Roman Numerals: Exploring the Numerals for 4, 6, 9 and Beyond

As a full-stack developer and professional coder, I‘ve always been fascinated by different numeral systems used throughout history. Roman numerals, in particular, stand out as an intriguing example of a system that was widely used for centuries but has since been largely replaced by the Hindu-Arabic numerals we use today.

In this article, we‘ll take an in-depth look at Roman numerals, with a special focus on the unique way they represent the numbers 4, 6, and 9. We‘ll explore the origins and evolution of this ancient numeral system, break down the rules and patterns, and even try our hand at some arithmetic.

What are Roman Numerals?

Roman numerals are a numeral system that originated in ancient Rome around the 1st century BC. The system uses combinations of letters from the Latin alphabet to represent values.

The seven basic symbols used in Roman numerals are:

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

The numerals are read from left to right, starting with the symbol representing the largest value. Symbols are additively combined to represent larger numbers. For example:

• II = 2 (1 + 1)
• VIII = 8 (5 + 1 + 1 + 1)
• XIII = 13 (10 + 1 + 1 + 1)
• XXVII = 27 (10 + 10 + 5 + 1 + 1)

One notable quirk of Roman numerals is the lack of a symbol for zero. The concept of zero as a number was not yet developed in Europe during the Roman era.

Subtractive Notation for 4 and 9

The numerals for 4, 40, 400, and 9, 90, 900 are where things get interesting. For these values, the system uses subtractive notation, placing a smaller value symbol before a larger one to indicate subtraction.

Here are the specific numerals and their subtractive combinations:

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

You can think of the notation as indicating "one less" than the next greater symbol. So IV is "one less than 5" and CM is "one hundred less than 1000."

The subtractive principle influenced the way other Roman numerals are constructed as well:

• VI = 6 (5 + 1, or "5 and 1")
• VII = 7 (5 + 2, or "5 and 2")
• VIII = 8 (5 + 3, or "5 and 3")
• XI = 11 (10 + 1, or "10 and 1")
• XII = 12 (10 + 2, or "10 and 2")
• XIII = 13 (10 + 3, or "10 and 3")

Complete Table of Roman Numerals to 1000

Here is a full table showing the Roman numerals for each value from 1 to 1000:

Hindu-Arabic	Roman	Hindu-Arabic	Roman	Hindu-Arabic	Roman
1	I	60	LX	500	D
2	II	70	LXX	600	DC
3	III	80	LXXX	700	DCC
4	IV	90	XC	800	DCCC
5	V	100	C	900	CM
6	VI	200	CC	1000	M
7	VII	300	CCC
8	VIII	400	CD
9	IX	500	D
10	X	600	DC
20	XX	700	DCC
30	XXX	800	DCCC
40	XL	900	CM
50	L	1000	M

From this table, a few key patterns emerge:

1. Powers of 10 up to 1000 use the symbols: I, X, C, M
2. Half of each power of 10 uses the next symbol: V, L, D
3. Symbols are additive up to 3 occurrences: III, XXX, CCC
4. Subtractive notation is used for 4 and 9: IV, IX, XL, XC, CD, CM

It‘s important to note that the subtractive forms like IV and IX can only be used in those specific cases. You would never write 99 as "IC" for example. The correct form is XCIX.

Representing Large Numbers

For numbers larger than 3,999, the Romans used a vinculum, or "overline", to indicate multiplication by 1000. Essentially, a line drawn above a numeral increases its value 1000-fold.

Here are the vinculum numerals up to 1 million:

Vinculum	Hindu-Arabic Value
V	5,000
X	10,000
L	50,000
C	100,000
D	500,000
M	1,000,000

These can be combined with the other numerals to create even larger values. For example:

• MMMCMXCIX = 3,999 (the largest value possible without vinculum)
• IV = 4,000
• MCDXLIV = 1,444
• CCCXXV = 325,000
• DCCLXVMMM = 765,000
• MMCDLXXXVIIICCCXLIX = 2,487,349

In theory, the vinculum allows for numbers up to 900,000,000 to be represented like so:

• DCCCCMMMMMMMMMMMMMM = 900,000,000

However, in actual historical practice, it was very rare for numbers beyond the millions to be written out.

Converting Years and Other Numbers

To convert a number like a year into Roman numerals, break it down by place value and translate each part according to the tables above.

For example, let‘s convert the year 1776 to Roman numerals:

1776 = 1000 + 700 + 70 + 6
M + DCC + LXX + VI
MDCCLXXVI

A few more examples:

• 2023 = MMXXIII
• 1492 = MCDXCII
• 476 = CDLXXVI
• 1999 = MCMXCIX
• 2520 = MMDXX

Basic Arithmetic with Roman Numerals

Addition

Adding Roman numerals is straightforward. Combine the numerals and simplify from left to right, making sure to use subtractive notation when needed.

Examples:

• XIV + VI = XX
• XXIX + XI = XL
• CCXLVI + DCXIX = DCCCLXV
• MMMDCCXXIV + CCLXXXVI = MMMMX

Subtraction

To subtract, remove the symbols of the smaller number from left to right.

Examples:

• XC – X = LXXX
• MM – XIV = MCMLXXXVI
• MCD – LXXIX = MCCCXXI

Multiplication and Division

Multiplication and division with Roman numerals is possible but cumbersome. The Romans preferred using an abacus for complex calculations.

To multiply, add the first number to itself the second number of times. For example:

XII × IV = XII + XII + XII + XII = XLVIII

Division works by repeatedly subtracting the divisor from the dividend until no remainder is left. The number of subtractions is the quotient.

LXXII ÷ XII = VI, because:

• LXXII – XII = LX
• LX – XII = XLVIII
• XLVIII – XII = XXXVI
• XXXVI – XII = XXIV
• XXIV – XII = XII
• XII – XII = 0

As you can see, we subtracted XII six times to reach zero, so LXXII ÷ XII = VI.

Needless to say, multiplication and division by larger values quickly becomes unwieldy and impractical with Roman numerals. This limited their usefulness for complex mathematics and commerce.

Interesting Facts and Quirks

• The largest number ever used in an official Roman document is MMMCMXCIX (3,999).
• Most clock faces use IIII for 4 o‘clock instead of IV. This is to maintain visual symmetry and avoid confusion at a quick glance.
• The year of construction on cornerstones and monuments is often written in Roman numerals. This can make old buildings hard to date!
• Roman numerals are used in numbered lists as an alternative to Arabic numbers or letters.
• The world‘s largest Roman numeral is on the Colosseum in Rome. It says "MMDCCLIV" which means 2754.

Legacy and Modern Usage

Despite being long outdated as a practical numeral system, Roman numerals still appear in a variety of modern contexts:

• Page numbers for prefaces, forewords, and introductions in books
• Copyright dates and production years for films, television, and video games
• Generational suffixes for monarchs and popes, like Queen Elizabeth II or Pope Benedict XVI
• Quarter and semester numbers in school systems
• Sequential labels for major sporting and cultural events like the Olympics or Super Bowl
• Naming conventions for spacecraft and satellites

For us as developers, working with Roman numerals can feel a bit like dealing with a legacy code base. The system can seem arcane and counter-intuitive from a modern perspective. But by understanding its rules and patterns, we can learn to navigate and even appreciate its unique characteristics.

In the end, Roman numerals are a tangible link to computing‘s ancient past. As coding and mathematics continue to evolve, it‘s both grounding and humbling to remember the primitive systems that preceded our current tools and knowledge.

Though the Romans would surely be baffled by the complexity and scale of information we work with today, one likes to think they would recognize something of their own ingenuity in the digital systems we now use to understand and order the universe.

I hope you‘ve enjoyed this deep dive into the world of Roman numerals. The next time you see an I, V, or X marking the credits of a film or the cover of a book, take a moment to appreciate the history and legacy of this fascinating numeral system.