Introduction^[1]

Roman numerals—whether you like them or not—represent one of the most enduring legacies of ancient Rome. The numbering system originated over two thousand years ago and persisted as the primary numeric script across Europe well into the Late Middle Ages. As I explain later, unlike modern numeral systems that rely on positional notation, Roman numerals are a non-positional, additive-subtractive system based on letters from the Latin alphabet. This numbering system, with its distinctive symbols—I, V, X, L, C, D, and M—is still recognised and used in some contexts even today.

The Roman Empire may have fallen many centuries ago, but the numeral system it developed has left a lasting cultural imprint. Roman numerals can still be found in architecture, clock faces, book chapters, and the names of monarchs and popes, underscoring their symbolic power and elegance. The continued use of Roman numerals is a testament not only to their historical significance but also to their adaptability and appeal.

This paper examines the origins, evolution, and applications of Roman numerals. It begins with a discussion of their roots in the Etruscan numeral system and explores how the Romans adapted these symbols to fit their language and administrative needs. The paper then examines the structure and functionality of Roman numerals, including their unique use of additive and subtractive notation, the lack of place value, and the absence of a zero. This section highlights both the practical benefits and limitations of the system, particularly in contrast with the Arabic numeral system that eventually replaced it.

Subsequent sections will explore the diverse applications of Roman numerals in ancient Rome, from commerce and governance to the military and public inscriptions. Through historical examples, we will see how Roman numerals found their way into nearly every aspect of Roman life and culture, making them a vital tool for the administration of the vast Roman Empire. The paper will also address regional variants and the flexibility of notation across the Empire, as well as the gradual transition to Arabic numerals in medieval Europe. The paper concludes with a look at the modern legacy of Roman numerals, examining their use in ceremonial, decorative, and institutional contexts.

Picture: Roman numerals inscribed over Admiralty Arch, London: “In the tenth year of King Edward VII / to Queen Victoria (from) most grateful citizens, 1910”
Citation: Admiralty Arch. (2024, June 29). In Wikipedia. https://en.wikipedia.org/wiki/Admiralty_Arch

First Things First

This paper aims to provide a comprehensive understanding of Roman numerals by tracing their history, structure, uses, and cultural significance. Roman numerals are often viewed as an antiquated system, yet their persistence and adaptability offer insights into how symbols acquire meaning beyond their practical applications. Frequent questions asked about Roman numerals include:

• What are their origins?
• How did they develop from earlier systems?
• How were they structured, and what practical advantages or limitations did this structure present?
• In what ways were they used across different areas of Roman life, and what does this reveal about Roman society?
• Why did they persist for so long, even after more efficient systems became available?
• How and why have they continued to be used in modern times?

The structure of this paper follows a chronological and thematic approach. Each section will delve into a specific aspect of Roman numerals, starting with their origins and moving through their practical applications, variations, decline, and modern legacy.

References
The bibliological sources quoted in the following text come from:

• Barnett, Jo Ellen. Time’s Pendulum: From Sundials to Atomic Clocks, the Fascinating History of Timekeeping and How Our Discoveries Changed the World. New York: Harcourt, 1998.
• British Horological Institute. “Records of the Wells Cathedral Clock.” Horological Journal 16, no. 2 (1974): 34–35.
• Burton, David M. The History of Mathematics: An Introduction. 7^th ed. New York: McGraw-Hill, 2011.
• Medieval Notations and Records: A Comparative Study. Cambridge University Press, 1967.
• Chisholm, James. Roman Inscriptions in the Western Empire. Oxford University Press, 1929.
• Cipolla, Carlo M. Clocks and Culture, 1300–1700. W. W. Norton & Company, 2003.
• D’Ooge, Benjamin L. Latin for Beginners. Ginn & Company, 1921.
• Encyclopaedia Britannica. “Roman Numerals.” Accessed October 30, 2023, https://www.britannica.com/topic/Roman-numerals
• Haynes, Sybille. Etruscan Civilization: A Cultural History. British Museum Press, 2000.
• Ifrah, Georges. The Universal History of Numbers: From Prehistory to the Invention of the Computer. Translated by David Bellos. Wiley, 1998.
• MacDonald, William. The Architecture of the Roman Empire, Volume II: An Urban Appraisal. New Haven: Yale University Press, 1982.
• Menninger, Karl. Number Systems and Their Development in History. New York: Dover Publications, 1992.
• Oxford English Dictionary. “Roman Numerals.” Accessed October 30, 2018.
• Rooney, Anne. The Story of Mathematics. Arcturus Publishing, 2009.
• Seife, Charles. Zero: The Biography of a Dangerous Idea. Viking, 2000.
• UNESCO Encyclopedia of Mathematics. “Roman Numerals.” Last modified 2020. https://www.unesco.org/encyclopedia/roman-numerals

Other sources are listed at the end of this paper.

Quick Introduction to Roman Numerals

• Basic Symbols and Their Values

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

• Additive and Subtractive Principles
  • The additive principle is straightforward: numbers are formed by adding the values of symbols. For example, VII is 7 (5 + 1 + 1).
  • The subtractive principle avoids four characters being repeated in succession. For instance, 4 is IV (5 – 1), not IIII (but see below). Similarly, 9 is IX (10 – 1).
• Large Numbers
  • Romans had a system to represent larger numbers by placing a line (vinculum) over a numeral, indicating it should be multiplied by 1,000. For example, V̅ represents 5,000.
• Non-Standard Form
  • Inscriptions and manuscripts from different periods show variations. For example, instead of IV for 4, IIII was also used, especially on clock faces.
• Modern Uses
  • Roman numerals are seen on clock faces, often using IIII for 4 instead of IV.
  • Monarchs and Popes use Roman numerals to denote order (e.g., Louis XIV of France).
  • Sporting events like the Super Bowl use Roman numerals for numbering (e.g., Super Bowl LII).
  • Annual events and editions of various competitions often use Roman numerals to indicate sequence.
• Influence on Later Numeral Systems
  • Roman numerals influenced the numeral systems of later European cultures. Although they were eventually replaced by the more efficient Hindu-Arabic numeral system, their legacy is evident in various modern applications.
• Educational and Cultural Significance
  • Roman numerals are still taught in schools as part of historical education.
  • They are used in cultural contexts, such as the naming of sequels in film and literature.
• Repeated Symbols
  • Symbols can be repeated up to three times to increase the value. For example, III = 3 or XXX = 30.
  • The symbols V, L, and D are never repeated.
• Subtractive Combinations
  • The subtractive combination is used to avoid four consecutive characters. For instance:

IV = 4 (5 – 1)    IX = 9 (10 – 1)    XL = 40 (50 – 10)    XC = 90 (100 – 10)    CD = 400 (500 – 100)    CM = 900 (1000 – 100)

• Additive Combinations
  • Numbers are generally written in descending order from left to right, adding the values. For example: VI = 6 (5 + 1) or XIII = 13 (10 + 1 + 1 + 1)
• Writing Large Numbers
  • To represent large numbers, a bar over a numeral indicates multiplication by 1,000. For example: V̅ = 5,000

Methodology and Sources
In compiling this paper, I have researched and drawn on a combination of primary sources and secondary analyses to provide a nuanced understanding of Roman numerals. Primary sources include ancient inscriptions, coins, and monumental inscriptions from across the Roman Empire, which provide direct evidence of how Roman numerals were used in context. Notable examples include the Fasti Antiates Maiores^[2], a pre-Julian Roman calendar that illustrates the early use of Roman numerals, and the inscriptions on the Arch of Titus^[3] and the Colosseum^[4]. Secondary sources include historical analyses, mathematical studies, and scholarly works that examine Roman numerals from various angles. Key references include Etruscan Civilization: A Cultural History by Sybille Haynes^[5], which provides insights into the Etruscan influence on Roman numerals, and The Universal History of Numbers by Georges Ifrah^[6], which explores the development and spread of numeral systems across civilisations. By bringing together these sources, I hope it provides a well-rounded view of Roman numerals and their significance.

Origins and Early Development of Roman Numerals

The Influence of Etruscan Numerals
The origins of Roman numerals are closely tied to the Etruscan civilisation, which inhabited north-central Italy before the rise of Rome. The Etruscans^[7], known for their advanced culture and engineering, had their own numbering system that likely inspired early Roman numerals. Archaeological evidence, such as Etruscan tomb inscriptions and artefacts, shows that the Etruscans used a system of symbols similar to the Romans – vertical strokes represented units, while V-shaped symbols likely represented groups of five.

The Etruscan numeral system is significant because it introduced the use of non-decimal groupings, such as symbols for five, ten, and fifty. Unlike the decimal-based systems of the Greeks and Egyptians, the Etruscan system allowed for easy addition of values without requiring a place-value structure. The Romans adapted these symbols, modifying and simplifying them to create their own unique system. This adaptation process illustrates the Romans’ practical approach to innovation, as they frequently borrowed and improved upon ideas from neighbouring cultures (Haynes 2000).

Visual and Practical Symbolism
Roman numerals have a visual simplicity that made them easy to carve into stone and metal, as well as to write on papyrus and other materials. Symbols like I, V, and X were easy to create and recognise, which would have been particularly useful in a society that relied on written inscriptions for official communication. The symbol V may have been chosen to represent five because of its resemblance to a human hand with five fingers, while X could symbolise two V’s joined together to represent ten (Rooney 2009). These visual choices not only made Roman numerals intuitive but also connected them to the physical world in a way that was accessible to both literate and illiterate people in Roman society.

Integration with the Roman Alphabet
One of the most distinctive aspects of Roman numerals is their integration with the Latin alphabet. Unlike other ancient systems that used unique symbols or cyphers, Roman numerals repurposed letters to function as numbers. This integration may have helped reinforce the authority of the Latin language across the Empire, as numbers were presented in the same alphabet used for laws, official edicts, and literature.

This connection between numerals and language likely added to the symbolic power of Roman numerals. Since letters had multiple uses, they were often imbued with multiple meanings, giving the numerals a level of cultural resonance beyond their mathematical utility. The letter C, for example, represented 100, but also started the Latin word centum, meaning “hundred.” Similarly, M represented 1,000, aligning with mille, the Latin word for “thousand.” This dual function of letters and numerals may have reinforced their mnemonic value, making the system more intuitive to Roman users (D’Ooge 1921).

Earliest Archaeological Evidence
As previously mentioned, the oldest evidence of Roman numerals can be found on artefacts like the Fasti Antiates Maiores, the ancient calendar that dates back to around 60 BC. This calendar uses Roman numerals to denote days and months, showcasing an early form of the numeral system that would become standardised over the next few centuries. Other early examples include the Arch of Titus and inscriptions on Roman coins, which provide a glimpse into how Roman numerals were used in public and administrative settings (MacDonald 1982).

These early inscriptions reveal that Roman numerals were not merely a mathematical tool but a central feature of Roman public life. The use of Roman numerals on monuments and coins helped establish a shared system of value and timekeeping across the Empire, linking different regions under a common language of numbers.

Structure and Functionality of Roman Numerals

Roman numerals are unique in their structure, blending additive and subtractive notation in a non-positional system. Unlike our modern Arabic numerals, which use place value and a zero to represent large numbers efficiently, Roman numerals represent values through a straightforward combination of symbols. This section will explore how Roman numerals are built, discuss the practical implications of their additive and subtractive rules, and examine the impact of the absence of zero and place value.

Additive and Subtractive Notation
The Roman numeral system primarily operates through additive notation. Symbols are placed in descending order, with each letter representing a fixed value: I (1), V (5), X (10), L (50), C (100), D (500), and M (1,000). In this format, numbers are built by combining symbols to reach the desired total. For example, the number 7 is written as VII (5 + 1 + 1), and 30 is written as XXX (10 + 10 + 10). This approach makes Roman numerals relatively intuitive, as each symbol’s contribution is immediately evident.

However, to avoid lengthy or cumbersome representations, the Roman numeral system incorporates subtractive notation for certain numbers. When a smaller numeral precedes a larger one, it is subtracted from the larger value. This practice is most commonly seen with 4 (IV, “one less than five”) and 9 (IX, “one less than ten”), as well as for 40 (XL), 90 (XC), 400 (CD), and 900 (CM).

The subtractive rule, however, was not applied consistently until later in Roman history. IIII was often used for 4, particularly in earlier inscriptions and in specific contexts, such as clock faces, where IIII is still preferred today. Similarly, VIIII for 9 was found in various inscriptions and documents, indicating that early Roman numerals allowed more flexibility in notation (Barnett 1998, 34).

The Impact of Subtractive Notation
The subtractive principle likely emerged as a practical response to avoid repetitive symbols and to simplify reading and writing. By the time the Roman numeral system reached maturity, subtractive notation had become the standard for key values, especially 4 and 9. Scholars have suggested that the subtractive rule also helped with mental arithmetic, as subtractive forms required less mental processing than additive sequences with long strings of symbols (Menninger 1992, 142).

However, this notation also introduced complexities that were not present in other systems, such as the Greek alphabetic numerals, which were purely additive. Roman numerals required users to recognise both addition and subtraction within a single string of symbols, making the system slightly more challenging to learn and use. In spite of these challenges, the subtractive notation persisted, as it allowed for a more elegant and less repetitive system, especially in monumental inscriptions where visual clarity was important.

Absence of Place Value and Zero
One of the most defining limitations of Roman numerals is the absence of place value and zero. Unlike the Arabic numeral system, which uses a positional notation to differentiate between numbers such as 5, 50, and 500 based on the digit’s position, Roman numerals assign fixed values to each symbol. The letter I always means 1, regardless of where it appears, while C always represents 100. As a result, Roman numerals cannot easily represent large numbers without an unwieldy accumulation of symbols.

This lack of positional value has two major implications:

• Cumbersome Representations of Large Numbers: For large numbers, Roman numerals become increasingly impractical. For example, the number 3,888 must be written as MMMDCCCLXXXVIII. This lack of positional efficiency limited the practicality of Roman numerals, especially in scientific or commercial contexts where complex calculations or very large numbers were necessary.
• Impossibility of Representing Zero: The Roman numeral system does not include a symbol for zero, as Roman numerals were primarily used to count tangible quantities—goods, soldiers, distances—where “nothing” was not an applicable concept.

The absence of zero had a significant impact on Roman mathematics and commerce, as zero plays a crucial role in algebra, place-value calculation, and advanced mathematics. In medieval times, when the concept of zero was introduced to Europe through the spread of Arabic numerals, Roman numerals remained ill-suited to advanced calculations and were gradually supplanted (Ifrah 1998, 145).

Workarounds and Practical Adjustments
Despite these limitations, the Romans found creative ways to use their numeral system effectively. In fields like commerce and engineering, Romans often used tallying methods or abacuses for calculations rather than relying solely on written numerals. Roman numerals were more of a shorthand for representing values rather than for performing mathematical operations. For example, tally marks were used in bookkeeping, and an abacus provided a practical tool for adding, subtracting, and multiplying without needing a written representation for every step.

The Romans also employed special symbols to indicate large values. The apostrophus notation (the ancient way of writing Roman numerals that uses an apostrophe-like symbol used a symbol resembling a “C” combined with an inverted “C” (IↃ) to represent 500, while CIↃ symbolised 1,000 (Chisholm 1929). Additional C and Ↄ pairs could be added to multiply the value by ten or one hundred, thus enabling the representation of very large numbers without a positional system.

Another workaround was the use of the vinculum notation, which placed a bar (or overline) above a numeral to indicate multiplication by 1,000. For example, V̅ represented 5,000, and X̅ represented 10,000. This practice, which became common in the late Republic and continued into the medieval period, allowed the Romans to denote large values more concisely. However, the vinculum notation was used primarily in specific contexts, such as monumental inscriptions and official documents, rather than in everyday commerce or communication.

Flexibility in Notation and Regional Variations
The Roman numeral system was not as rigidly standardised as modern numeral systems, which allowed for regional and temporal variations. In early Roman inscriptions, certain numbers could be represented in multiple ways. For example, 18 was sometimes written as XIIX rather than the more standard XVIII, and 99 could appear as IC or XCIX (Fasti Antiates Maiores, cited in Chisholm 1929). These variations reflect the adaptability of the system but also contributed to inconsistencies that could be confusing for modern readers.

Over time, certain forms became preferred in official documents, while others persisted in regional dialects or local inscriptions. This flexibility highlights the pragmatic nature of Roman numerals; they were not bound by strict rules but were adapted as needed to suit different contexts and audiences. This adaptability was both a strength and a weakness, as it allowed for easy use but sometimes made Roman numerals inconsistent or ambiguous.

Summary of the Roman Numeral System’s Functionality
In summary, Roman numerals provided a versatile, though limited, system of representing numbers. The additive and subtractive notation allowed for a compact representation of certain values, while the absence of place value and zero imposed practical constraints. Despite these limitations, the Romans developed workarounds, including the use of abacuses, apostrophus and vinculum notation, and local adaptations. The Roman numeral system served its purpose in a largely non-mathematical society where written numbers were primarily for labelling, counting, and simple arithmetic.

Although the Roman numeral system lacks the mathematical efficiency of the Arabic numeral system, its structure was sufficient for Roman needs and has contributed to its lasting cultural legacy. Today, Roman numerals are viewed as a symbol of classical antiquity, their design and structure connecting us to a time when numbers were deeply embedded in the visual and linguistic fabric of daily life.

Applications in Ancient Rome

Roman numerals were not just a numbering system but also a fundamental aspect of Roman culture, playing a central role in the Roman Empire. From commerce and administration to military organisation and public inscriptions, Roman numerals served as a practical tool for managing a vast and diverse society. The different ways Roman numerals were used in ancient Rome is explained below showing how they supported the Empire’s economic, administrative, and military needs and helped establish a cohesive visual language in public spaces.

Use in Commerce and Trade
In the commercial life of ancient Rome, Roman numerals were essential for recording transactions, measuring goods, and calculating taxes. Trade was an important part of the Roman economy, especially given the Empire’s extensive network of trade routes that connected the Mediterranean, Africa, the Middle East, and even parts of Asia. Merchants and traders used Roman numerals to mark goods, calculate prices, and tally sums. Records from Roman Egypt, for example, reveal the use of Roman numerals in papyrus receipts and account books documenting the sale of grain, livestock, and other goods (Rathbone 1991, 122-123).

On physical goods, Roman numerals were sometimes inscribed to indicate quantities or weights. Roman coinage provides clear examples of this, as coins often bore Roman numerals to signify their denomination or value. The use of symbols like I for one unit of currency or X for ten units helped standardise trade across the Empire. Coins such as the denarius, sestertius, and as displayed values in Roman numerals, allowed for a uniform system of currency that facilitated transactions across different regions and cultures within the Empire (Burnett 2004, 38).

Additionally, in larger commercial transactions, tallying and counting boards were often employed to simplify calculations. Roman merchants and tax collectors likely used counting tables or abacuses in conjunction with Roman numerals, as these tools provided a way to handle complex calculations without needing a more flexible written system like Arabic numerals. Roman numerals, in this context, were largely a shorthand for final sums rather than the method used for calculation itself.

Legal and Governmental Uses
Roman numerals were indispensable in the administrative and legal functions of the Empire. As Rome grew from a city-state to a vast empire, the need for an organised system of governance became critical. Roman numerals were used in official documents, laws, tax records, and censuses, helping the Roman state maintain control over a large, diverse and burgeoning population.

One of the most significant examples of Roman numerals in administration was in the census, a critical process for tracking the population and wealth of Roman citizens. The census, conducted every five years, recorded information on property, wealth, and household composition. Roman numerals were used to record the results, which were essential for determining citizens’ tax obligations and military service requirements (Nicolet 1991, 67). Inscriptions from various regions of the Empire, including Italy, Gaul, and Britannia, provide evidence of census records written in Roman numerals, illustrating how the system helped the Empire administer its territories effectively.

Roman law also relied on Roman numerals, as legal documents, edicts, and codes often enumerated laws, penalties, and regulations. The famous Twelve Tables6F^[8], the earliest codification of Roman law, used Roman numerals to number each table and clause, establishing a precedent for structuring legal texts that would continue throughout Roman history (Crawford 1996, 56). This use of numerals to categorise and organise laws made it easier for Roman citizens and magistrates to reference specific regulations.

Furthermore, Roman numerals were used in the titles of officials and military ranks, which served to reinforce the social hierarchy within Roman society. For example, high-ranking magistrates and governors were often designated by titles that included numerals indicating their seniority or level of authority. This structured use of numerals emphasised the organised and hierarchical nature of Roman government.

Military Usage and Legion Identification
The Roman military was one of the most powerful forces in the ancient world, and Roman numerals were instrumental in the organisation and identification of military units. The Roman army was divided into legions, each with a specific numeric designation. Roman numerals were used to identify each legion, giving them both a sense of unity and a distinct identity. For example, Legio X Gemina was the 10^th legion, and Legio III Gallica was the 3^rd legion stationed in Gaul. These numeric designations were prominently displayed on banners, armour, and other military equipment, reinforcing the order and discipline for which the Roman army was famous (Keppie 1984, 96).

Each legion was composed of several cohorts, which were further divided into centuries, led by centurions. The centurions themselves were identified by ranks that often included Roman numerals to denote seniority and authority within the legion. For instance, a Primus Pilus was the highest-ranking centurion in a legion, while other centurions had specific titles that could include numeric designations indicating their position within the command structure. Roman numerals thus played a critical role in organising the complex hierarchical structure of the Roman military.

The use of Roman numerals in military contexts also extended to construction and engineering projects. Roman soldiers, particularly those in the legions, were responsible for building roads, forts, and other infrastructure. Roman numerals were often inscribed on stones and bricks to mark sections, record measurements, or indicate the legion that had completed the construction work. Archaeologists have uncovered numerous stones with inscriptions like LEG VI FERR, indicating that the 6^th Legion Ferrata was responsible for the construction of a particular section of a wall or road. These inscriptions not only helped organise military construction projects but also served as a form of propaganda, reinforcing the presence and achievements of the Roman army in the provinces (Heath 1995, 128).

Public Inscriptions and Monuments
One of the most enduring applications of Roman numerals was in public inscriptions on monuments, temples, triumphal arches, and other public buildings. Roman numerals were frequently used to commemorate significant dates, mark dedications, and record the names and titles of patrons or emperors responsible for commissioning a structure. These inscriptions served both an informational and propagandistic function, as they publicly displayed the power and accomplishments of the ruling elite.

A prominent example of Roman numerals in public inscriptions is the Arch of Titus, constructed in 81 AD to commemorate Emperor Titus’s victory in the Jewish War. The inscription on the arch includes the date of dedication, written in Roman numerals, as well as the official titles of Titus, underscoring his authority and achievements. Such inscriptions not only recorded historical events but also reinforced the legitimacy of the emperor by publicly displaying his accomplishments in a highly visible and permanent medium (MacDonald 1982, 84).

Roman numerals also appear on imperial coinage, statues, and altars, where they often show the regnal years of emperors. Coins minted during the reign of an emperor might bear inscriptions like IMP VESP VII, meaning that Emperor Vespasian held the title of Imperator for the seventh time. These regnal titles, marked by numerals, helped unify the Empire under a single system of time-keeping that was accessible to citizens across regions. This use of Roman numerals on coins allowed the state to disseminate imperial authority and chronology throughout the Empire (Burnett 2004, 47).

In addition, Roman numerals were frequently used in funerary inscriptions on tombstones and epitaphs, where they recorded the age of the deceased and the year of death. Inscriptions like ANNOS XX (meaning “aged 20 years”) or D M (Dis Manibus, “to the spirits of the departed”) accompanied by Roman numerals served as a way to memorialise individuals while also embedding Roman cultural symbols into the fabric of daily life. These inscriptions demonstrate how Roman numerals permeated even the private and sacred aspects of Roman society, leaving a lasting legacy on the physical landscape of the Empire.

Summary of Applications in Ancient Rome
Roman numerals were deeply integrated into the fabric of Roman society, from the everyday activities of commerce and administration to the highly structured world of the military and the public symbolism of monumental architecture. Through their use in trade, government, military, and public inscriptions, Roman numerals helped to structure Roman society and assert the unity and authority of the Empire.

Roman numerals were more than just numbers—they were symbols of Roman identity, projecting a sense of order, hierarchy, and cultural cohesion. Even today, the visibility of Roman numerals in inscriptions, coins, and monuments provides a window into how the Romans lived, worked, and commemorated their achievements, reminding us of the far-reaching influence of this ancient numbering system.

Variants and Regional Differences

The Roman numeral system, whilst relatively uniform in its core symbols and principles, showed significant flexibility and regional diversity. Unlike the rigid standardisation that characterises modern numeral systems, Roman numerals allowed for local adaptations and non-standard forms, particularly in inscriptions, manuscripts, and later medieval documents. This flexibility reflects both the pragmatic nature of Roman numerals and the influence of linguistic, regional, and cultural variations across the vast Roman Empire. In this section, we will examine common variants, non-standard medieval forms, and the cultural influences that shaped the use of Roman numerals in different regions.

Common Variants and Deviations in Ancient Rome
One of the most noticeable aspects of Roman numerals is the variation in representing certain numbers, particularly those that use subtractive notation, such as 4, 9, 40, and 90. The subtractive principle—where a smaller numeral precedes a larger one to indicate subtraction—became standard over time, but earlier forms often used additive notation instead.

For instance, the numeral IIII was frequently used to represent the number 4, rather than the more efficient IV. This is still visible today on many clock faces, where IIII remains a popular choice for aesthetic and traditional reasons. Similarly, VIIII was often used in place of IX to represent 9. The prevalence of additive forms like IIII and VIIII suggests that subtractive notation was not consistently applied, especially in early Roman inscriptions and informal documents (Barnett 1998, 45).

These additive forms may have had practical benefits. In large stone inscriptions, for example, the use of IIII instead of IV created visual symmetry, balancing the numerals on either side of the face of a clock or within the context of a monument. The use of IIII is also observed on Roman-era gate markers, such as those at the Colosseum, where entrances are labelled with numbers like XLIIII for 44, combining both subtractive (XL for 40) and additive forms (IIII for 4) within the same numeral (MacDonald 1982, 217). This combination illustrates the flexibility and adaptability of the system in practical applications.

Non-Standard Medieval Forms
In the medieval period, as the Roman Empire’s influence waned and local cultures reasserted their identities, non-standard forms of Roman numerals became more common. Scribes and stonecutters, particularly those in regions distant from Rome, introduced new variations to meet local needs or due to limited knowledge of the classical numeral forms. Some medieval manuscripts, for instance, display curious adaptations and even errors in the use of Roman numerals, reflecting the inconsistencies that arose as Roman culture blended with local traditions.

One prominent example is the use of lower-case letters for Roman numerals in medieval manuscripts, where i, ii, iii, and iv gradually replaced the traditional upper-case forms. Another unique adaptation was the addition of the letter “j” as a swash form of the final “i” in numbers, such as “iiij” for 4 or “vij” for 7. This practice likely emerged as a way to differentiate the final “i” in a sequence, possibly to prevent tampering or misinterpretation in documents like prescriptions or legal records, where precision was essential (Dreyfus 1983, 78).

Medieval manuscripts also reveal unusual additive forms for numbers that would traditionally use subtractive notation. For example:

– 9 was sometimes written as VIIII instead of IX.

– 99 could appear as IC or LXXXXVIIII.

– 90 might be represented as LXXXX instead of the more concise XC.

In some cases, scribes used compound forms to express numbers according to spoken language conventions rather than strict numerical rules. For instance, the number 18 was sometimes represented as IIXX rather than XVIII, a practice seen in certain regional inscriptions. This might have reflected the Latin way of expressing numbers as “two from twenty” (duodeviginti), echoing how these numbers were spoken (Chisholm 1929, 112).

Cultural Influences on Variants
The Roman numeral system, though originating in Rome, was adapted to suit various cultural and linguistic contexts as it spread across the Empire. For example, in regions with strong Celtic, Gallic, or Germanic influences, Roman numerals were sometimes mixed with local symbols or adapted to reflect local language structures. This resulted in inscriptions where Latinised forms of native terms for numbers were combined with Roman numerals, creating unique regional variants.

In certain inscriptions, particularly from Roman Gaul and Britannia, numerals were occasionally written in forms that diverged from classical standards, possibly reflecting local dialects or preferences. The combination of Latin words and numerals allowed people across the Empire to adapt Roman numerals to their linguistic context, integrating the numbering system into local customs without completely replacing indigenous traditions.

During the later Empire, as Latin began to diverge into the Romance languages, regional pronunciations may have influenced numeral variations. For example, French-speaking scribes in the medieval period sometimes used notations like IIIIXXXIX for 99, a structure that mirrored the French phrase “quatre-vingt-dix-neuf” (literally “four-twenties and nineteen”) (Helfer 2007, 143). This reflected the local way of expressing numbers rather than adhering to classical Latin forms, illustrating how linguistic evolution impacted the structure of Roman numerals over time.

Roman Numerals in the Byzantine Empire
The Byzantine Empire, the eastern continuation of the Roman Empire, provides a fascinating example of how Roman numerals evolved in different cultural contexts. While the Byzantines continued to use Greek for official and ecclesiastical purposes, they retained Roman numerals in certain administrative functions. Over time, however, they adapted the system by incorporating elements of Greek notation, leading to hybrid forms.

In Byzantine inscriptions, one might find Greek letters used alongside Roman numerals or adapted to fit Greek grammatical structures. This merging of numeral systems created unique forms, especially in contexts such as church inscriptions, where Greek and Latin influences coexisted. Byzantine coins and official documents sometimes reflect these mixed forms, showing how Roman numerals were modified to suit the cultural environment of the Eastern Roman Empire (Ostrogorsky 1969, 105).

Medieval Calendars and Alternative Forms
The medieval period also saw the use of Roman numerals in calendars, especially for calculating dates of Christian holidays like Easter. However, scribes and scholars occasionally used non-standard forms when recording dates. The medieval calendar of the Fasti Antiates Maiores, for example, used subtractive forms in some entries and additive forms in others, reflecting a lack of strict standardisation in numerical notation.

Additionally, chronograms—inscriptions that encode dates within a text by marking certain letters as numerals—became popular during the Renaissance and later. This practice involved highlighting Roman numerals within words to convey a date, often the year of a building’s completion or a notable event. Although chronograms used Roman numerals, they were often structured according to the creative impulses of the artist or writer, adding another layer of variability to the system.

Modern Echoes of Variability
Even today, the legacy of these variations persists in specific contexts. For instance, many clock faces still display IIII instead of IV for the number 4, a practice that may have originated in early Roman times or as a choice for visual symmetry on round clock faces. This additive form has become iconic, especially in traditional European clock design, and serves as a reminder of the flexibility that characterised the Roman numeral system (Cipolla 2003, 56).

Furthermore, certain monumental inscriptions and architectural elements retain non-standard forms as a deliberate aesthetic choice. For example, Admiralty Arch in London uses MDCCCCX for the year 1910, favouring an additive form (CCCC for 400) over the standard subtractive form (CM). These modern instances reflect the enduring influence of Roman numerals’ inherent adaptability and the aesthetic preferences associated with different forms.

Summary of Variants and Regional Differences
The Roman numeral system was never as standardised as modern numbering systems, which allowed for considerable regional and temporal variations. From the use of IIII on clocks to non-standard forms in medieval manuscripts, the flexibility of Roman numerals highlights their adaptability to different cultural contexts and practical needs. These variations reflect the influence of language, local customs, and practical considerations, underscoring the Roman numeral system’s resilience and malleability.

Today, the remnants of these variants are still visible, particularly in decorative and ceremonial contexts, where traditional forms are valued for their historical resonance. The continued use of Roman numerals in specific formats serves as a testament to their enduring cultural and aesthetic appeal.

Transition to Arabic Numerals

The transition from Roman numerals to Arabic numerals in Europe was neither quick nor was it straightforward. Despite the clear mathematical advantages of the Arabic numeral system, Roman numerals remained in common use for centuries, partly due to cultural resistance and the entrenched position of Roman numerals in European society. The factors that led to the gradual replacement of Roman numerals with Arabic numerals, the practical advantages of the new system, and the roles played by key figures like Fibonacci^[9] in promoting this change are covered below.

The Introduction of Arabic Numerals to Europe
Arabic numerals were introduced to Europe through the cultural and intellectual exchanges that took place between the Islamic world and the Christian kingdoms of Europe, particularly in Spain and Italy. By the 8^th century, the Islamic Golden Age^[10] had produced numerous advances in mathematics, science, and philosophy. Scholars in the House of Wisdom^[11] in Baghdad studied and translated Greek, Persian, and Indian texts, including works on mathematics. It was through this synthesis of knowledge that the Hindu-Arabic numeral system—originally developed in India—was refined and transmitted across the Islamic world (Ifrah 1998, 229).

By the 10^th century, scholars in al-Andalus (Islamic Spain) began introducing Arabic numerals to Western Europe through translations of mathematical texts. Latin translations of works by Islamic scholars such as al-Khwarizmi and al-Kindi brought the Hindu-Arabic numeral system^[12], along with the concept of zero, to European scholars. Al-Khwarizmi’s treatise, often referred to in Latin as Algoritmi de numero Indorum (The Calculation of Hindu Numbers), played a pivotal role in spreading the Arabic numeral system. The term algorithm itself derives from his name, signifying the importance of his contributions (Ifrah 1998, 234).

The Practical Advantages of Arabic Numerals
The Hindu-Arabic numeral system offered several practical advantages over Roman numerals, which eventually made it more appealing to mathematicians, merchants, and scholars. The most significant of these advantages included:

• Place Value and Positional Notation: Unlike Roman numerals, which assign a fixed value to each symbol, Arabic numerals use a place-value system. This allows a single symbol, such as 5, to represent different values depending on its position (e.g., 5, 50, 500). Place value made calculations more straightforward and efficient, particularly for large numbers, as there was no need for lengthy strings of symbols (Burnett 2004, 66).
• Inclusion of Zero: The concept of zero was revolutionary and did not exist in Roman numerals. Zero functions as both a placeholder in positional notation and as a number in its own right, enabling more complex calculations. The addition of zero to the number system allowed for significant advancements in algebra, calculus, and other branches of mathematics. It was also crucial for accounting and financial records, as it provided a clear way to represent “nothing” in calculations (Seife 2000, 45).
• Ease of Arithmetic Operations: Arabic numerals facilitate arithmetic operations like addition, subtraction, multiplication, and division, which are cumbersome with Roman numerals. For example, adding 158 and 274 is simple in Arabic numerals but requires laborious tallying in Roman numerals (CLVIII + CCLXXIV). This efficiency was particularly appealing to merchants, bankers, and scholars who needed to perform regular calculations (Menninger 1992, 257).

Cultural Resistance to Change
Despite its advantages, the adoption of Arabic numerals faced significant resistance in medieval Europe. Roman numerals were deeply embedded in European culture, especially in religious, legal, and governmental contexts. Latin documents, laws, and church records continued to use Roman numerals well into the medieval period, and the prestige associated with Roman civilisation lent Roman numerals an aura of authority and tradition. Additionally, many Europeans associated Arabic numerals with the Islamic world, which was often viewed with suspicion or hostility during the Crusades and other periods of religious conflict (Burnett 2004, 69).

Another factor contributing to the resistance was the unfamiliarity of the general population with the new numeral system. Roman numerals were easily recognisable, while Arabic numerals, especially the curved shapes of 2, 3, and 5, looked strange to medieval Europeans. Moreover, the lack of standardised symbols for Arabic numerals in early manuscripts led to confusion, as different regions experimented with their own forms. These factors combined to slow the spread of Arabic numerals in Europe, even as the benefits of the new system were increasingly recognised.

The Role of Fibonacci and the Liber Abaci
I’ve previously mentioned the Italian mathematician Leonardo of Pisa (Pisano), better known as Fibonacci. A key turning point in the acceptance of Arabic numerals in Europe was the publication of Liber Abaci (The Book of Calculation) in 1202 by Fibonacci. Legend has it that he travelled extensively through the Mediterranean, where he encountered the Arabic numeral system and recognised its potential advantages over Roman numerals. In Liber Abaci, he introduced the Hindu-Arabic numerals and demonstrated their practical applications in commerce, measurement, and finance.

Fibonacci’s book provided clear examples of how to use Arabic numerals for basic arithmetic and more complex calculations, such as calculating interest and converting currency. He argued that the new system would simplify business and trade, and his examples resonated with the growing merchant class in Italy. Italian city-states like Venice and Genoa were emerging as major trading centres and Fibonacci’s work appealed to merchants and bankers who needed an efficient system for calculating profits and managing records (Devlin 2011, 98).

A page of the Liber Abaci from the National Central Library. The list on the right shows the numbers 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 (the Fibonacci sequence). The 2, 8, and 9 resemble Arabic numerals more than Eastern Arabic numerals or Indian numerals.
Citation: Liber Abaci. (2024, July 28). In Wikipedia. https://en.wikipedia.org/wiki/Liber_Abaci

However, Fibonacci’s advocacy did not immediately result in widespread adoption. Although Liber Abaci was influential, many people, including scholars and officials, continued to prefer Roman numerals for official documents. Nonetheless, Fibonacci’s book laid the groundwork for the gradual spread of Arabic numerals in Italy and, eventually, across Europe.

The Gradual Shift and the Role of Printing
The widespread adoption of Arabic numerals in Europe accelerated with the invention of the printing press in the mid-15^th century. Before this, manuscripts were laboriously copied by hand, often by monks and scribes who were more comfortable with Roman numerals. The printing press made it possible to standardise texts, including numerals, and to disseminate them more widely. Books on mathematics, astronomy, and commerce increasingly used Arabic numerals, which became more familiar and accessible to the general population as literacy rates rose.

The adoption of Arabic numerals was also encouraged by prominent scholars of the Renaissance, who valued the practical applications of the new system. Figures like Nicolaus Copernicus and Johannes Kepler used Arabic numerals in their astronomical calculations, highlighting the system’s superiority in scientific contexts. As these scholars’ works spread through printed books, the numerals gained legitimacy and acceptance in intellectual and commercial circles (Ifrah 1998, 300).

By the 16^th century, Arabic numerals had largely supplanted Roman numerals in mathematical texts and commercial records. While Roman numerals continued to be used in specific contexts, such as titles of monarchs and inscriptions on buildings, Arabic numerals had become the preferred system for most practical purposes.

The Decline of Roman Numerals in Daily Use
The decline of Roman numerals in daily life was primarily due to their unsuitability for complex calculations. The increasing complexity of European economies, particularly with the rise of banking and international trade, required a more efficient system for bookkeeping and calculations. Arabic numerals, with their ease of use in mathematical operations, provided a solution that Roman numerals simply could not match.

As Europe entered the Age of Exploration (c. 1400 to 1600) and began establishing colonies across the globe, the logistical demands of managing far-flung trade networks further reinforced the need for a streamlined numeral system. Arabic numerals facilitated the development of accounting methods, enabling merchants and bankers to track profits, losses, and inventory with greater accuracy. In time, the majority of Europe’s commercial and financial institutions fully embraced the new numeral system, relegating Roman numerals to ceremonial and symbolic uses (Cipolla 2003, 72).

Enduring Symbolic Value of Roman Numerals
Although Arabic numerals replaced Roman numerals in most practical applications, Roman numerals continued to be used in legal documents, monumental inscriptions, and religious texts, where they were valued for their association with tradition and authority. Even today, Roman numerals are used to designate monarchs, popes, book chapters, clock faces, and film sequels. Their use in these contexts underscores their enduring symbolic power, linking modern institutions and cultural practices to the prestige and heritage of the Roman Empire.

In sum, the transition from Roman to Arabic numerals reflects a broader cultural shift in Europe, from a world governed by tradition to one driven by practical innovation. The adoption of Arabic numerals marked a significant step forward in European mathematics, commerce, and science, paving the way for the rapid technological and intellectual advancements of the Renaissance and beyond.

Summary of Transition to Arabic Numerals
The gradual replacement of Roman numerals with Arabic numerals in Europe was driven by the practical advantages of the new system, particularly its use of place value and zero. Despite cultural resistance, figures like Fibonacci and the rise of the printing press helped accelerate the spread of Arabic numerals. While Roman numerals retained symbolic and ceremonial value, Arabic numerals became essential for the increasingly complex economic and scientific needs of a rapidly developing Europe.

Modern Uses and Cultural Legacy

Roman numerals, though largely obsolete in contemporary everyday calculations, continue to hold a unique place in modern society. They are commonly used in ceremonial and formal contexts, where they evoke a sense of tradition and gravitas. From monarchs’ titles and architectural inscriptions to watch faces and movie sequels, Roman numerals have retained their cultural relevance, symbolising both the legacy of the Roman Empire and the enduring human appreciation for classical aesthetics. This section examines the various ways in which Roman numerals are used today and analyses the cultural and symbolic reasons behind their continued appeal.

Roman Numerals in Monarchical and Papal Titles
One of the most recognisable modern uses of Roman numerals is in the titles of monarchs and popes. Roman numerals are used to distinguish different rulers with the same name, as in Elizabeth II of the United Kingdom or Louis XIV of France. This tradition is especially prominent in European monarchies, where Roman numerals signify continuity and legitimacy, linking current rulers to a long line of predecessors. In the case of Elizabeth II, the numeral II represents her as the second queen named Elizabeth to ascend to the British throne, evoking a direct connection to Elizabeth I and the history of the British monarchy (Chambers 1999, 45). Prince Charles on succeeding his mother following her passing, became King Charles III.

In the Catholic Church, Roman numerals are used to distinguish different popes with the same papal name, such as John Paul II and Benedict XVI. This practice underscores the church’s long history, symbolising the unbroken line of leadership stretching back to Saint Peter. Roman numerals in these contexts act as markers of tradition, anchoring contemporary figures within the weighty legacy of their institutions. The use of Roman numerals for monarchs and popes thus serves not only a practical purpose but also reinforces the authority and historical continuity of these roles.

Use in Clock Faces and Timekeeping
Roman numerals are often seen on clock faces, particularly on traditional or ornamental timepieces. Many clockmakers, including renowned companies like Rolex and Cartier, continue to use Roman numerals on the dials of their watches and clocks, associating their products with luxury and elegance. Roman numerals lend a timeless, classical appearance to clock faces, which is why they are often preferred for decorative clocks in public buildings, historic monuments, and even on household mantel clocks.

Cartier Santos – steel/gold from 1988
Citation: Cartier (jeweller). (2024, October 5). In Wikipedia. https://en.wikipedia.org/wiki/Cartier_(jeweler)
Attribution: Noop1958, CC BY-SA 3.0 <https://creativecommons.org/licenses/by-sa/3.0&gt;, via Wikimedia Commons

Interestingly, many clock faces display IIII instead of IV for the number 4. This choice, often attributed to visual symmetry, is thought to balance the numeral arrangement on circular clock faces, especially since VIII (8) appears on the opposite side of the dial. The use of IIII may also reflect a historical preference, as the additive form was common in early Roman inscriptions. This detail exemplifies how Roman numerals have maintained not only their aesthetic appeal but also their stylistic nuances, even in modern contexts (Cipolla 2003, 78).

In addition to clocks, Roman numerals are used in other forms of timekeeping. For example, they are often employed in sundials and astronomical clocks, where their classical associations enhance the ancient and scholarly appearance of these instruments. By appearing on clocks and timepieces, Roman numerals continue to remind us of the passing of time in a format that recalls the ancient world, linking our everyday experience to a broader historical context.

Architectural Inscriptions and Cornerstones
Roman numerals are frequently used in architectural inscriptions and on cornerstones to mark the dates of building construction, renovations, or dedications. This practice is particularly common on public buildings, monuments, and commemorative plaques. For example, the Statue of Liberty in the United States has the year MDCCLXXVI (1776) inscribed on its tablet, signifying the date of American independence. Similarly, many historical buildings in Europe and North America have cornerstones bearing Roman numerals, a feature that lends a sense of permanence and gravitas to these structures (Harris 2014, 123).

Inscriptions in Roman numerals are especially prominent in neoclassical architecture, where they serve as a visual reference to the Roman Empire and its legacy. Buildings constructed in the 18^th and 19^th centuries, when neoclassicism was at its height, frequently used Roman numerals in dedications and plaques as a way of emulating the grandeur of ancient Rome. Today, these inscriptions remain as historical markers, connecting modern viewers to the ideals and aesthetic values of the neoclassical period.

Roman numerals are also used on war memorials, such as the Arc de Triomphe in Paris, where the dates of significant battles and the names of military leaders are inscribed. The use of Roman numerals on these memorials imparts an aura of solemnity and dignity to the inscriptions, reflecting the honour and respect due to those who served. Roman numerals thus continue to serve as a symbol of commemoration, marking important events and dates in a format that feels timeless and universally resonant.

Entertainment and Popular Culture
In modern entertainment, Roman numerals are widely used to indicate sequels and instalments in film, television, and video game series. For example, movie franchises like Rocky, Star Wars, and the Fast and Furious series use Roman numerals to denote sequels, as in Rocky II and Star Wars Episode IV. Roman numerals lend a sense of grandeur and importance to these titles, suggesting continuity and an established legacy. For instance, Super Bowl events are numbered with Roman numerals (e.g., Super Bowl LIII), a tradition that conveys the significance of each annual event and its connection to a larger tradition (Brown 2015, 88).

In the world of video games, Roman numerals are similarly used to give sequels a sense of importance and formality. Games such as Final Fantasy VII and Grand Theft Auto V use Roman numerals to indicate their place in a long-running series, creating a perception of continuity and respect for tradition. This practice is particularly effective for franchises that have built a loyal fanbase, as it suggests that each new instalment is part of a larger narrative or universe.

Beyond sequels, Roman numerals are also commonly used in sports and pageantry. For example, the Olympic Games use Roman numerals to denote each Olympiad, with the 2024 Summer Olympics designated as the Games of the XXXIII Olympiad. Roman numerals confer a sense of prestige on these events, linking them to an ancient tradition of athletic competition. In this way, Roman numerals continue to serve as a cultural bridge, connecting contemporary events with historical roots.

Academic and Organisational Uses
Roman numerals are often used in academic contexts to organise information hierarchically, especially in legal and scholarly documents. In law, Roman numerals are used to indicate sections, subsections, and clauses, providing a structured outline that is clear and orderly. For example, the United States Constitution is divided into articles and sections, many of which are numbered with Roman numerals (e.g., Article I, Section 2). This practice is also common in scientific papers, where Roman numerals are used to number chapters, sections, or appendices, helping readers navigate complex documents (Oxford English Dictionary, 2018).

In academia, Roman numerals are frequently used to label book chapters and prefaces, as well as appendices in longer works. This numbering style lends a formal and scholarly tone, reflecting the traditional association of Roman numerals with knowledge and learning. Roman numerals are also used in outlines and organisational charts to denote hierarchical levels, adding a sense of structure and gravitas that enhances the document’s authority.

Symbolic Value and Cultural Appeal
Roman numerals endure in modern society not only because of their aesthetic appeal but also due to their symbolic associations with classicism, authority, and timelessness. The use of Roman numerals invokes the cultural heritage of the Roman Empire, a civilisation associated with power, knowledge, and order. This symbolic value is particularly potent in Western cultures, where the legacy of Rome is deeply embedded in legal, architectural, and educational traditions.

Roman numerals also provide a connection to the past that is visually distinctive. Unlike Arabic numerals, which are ubiquitous and practical, Roman numerals stand out as markers of formality and tradition. Their use in titles, dates, and decorative contexts creates an air of permanence and continuity, suggesting that the values or events they mark are significant and enduring. In a world dominated by fast-paced change, Roman numerals offer a link to the ancient and the immutable, lending a sense of stability and history.

Moreover, Roman numerals are widely recognised and understood, even by people unfamiliar with their mathematical rules. Their simple, iconic forms make them easy to read and identify, while their limited set of symbols (I, V, X, L, C, D, M) allows for an intuitive understanding of numbers in specific contexts. This combination of familiarity and distinctiveness has helped Roman numerals remain relevant in modern society, where they continue to signify heritage and continuity.

Summary of Modern Uses and Cultural Legacy
Roman numerals have transcended their original practical function to become powerful symbols of tradition, authority, and continuity. From monarchical titles and architectural inscriptions to clocks and popular culture, they continue to be used in contexts where a sense of permanence or gravitas is desired. The enduring appeal of Roman numerals lies in their connection to the classical past and their ability to evoke a sense of timelessness, providing a cultural bridge between ancient and modern worlds.

Mathematical Properties and Limitations

Roman numerals, with their unique additive and subtractive structure, are a testament to the ingenuity of the Roman Empire. However, as a non-positional numbering system lacking a symbol for zero, Roman numerals presented significant limitations when it came to mathematical calculations. This section explores the mathematical properties of Roman numerals and the challenges they posed for arithmetic operations. It shows how the Romans attempted to overcome these limitations through tools like the abacus and various workarounds.

Additive and Subtractive Notation
At the heart of the Roman numeral system is its additive-subtractive structure. Roman numerals represent values by combining letters with fixed numerical values in specific sequences. Most numbers are formed additively by placing symbols from largest to smallest in descending order (e.g., VIII for 8, or XII for 12). However, when a smaller numeral precedes a larger one, it is subtracted, a rule applied to numbers such as 4 (IV), 9 (IX), 40 (XL), 90 (XC), 400 (CD), and 900 (CM).

This structure allowed Roman numerals to be relatively compact for certain numbers, yet it created an inconsistency that could complicate interpretation, particularly when reading longer sequences. For example, 39 is represented as XXXIX (30 + 9), whereas 246 is written as CCXLVI (200 + 40 + 6). This combination of additive and subtractive notation made Roman numerals somewhat intuitive for representing values but less practical for performing arithmetic operations, especially compared to positional systems like Arabic numerals, which assign value based on digit position.

Limitations of the Non-Positional System
The Roman numeral system’s lack of positional notation was one of its most significant limitations. In positional systems, like the modern Arabic numeral system, the position of each digit determines its value based on powers of ten. For example, in the number 345, the “3” represents 300 because it is in the hundreds place. In Roman numerals, however, there is no positional value: each symbol has a fixed value regardless of its position, and numbers are simply built by combining these symbols.

This non-positional approach imposed practical constraints, particularly as numbers grew larger. Representing large numbers required increasingly lengthy combinations of symbols. For example, 3,888 had to be written as MMMDCCCLXXXVIII, with each numeral contributing to the total in a cumbersome way. This limitation made Roman numerals unsuitable for expressing large numbers efficiently, and it hindered the system’s usability in fields that required detailed calculations, such as commerce, science, and engineering (Menninger 1992, 212).

The Absence of Zero and Its Implications
Another major limitation of Roman numerals was the absence of zero. The concept of zero as a placeholder or as a number in its own right did not exist in Roman mathematics. Zero, which is essential in positional systems for denoting an empty value in a given place, was unknown to the Romans and most other ancient civilisations, except the Indians. In Roman numerals, there was simply no way to represent “nothing.” Consequently, mathematical operations involving zero were impossible, and complex calculations required cumbersome workarounds or additional counting aids.

The absence of zero also had implications for accounting and record-keeping. In financial calculations, zero allows for accurate representations of quantities or balances. For example, in a ledger, a zero clearly indicates a balance or lack of stock, something the Romans had to convey either through written words (e.g., nulla, meaning “none”) or by leaving spaces blank, which could lead to ambiguity. The absence of zero limited the precision and clarity of Roman accounting systems, which ultimately led to the adoption of the Hindu-Arabic system when it became available.

Challenges in Arithmetic Operations
Performing arithmetic operations with Roman numerals was a complex and labour-intensive process, particularly for multiplication, division, and calculations involving large numbers. Simple addition and subtraction were manageable, as Roman numerals could be broken down into component parts, but multiplication and division were far more difficult due to the lack of positional value and zero.

For example, to add XLII (42) and XXIII (23), one would need to mentally or visually combine the values:

– Break down XLII as 40 + 2 and XXIII as 20 + 3.

– Sum each place (40 + 20 = 60; 2 + 3 = 5) to arrive at LXV (65).

While this approach works for small numbers, it becomes unwieldy with larger numbers or more complex calculations. Multiplication and division, which require iterative processes and place values, were especially challenging in the Roman numeral system. Without positional notation, there was no efficient way to multiply or divide numbers, which is why the Romans developed alternative methods for handling complex calculations, such as the abacus and counting boards.

Carnuntum Open Air Museum: Abacus
“Carnuntum Open Air Museum: Abacus” by reinholdbehringer is licensed under CC BY-NC-SA 2.0.

The Abacus and Counting Boards
To perform calculations more efficiently, the Romans relied on the abacus, a counting device that helped bridge the gap left by the limitations of their numeral system. Roman abacuses used a series of grooves or beads to represent units, tens, hundreds, and so on, allowing users to perform arithmetic by moving counters rather than writing out lengthy combinations of numerals. The abacus allowed for faster addition, subtraction, and even some forms of multiplication and division.

The Roman abacus typically consisted of a board with grooves or lines and counters, sometimes made of metal or stone. Counters could be moved up and down the board to represent different values, and the user could easily add or subtract by shifting counters. The Roman abacus was a powerful tool, as it allowed merchants, tax collectors, and accountants to handle large calculations without needing to write out long strings of Roman numerals. The abacus was also portable, making it a practical solution for use in various contexts, from market transactions to military logistics (Smith 1958, 56).

Counting boards were another common tool used in conjunction with Roman numerals. These boards, similar to the abacus but often simpler in design, featured lines or columns where tokens could be placed to represent different values. Counting boards facilitated basic calculations by enabling users to move tokens across columns, each representing a specific power of ten. While counting boards did not solve all the limitations of the Roman numeral system, they provided a workaround that allowed for practical calculations without the need for an entirely different numeral system.

Roman Fractions and the Duodecimal System
The Roman numeral system also incorporated a duodecimal (base-12) structure for fractions, particularly in contexts like measurement and coinage. The base-12 system was advantageous because 12 can be easily divided into halves, thirds, and quarters, making it useful for everyday measurements. For example:

• The Romans used the uncia (one-twelfth) to divide the as (a unit of currency), with each uncia representing one-twelfth of the as.
• Roman fractions were typically indicated by dots or symbols. The symbol S represented a semis (half), while additional dots indicated smaller fractions, such as four dots (····) for one-third (triens).

This duodecimal approach to fractions was practical for handling common divisions in trade and daily life, even if it limited the system’s adaptability to other forms of calculation. However, the lack of a flexible notation for fractions hindered its usefulness in scientific calculations, which became a necessity as European mathematics evolved.

Limitations in Scientific and Advanced Mathematics
The limitations of Roman numerals became particularly evident in scientific and mathematical fields as European scholarship developed. While Roman numerals were sufficient for the daily needs of commerce, construction, and administration, they were impractical for advanced calculations in astronomy, engineering, and geometry. For example, calculating the circumference of a circle, which requires multiplication, division, and the concept of pi, was difficult to express and compute with Roman numerals. As Europe entered the Renaissance (early 1300s and ending in the early 1600s), the need for a more flexible and efficient numeral system became clear, particularly among scientists and mathematicians.

Roman numerals also posed challenges for accounting and financial calculations, especially as European economies became more complex. Trade routes, banking practices, and international commerce required precise accounting methods that Roman numerals could not adequately support. Banking, in particular, demanded accurate records of debts, interest, and other financial calculations. This demand for accuracy and flexibility further drove the adoption of Arabic numerals, which facilitated the development of double-entry bookkeeping and other accounting practices that were essential to the growth of commerce in Renaissance Europe (Ifrah 1998, 310).

Summary of the Mathematical Properties and Limitations of Roman Numerals
In summary, Roman numerals served the needs of the Roman Empire well, especially in the context of counting, labelling, and simple arithmetic. However, their non-positional nature, the absence of zero, and their cumbersome structure for larger numbers made them inadequate for more advanced calculations. Roman numerals limited the ability to perform complex arithmetic and hindered the development of mathematical and scientific inquiry in medieval Europe.

The Romans devised tools such as the abacus and counting boards to compensate for these limitations, but ultimately, the arrival of Arabic numerals and their positional system proved far more efficient. The transition to Arabic numerals marked a turning point in European mathematics, enabling a level of sophistication and precision that the Roman system could not achieve.

Roman Numerals in Comparative Perspective

The Roman numeral system, while effective for its purposes within the Roman Empire, was only one of several numbering systems used in the ancient world. Other civilisations, such as the Egyptians, Babylonians, and Greeks, developed their own unique systems, each with its strengths, limitations, and cultural significance. This section explores the distinctive features of these ancient systems, comparing them to Roman numerals to highlight the unique aspects of the Roman system and the reasons behind its persistence in Western culture.

Comparison with the Egyptian Numeral System
The Egyptian numeral system dates back to around 3000 BC and used a hieroglyphic representation for numbers. Like Roman numerals, the Egyptian system was non-positional and relied on additive principles. The Egyptians used distinct symbols for each power of ten: a vertical line for 1, an arch or heel bone for 10, a coil for 100, a lotus flower for 1,000, and so on. Each symbol could be repeated up to nine times, allowing the Egyptians to represent large numbers efficiently (Menninger 1992, 45).

For example, the number 276 would be represented by two coils (200), seven heel bones (70), and six vertical lines (6). The Egyptians could express large values by simply repeating symbols, much like the Roman system. However, unlike the Romans, the Egyptians did not have a subtractive principle, so their system was entirely additive.

The strengths of the Egyptian System were:

• Simplicity in Representation: The Egyptian system allowed for straightforward representation of numbers by combining a limited set of symbols.
• Effective for Counting and Recording: The system was well-suited to Egyptian administrative needs, especially for accounting in agriculture, trade, and construction.

The limitations of the Egyptian System were:

• Lack of Place Value: Like Roman numerals, the Egyptian system lacked positional notation, making arithmetic operations complex and unwieldy.
• Limited Flexibility for Calculation: Without positional value or a zero, the system struggled with large numbers and complex calculations.

In comparison, Roman numerals were more versatile in some respects, particularly due to the subtractive notation that allowed for slightly shorter numeral representations. However, both systems faced challenges in performing arithmetic, particularly for multiplication and division.

Comparison with the Babylonian Numeral System
The Babylonian numeral system was highly advanced for its time and unique in its approach. Developed around 2000 BC, the Babylonian system used a base-60 (sexagesimal) structure rather than the base-10 structure that most ancient cultures used. This system, known as the sexagesimal system, is still partially in use today for measuring time (60 seconds per minute, 60 minutes per hour) and angles (360 degrees in a circle).

The Babylonians used a combination of two symbols—one for units and one for tens—to represent numbers from 1 to 59, after which they would reset, increasing the value of the next position by a factor of 60. This made the Babylonian system a positional numeral system and thus more versatile for calculations than Roman numerals or the Egyptian system.

The strengths of the Babylonian System were:

• Positional Notation: The use of place value made it easier to perform arithmetic operations, such as addition and multiplication, especially compared to the Roman and Egyptian systems.
• Base-60 Flexibility: The sexagesimal system’s divisibility made it highly flexible for fractions and calculations involving subdivisions, which was particularly advantageous for astronomical calculations.

The limitations of the Babylonian System were:

• Complexity of Base-60: The base-60 system could be complex and less intuitive, especially for basic arithmetic.
• Ambiguity Due to Lack of Zero: While the Babylonians did eventually introduce a symbol to act as a placeholder, early versions of the system lacked zero, which led to potential ambiguities in number representation.

In comparison, Roman numerals were simpler and easier to learn for basic counting and record-keeping, which were the primary needs of the Roman Empire. However, the Babylonian system’s positional notation and the flexibility of base-60 made it far superior for complex calculations, particularly in fields like astronomy.

Comparison with the Greek Numeral System
The Greek numeral system (also known as the Ionian or alphabetic numeral system), developed around 450 BC, was a quasi-decimal system that used the Greek alphabet to represent numbers. Each letter of the Greek alphabet corresponded to a different number, with the first nine letters representing 1 through 9, the next nine letters representing 10 through 90, and the final nine letters representing 100 through 900. This allowed the Greeks to represent numbers up to 999 with a relatively compact notation.

For larger numbers, the Greeks would place a mark, called keraia, next to the numeral to indicate thousands, ten thousands, and so on. This system was particularly well-suited to the needs of Greek mathematicians, as it allowed for efficient notation of moderately large numbers and was relatively compact.

The strengths of the Greek System were:

• Compact Notation: The Greek system allowed for shorter numeral representations, particularly for numbers in the hundreds and thousands.
• Use in Mathematics and Science: The Greeks developed advanced mathematical and geometric theories, and their numeral system, though limited, was more compact than Roman numerals, making it easier to work with for scholarly purposes.

The limitations of the Greek System were:

• Lack of Positional Notation and Zero: Like Roman numerals, the Greek system did not use place value or zero, which limited its usefulness for complex calculations.
• Dependency on the Greek Alphabet: Since the system relied on the Greek alphabet, it could be difficult for non-Greek speakers to learn, reducing its portability.

Compared to the Roman system, the Greek numeral system was more compact, especially for larger numbers. However, it lacked place value and zero, which restricted its applications. Both the Greek and Roman systems were adequate for their respective cultures but were ultimately less versatile than positional systems like the Babylonian and Hindu-Arabic numerals.

Comparison with the Hindu-Arabic Numeral System
The Hindu-Arabic numeral system, originating in India and later transmitted to Europe through the Islamic world, represents the pinnacle of ancient numeral development. This system, based on a base-10 positional structure, uses ten symbols (0–9) and includes zero both as a number and as a placeholder. The Hindu-Arabic system became the global standard due to its efficiency and adaptability.

The strengths of the Hindu-Arabic System were:

• Positional Notation with Zero: The introduction of zero and place value allowed for easy representation of large numbers and efficient calculations.
• Universality and Simplicity: With only ten symbols, the Hindu-Arabic system is relatively simple and universally applicable, making it easy to learn and use across different cultures.
• Efficiency in Arithmetic: The system’s design facilitates complex calculations, including multiplication, division, and algebra, making it ideal for advanced mathematics, science, and commerce.

The limitations of the Hindu-Arabic System were:

• Initial Cultural Resistance in Europe: Early on, Arabic numerals were viewed with suspicion in medieval Europe, particularly due to their association with Islamic culture. This resistance slowed their adoption, even though the system was mathematically superior.

In comparison to Roman numerals, the Hindu-Arabic system was vastly more efficient, versatile, and suited to complex calculations. Once introduced to Europe, it eventually replaced Roman numerals in most practical applications, marking a major advancement in mathematics and commerce.

Cultural and Practical Differences
Each ancient numeral system developed in response to the unique needs and values of its culture:

• The Romans used numerals primarily for record-keeping, counting, and labelling, purposes that did not demand a complex mathematical system. Their non-positional, additive-subtractive notation was suited to everyday tasks but lacked the flexibility for advanced mathematics.
• The Babylonians, with their base-60 positional system, focused on astronomy and detailed calculations, which their numeral system facilitated.
• The Greeks valued compactness and clarity, developing a system that, while limited, was effective for mathematical exploration and philosophical inquiry.
• The Egyptians focused on practical applications in trade, construction, and administration, with a system that prioritised simplicity in recording and counting over complex calculations.

The eventual dominance of the Hindu-Arabic numeral system was not only due to its mathematical superiority but also to the growth of trade, commerce, and scientific inquiry in the medieval and Renaissance periods. As Europe’s needs evolved, so too did its acceptance of a numeral system that allowed for greater precision and adaptability.

Summary of Numerals in Comparative Perspective
When compared with other ancient numeral systems, Roman numerals reveal both strengths and weaknesses. While their additive-subtractive structure was adequate for Rome’s practical needs, it was ultimately limited in scope. Systems like the Babylonian and Hindu-Arabic numerals offered greater versatility for scientific and commercial calculations, and their positional notation allowed them to outpace the Roman system in effectiveness and efficiency. In the end, the Hindu-Arabic system, with its positional notation and use of zero, set the standard for modern mathematics.

St. Peter’s Basilica (Vatican City)
Maderno’s façade, with the statues of Saint Peter (left) and Saint Paul (right) flanking the entrance stairs
Citation: St. Peter’s Basilica. (2024, October 26). In Wikipedia. https://en.wikipedia.org/wiki/St._Peter%27s_Basilica
Attribution: Jean-Pol GRANDMONT, CC BY-SA 3.0 <https://creativecommons.org/licenses/by-sa/3.0&gt;, via Wikimedia Commons
This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.

Conclusion

Roman numerals represent a fascinating chapter in the history of mathematics and civilisation. Originating in ancient Rome and deeply integrated into the fabric of Roman society, these numerals served a variety of practical purposes, from commerce and governance to public inscriptions and military organisation. Unlike modern numerals, Roman numerals are a non-positional system, relying on a simple set of symbols—I, V, X, L, C, D, and M—and combining additive and subtractive notation to represent values. While this structure was adequate for the needs of the Roman Empire, it imposed limitations that ultimately led to the adoption of more flexible systems as Europe advanced.

The Evolution and Limitations of Roman Numerals
Throughout the Roman Empire, numerals were used to administer a vast and complex society. Inscriptions on monuments, records of census data, and the numbering of military legions all demonstrate how Roman numerals permeated daily life. However, the absence of place value and a symbol for zero limited the system’s capacity for advanced calculations, making it challenging to perform complex arithmetic or work with large numbers efficiently. While the Romans developed tools such as the abacus to overcome these limitations, the system remained cumbersome compared to the base-10, positional structure of the later Hindu-Arabic numeral system.

In a broader historical context, Roman numerals were one of several numbering systems developed by ancient civilisations, each shaped by the unique needs of its society. Compared to the Egyptian, Babylonian, and Greek numeral systems, Roman numerals were relatively simple and pragmatic, well-suited to the practical tasks of counting, recording, and labelling. However, as Europe’s intellectual and economic needs evolved—particularly during the Renaissance—Roman numerals proved inadequate for the increasingly sophisticated calculations required in commerce, science, and engineering.

The Shift to Arabic Numerals and the Legacy of Roman Numerals
The gradual replacement of Roman numerals with Arabic numerals was a watershed moment in European mathematics and commerce. Arabic numerals, with their positional value and the inclusion of zero, allowed for a new level of precision and efficiency in calculations. This system, introduced to Europe by scholars like Fibonacci and spread by the printing press, eventually supplanted Roman numerals in most practical applications. By the 16^th century, Arabic numerals had become the standard in commerce, science, and daily life, enabling advancements in accounting, navigation, and technology that were previously impossible.

Despite this shift, Roman numerals have retained a distinct cultural and symbolic role. Today, they are used in formal, decorative, and ceremonial contexts, where they evoke a sense of tradition, authority, and continuity. From monarchical titles and clock faces to movie sequels and sports events like the Olympics and Super Bowl, Roman numerals are used to mark significant moments and figures, bridging the modern world with the ancient past. Their continued presence in architecture, academic documents, and cultural artefacts speaks to their lasting impact and their symbolic value as markers of history and heritage.

Reflecting on the Enduring Appeal of Roman Numerals
The endurance of Roman numerals lies in their simplicity, visual distinctiveness, and association with classical antiquity. As a non-positional system with only seven core symbols, Roman numerals are accessible and easily recognisable. They are imbued with a sense of permanence and grandeur, qualities that make them particularly suitable for commemorative and decorative purposes. In many ways, Roman numerals have transcended their original function as a practical numbering system, evolving into symbols of continuity and timelessness.

In an increasingly digital world, where Arabic numerals dominate nearly every practical aspect of life, Roman numerals continue to serve as cultural touchstones, reminding us of the achievements of ancient Rome and the influence of its civilisation on Western history. Their use in modern contexts, from film titles to historical buildings, reinforces a connection to the past, reflecting a human desire to honour tradition even as society progresses.

Final Thoughts

Roman numerals have withstood the test of time, surviving long after their practical utility was eclipsed by more efficient systems. They are a testament to the Roman Empire’s impact on Western civilisation, not only in terms of governance, law, and architecture but also in the realm of symbolic representation. Although they no longer serve as a primary means of counting or calculation, Roman numerals retain an honoured place in our cultural lexicon, carrying with them the legacy of a world that valued order, authority, and tradition.

We encounter these ancient symbols in our everyday lives. As we walk past historical buildings with cornerstones marked MCM for 1900, see the date MDCCLXXVI (1776) on the Statue of Liberty, or watch the latest Olympic Games, we’re reminded of the enduring legacy of Roman numerals in marking time and commemorating tradition. In an era dominated by speed and change, Roman numerals remind us of a time when numbers were inscribed in stone, intended to endure across the centuries. They represent an enduring legacy of a civilisation that, in many ways, still shapes our world today.

Picture: Detail of tablet with the inscription ‘July IV. MDCCLXXVI’ Statue of Liberty, Liberty Island, Manhattan, New York, USA
Attribution: Ludovic Bertron from New York City, USA, CC BY 2.0 <https://creativecommons.org/licenses/by/2.0&gt;, via Wikimedia Commons
This file is licensed under the Creative Commons Attribution 2.0 Generic license.
File URL: https://upload.wikimedia.org/wikipedia/commons/f/f2/Miss_Liberty_IV_%283010903452%29.jpg

Remember: Use these three rules for writing Roman numerals:

I     Roman numerals are written from largest to smallest, from left to right. Add up the value of each symbol.

II   Only I, X, C and M can be repeated. Never repeat a symbol more than three consecutive times.

III When a smaller numeral is to the left of a bigger numeral, subtract it. when it appears to the right of the largest number(s), it should be added.

Roman Numerals Explained in Simple Terms

 
Artistic impression of a young teacher.
Drawn by DALL-E, a subset of ChatGPT, March 2024.

Imagine numbers made of letters! That’s what Roman numerals are—a special way of using letters like “I,” “V,” “X,” and “M” to write numbers. These numbers come from ancient Rome, over 2,000 years ago, when the Romans ruled a big empire across Europe, parts of Asia, and North Africa. They didn’t have the same numbers we use today, like 1, 2, and 3, so they created their own system. It’s like their own secret code that they used in all kinds of ways!

How They Work
Here’s a cool thing about Roman numerals: each letter stands for a certain number. For example:

– “I” means 1

– “V” means 5

– “X” means 10

– “L” means 50

– “C” means 100

– “D” means 500

– “M” means 1,000

To make numbers, they would put these letters together. If a smaller number is before a bigger one, you subtract it; if it’s after, you add it. For example:

– IV means 4 (5 – 1)

– VI means 6 (5 + 1)

– XI means 11 (10 + 1)

Why It’s Still Important
Even though we don’t use Roman numerals for maths today, we still see them all around us. They remind us of history and are used on old buildings, clocks, and even in the names of kings and queens! The Romans left such a big mark on the world that we still honour them by using their numbers in special places.

So next time you see Roman numerals, like “XII” on a clock or “MMXXIV” (2024) on a building, you’ll know it’s a little piece of history connecting us to the ancient Romans!

Sources and Further Reading

• https://byjus.com/maths/roman-numerals/
• https://en.wikipedia.org/wiki/Arch_of_Titus
• https://en.wikipedia.org/wiki/Fasti_Antiates_Maiore
• https://en.wikipedia.org/wiki/Roman_numerals
• https://historylearning.com/a-history-of-ancient-rome/education-in-ancient-rome/
• https://historylearning.com/a-history-of-ancient-rome/history-of-roman-numerals/
• https://romannumerals.guide/how-roman-numerals-work
• https://worldhistoryedu.com/history-of-the-roman-numerals/
• https://www.britannica.com/topic/Roman-numeral
• https://www.encyclopedia.com/science/encyclopedias-almanacs-transcripts-and-maps/roman-numerals-their-origins-impact-and-limitations
• https://www.historyhit.com/the-complete-guide-to-roman-numerals/
• https://www.mathsisfun.com/roman-numerals.html
• https://www.sciencefocus.com/science/how-did-the-romans-do-arithmetic-with-their-numerals
• https://www.splashlearn.com/math-vocabulary/roman-numerals
• https://www.storyofmathematics.com/roman.html/
• UNESCO Encyclopedia of Mathematics. “Roman Numerals.” at https://www.unesco.org/encyclopedia/roman-numerals
• YouTube Video: A Brief History of Roman Numerals at https://www.youtube.com/watch?v=7TCosugReUw
• YouTube Video: Roman Numerals Explained With Many Examples at https://www.youtube.com/watch?v=MOIJxae_7Pk

Bibliography

• Fibonacci’s Liber Abaci: A Translation into Modern English of Leonardo Pisano’s Book of Calculation (Sources and Studies in the History of Mathematics and Physical Sciences), by Laurence Sigler, published by Springer, available from https://www.amazon.co.uk/Fibonaccis-Liber-Abaci-Translation-Calculation/dp/0387954198
• A Brief Introduction to Ancient Counting Systems For The Non-Mathematician, by David Cycleback (Author), published by Lulu.com, available from https://www.amazon.co.uk/Introduction-Ancient-Counting-Systems-Non-Mathematicians/dp/1312100842
• Ancient Literacy, by William V. Harris, published by Harvard University Press, available from https://www.amazon.co.uk/Ancient-Literacy-British-Museum-William/dp/0674033817
• Ancient Rome: Social and Historical Documents from the Early Republic to the Death of Augustus, by Matthew Dillon and Lynda Garland, published by Routledge, available from https://www.amazon.co.uk/Ancient-Rome-Historical-Documents-Sourcebooks/dp/0415726999/
• Clocks and Culture, 1300–1700, by Carlo M. Cipolla, published by W. W. Norton & Company, available from https://www.amazon.co.uk/Clocks-Culture-1300-1700-Norton-Library/dp/0393324435/
• Etruscan Civilization: A Cultural History, by Sybille Haynes, published by Getty Publications, available from https://www.amazon.co.uk/Etruscan-Civilisation-Cultural-Sybille-Haynes/dp/0892366001/
• From Zero to Infinity: What Makes Numbers Interesting, by Constance Reid, published by A K Peters/CRC Press, available from https://www.amazon.co.uk/Zero-Infinity-Makes-Numbers-Interesting/dp/1568812736
• Handbook to Life in Ancient Rome, by Lesley Adkins and Roy A. Adkins, published by Oxford University Press, available from https://www.amazon.co.uk/Handbook-Life-Ancient-Lesley-Adkins/dp/0195123328/
• Number Words and Number Symbols: A Cultural History of Numbers, by Karl Menninger, translated by Paul Broneer, published by Dover Publications Inc., available from https://www.amazon.co.uk/Number-Words-Symbols-Cultural-History/dp/0486270963/
• Roman Society and Roman Law in the New Testament, by A.N. Sherwin-White, available from https://www.amazon.co.uk/Roman-Society-Testament-Sherwin-White-1992-07-02/dp/B01K13GQT4/
• The Ancient Roman World, by Ronald Mellor and Marni McGee, published by OUP USA, available from https://www.amazon.co.uk/Ancient-Roman-Mellor-Ronald-Hardcover/dp/B00OL3UN9K/
• The Number Concept: Its Origin and Development, by Levi Leonard Conant, published by Legare Street Press, available from https://www.amazon.co.uk/Number-Concept-Its-Origin-Development/dp/1017628092/
• The Oxford Handbook of Latin Palaeography, by Frank T. Coulson and Robert G. Babcock, published by OUP USA, available from https://www.amazon.co.uk/Oxford-Handbook-Palaeography-OXFORD-HANDBOOKS/dp/0197689701/
• The Oxford Handbook of Roman Epigraphy, by Christer Bruun and Jonathan Edmondson, published by Oxford University Press, available from https://www.amazon.co.uk/Oxford-Handbook-Roman-Epigraphy/dp/0190860308/
• The Roman Way, by Edith Hamilton, published by W. W. Norton & Co., available from: https://www.amazon.co.uk/Roman-Way-Edith-Hamilton-ebook/dp/B06XPPQM34/
• The Romans: An Introduction, by Abigail Graham and Antony Kamm, published by Routledge, available from https://www.amazon.co.uk/Romans-Introduction-Peoples-Ancient-World-dp-1138543896/dp/1138543896/ref=dp_ob_title_bk

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End Notes and Explanations

1. Source: Compiled from my research using information available at the sources stated throughout the text, together with information provided by machine-generated artificial intelligence at: bing.com [chat] and https://chat.openai.com. Text used includes that on Wikipedia websites is available under the Creative Commons Attribution-ShareAlike License 4.0; additional terms may apply. By using those websites, I have agreed to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organisation. ↑
2. Further Information: The Fasti Antiates Maiores is a painted wall calendar from the late Roman Republic, the oldest archaeologically attested local Roman calendar and the only such calendar known from before the Julian calendar reforms. It was created between 84 and 55 BC and discovered in 1915 at Anzio (ancient Antium) in a crypt next to the coast. It is now located in the Palazzo Massimo alle Terme in Rome, part of the Museo Nazionale Romano. The Fasti Antiates Maiores consist of two fragments of the 13-month calendar and the List of Roman consuls. The 1.16 m high and 2.5 m wide calendar contains the leap month Mensis Intercalaris in addition to the twelve months. The calendar takes the form of a table with 13 columns, each of which is a month labelled with an abbreviation of its name. Still legible are the following: IAN for Ianuarius (January), FEB for Februarius (February), APR for Aprilis (April), IVN for Iunius (June), and SEP for September. Each column consists of several rows containing the days as well as a supplementary row indicating the total number of days in the month. Still visible are XXIIX (28) for February, XXIX (29) for April, June and partly for August, XXXI (31) (partial) for May and October.Sources: Wiseman, Timothy Peter (2004). The Myths of Rome. University of Exeter Press. p. 63. ISBN 978-0-85989-703-7, Sears, Gareth; Keegan, Peter; Laurence, Ray (18 July 2013). Written Space in the Latin West, 200 BC to AD 300. A&C Black. p. IX. ISBN 978-1-4411-8876-2, and Zautner, Andreas E. (16 February 2021). The Lunisolar Calendar of the Germanic Peoples: Reconstruction of a bound moon calendar from ancient, medieval and early modern sources. BoD – Books on Demand. p. 27. ISBN 978-3-7534-0723-4. and cited at: https://en.wikipedia.org/wiki/Fasti_Antiates_Maiore ↑
3. Further Information: The Arch of Titus (Italian: Arco di Tito; Latin: Arcus Tītī) is a 1^st century AD honorific arch. It is located on the Via Sacra, Rome, just to the south-east of the Roman Forum. It was constructed in c. 81 AD by Emperor Domitian shortly after the death of his older brother Titus to commemorate Titus’s official deification or consecratio and the victory of Titus together with their father, Vespasian, over the Jewish rebellion in Judaea. It became a symbol of the Jewish diaspora, and the menorah depicted on the arch served as the model for the menorah used as the emblem of the State of Israel. The arch has provided the general model for many triumphal arches erected since the 16th century. It is the inspiration for the Arc de Triomphe in Paris. Cited at: https://en.wikipedia.org/wiki/Arch_of_Titus ↑
4. Further Information: The Colosseum, also known as the Flavian Amphitheatre in Rome, has an interesting connection with Roman numerals. They were prominently used in the building’s design and organisation, especially in helping manage large crowds. The main ways Roman numerals were used in the Colosseum were:Gate Numbering for Organisation of Spectators – The Colosseum had 80 entrances, with 76 numbered arches for the general public to enter and exit. Above each of these entrance arches, Roman numerals were inscribed to indicate the entrance number, guiding spectators to their assigned sections. This system allowed for a controlled flow of up to 50,000 to 80,000 spectators. The Roman numerals provided a clear and organised way to help people find their seats, similar to how modern stadiums use numbered sections. For example, the main entrances at the northern and southern ends of the arena, used by the emperor and other VIPs, were marked with larger, more prominent Roman numerals.

   Internal Seating Arrangements – Inside the Colosseum, Roman numerals were also used to designate seating areas according to the social hierarchy. Spectators were seated according to their social rank, with the best seats closest to the arena reserved for senators, equestrians (knights), and wealthy citizens. Ordinary Roman citizens and lower classes were seated higher up. Roman numerals marked these sections, ensuring that people could quickly find their places based on their social status.

   In these ways, Roman numerals were essential for organising large crowds, maintaining order, and reinforcing the social hierarchy at events in the Colosseum. ↑

5. Further Information: The book Etruscan Civilization – A Cultural History by Sybille Haynes relates to Roman numerals because it explores the Etruscans’ influence on early Roman culture, including aspects of their numerical system. The Etruscans, who inhabited central Italy before the rise of Rome, used their own numbering system, which the Romans likely adapted and simplified. The Etruscans used symbols similar to Roman numerals, especially for key values like 1, 5, 10, 50, and 100. Archaeological evidence, such as Etruscan tomb inscriptions and artefacts, show that they represented numbers with repeated symbols, which may have inspired the Romans to adopt a similar additive system. The Etruscan symbol for five, for example, was likely the origin of the Roman “V,” and their symbol for ten influenced the Roman “X.”Haynes’ book provides context for understanding how Roman numerals may have evolved from Etruscan numerals, showing the cultural exchange between the two civilisations and the Etruscans’ role in shaping early Roman practices. ↑
6. Further Information: The book The Universal History of Numbers by Georges Ifrah is relevant to the study of Roman numerals because it traces the development of numeral systems across different cultures, including the Romans. Ifrah’s work provides a detailed look at how the Roman numeral system fits within the broader history of numbers, exploring its origins, structure, and limitations. In particular, Ifrah discusses how the Roman numeral system likely developed from earlier influences, including the Etruscan numeral system. He also explains how Roman numerals were effective for certain uses, like labelling and counting, but had significant limitations due to their lack of place value and zero. These limitations eventually led to the adoption of the more efficient Hindu-Arabic numeral system in Europe.Overall, Ifrah’s book places Roman numerals in the context of global numeral systems, showing how they served their purpose in Roman society but were ultimately replaced by more advanced systems better suited for complex calculations. ↑
7. Explanation: Etruria today corresponds to parts of central Italy, especially Tuscany, where the Etruscan heritage continues to be celebrated and studied. The Etruscans’ impact on Roman civilisation has left a lasting legacy in Italian culture, art, and history. ↑
8. Explanation: The Twelve Tables were the first written code of laws in ancient Rome, created around 450 BC. They were a set of laws that were publicly displayed on bronze tablets in the Roman Forum, so that all Roman citizens could know and refer to them. This was a significant step in establishing a fair and transparent legal system for the Roman people. The Twelve Tables were displayed on bronze tablets in the Roman Forum. The Twelve Tables covered fundamental aspects of civil law, criminal law, and property rights, laying the groundwork for Rome’s legal system and influencing Western legal traditions:Court Procedures, Trials and Appeals, Debt and Property Rights, Family Rights and Authority, Inheritance and Guardianship, Ownership and Possession, Land and Property Rights, Personal Injury and Damages, Public Law and Citizens’ Rights, Religious Practices and Burial Rights and Supplementary Laws and Prohibitions. For further detailed information and insights into the origins, content, and impact of the Twelve Tables. Good sources are Frier, Bruce W. The Rise of the Roman Jurists: Studies in Cicero’s Pro Caecina. Princeton University Press, 1985, or, for an accessible overview: Crawford, Michael (ed.). Roman Statutes, Volume I, Institute of Classical Studies, 1996. ↑
9. Explanation: Fibonacci is best known for a mathematical curiosity known as the Fibonacci sequence is a series of numbers in which each number after the first two is the sum of the two preceding ones, starting with 0 and 1. It goes as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. This sequence was introduced to the Western world by the Italian mathematician Leonardo of Pisa, known as Fibonacci, in his 1202 book Liber Abaci (The Book of Calculation), although the sequence itself had been previously known in Indian mathematics. Each number in the sequence can be represented as F(n) = F(n-1) + F(n-2), where F(n) is the nth Fibonacci number, with F(0) = 0 and F(1) = 1. This recursive pattern creates a sequence that grows exponentially, often seen in various aspects of nature, such as the arrangement of leaves, the branching of trees, the spiral patterns of shells, and even the breeding patterns of certain animals.An interesting feature of the Fibonacci sequence is its connection to the golden ratio, an irrational number approximately equal to 1.6180339887… As you progress through the sequence, the ratio of successive Fibonacci numbers (such as 13/8 or 21/13) approaches the golden ratio. This relationship highlights the sequence’s unique mathematical beauty and why it often appears in nature, art, and architecture.

   The Fibonacci sequence remains relevant in modern mathematics and computer science, with applications in algorithms, data structures, and financial markets, where it appears in technical analysis models. It is celebrated for its elegance, versatility, and ability to represent growth and patterns across different disciplines naturally. The Fibonacci sequence doesn’t have a direct connection to Roman numerals, as Roman numerals are not structured in a way that naturally supports or expresses the Fibonacci pattern. Roman numerals are primarily additive and subtractive symbols, such as I, V, X, L, C, D, and M, representing 1, 5, 10, 50, 100, 500, and 1,000. They lack a place-value system, making it challenging to represent a sequence like Fibonacci’s in a straightforward or efficient way. However, the Fibonacci sequence and Roman numerals can intersect in historical studies, where Fibonacci’s introduction of the sequence to the West through Liber Abaci also promoted the adoption of Hindu-Arabic numerals over the traditional Roman system. This book showed the utility of positional decimal systems for complex calculations, which ultimately helped phase out Roman numerals in Europe for most practical mathematics. ↑

10. Explanation: The Islamic Golden Age was a period of cultural, scientific, and intellectual flourishing in the Islamic world, roughly from the 8^th to the 14^th century. It began during the Abbasid Caliphate, particularly under Caliph Harun al-Rashid and his successors. Scholars in fields like mathematics, medicine, astronomy, philosophy, and literature made groundbreaking advancements, often building on knowledge from Greek, Persian, Indian, and other ancient civilisations. This period saw the development of algebra, advancements in medicine and optics, and significant contributions to philosophy and literature. The knowledge produced during this era was later translated into Latin and contributed to the European Renaissance. ↑
11. Explanation: The House of Wisdom (Bayt al-Hikma) was a major intellectual centre in Baghdad during the Islamic Golden Age. Founded in the 9^th century by Caliph al-Ma’mun, it served as a library, translation institute, and research centre where scholars from diverse backgrounds worked together. They translated important texts from Greek, Persian, and Sanskrit into Arabic, preserving and expanding knowledge in fields such as mathematics, astronomy, medicine, and philosophy. The House of Wisdom was instrumental in advancing scientific and philosophical thought, and its scholars helped shape the intellectual legacy of the Islamic Golden Age. ↑
12. Explanation: The Hindu-Arabic numeral system is a decimal (base ten) positional numeral system that is the most common system for the symbolic representation of numbers in the world today. It originated in India around the 6^th century AD and was later transmitted to the Islamic world, and then to Europe. The system was revolutionary because it included a symbol for zero and its positional notation system, which allowed for the concise representation of large and small numbers. This greatly simplified arithmetic operations, including multiplication and division. The system uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, which are combined to form larger numbers. The value of the digit depends on its position, with each position representing a power of ten. This system made calculations easier and more accurate, and it was crucial for the development of mathematics, science, and commerce. It eventually replaced other numeral systems in Europe and worldwide due to its efficiency, particularly after its adaptation in the work of Fibonacci, who introduced it to the Western world through his book Liber Abaci in 1202. ↑