome. In the wake of Roman conquests, most of Europe adopted the Roman numeral system and used it throughout the Middle Ages. Accordingly, theoretical mathematical advances were likewise also stunted throughout most of Western civilization for nearly 1,000 years. The absence of zero and irrational numbers, impractical and inaccurate fractions, and difficulties with multiplication and division prevented the Romans and the Europeans who later used the system from making advances in number theory and geometry as the Greeks had done in the Pythagorean and Euclidean schools.

During these mathematical Dark Ages, advancements in these fields were made by Middle Eastern and Indian subcontinent civilizations. With the innovation of zero place use within the Hindu-Arabic place-value system, great advances were made in these regions in the fields of geometry, number theory, and the invention and advancement of algebra.

Regardless of the Roman numeral system's limitations, the existing archaeological record establishes that the Romans were able to overcome many of those limitations with regard to the practicalities of construction. Roman roads and aqueducts remain as a testament to the engineering feats that the Romans were able to accomplish with their flawed system. Although Roman numerals are no longer a necessary component of mathematics, they are an important part of the history of the development of Western civilization. Modern numerals remain aesthetically important because of their widespread artistic use in art, architecture, and printing.

JOSEPH P. HYDER