Rules for Writing Roman Numerals

Introduction

This page reviews the rules for forming Roman numerals.

The Basic Numerals

In order to learn the rules of forming Roman numerals, we need to identify the seven basic numerals:

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

It is best to think of Roman numerals in terms of groups of letters. These groups of letters should be equated to writing decimal numbers in expanded form. For example, the number 1,234 can be written in expanded form as 1000+200+30+4. Thinking about Roman numerals in this way will make it much easier to convert between the decimal and Roman systems.

Outside of the seven basic numerals, there are only 23 groupings of numerals, that when combined properly, can form all the Roman numerals. The 23 groupings and 7 basic numerals are listed in the table below. By inspecting the table, we can make observations that will lead us to deriving the rules of writing decimal numbers in Roman numeral form.

The Basic Rules

There are three basic rules for forming Roman numerals. We will review them by making observations from the table above.

Observation 1

In any given grouping of numerals, notice that only the M, C, X, and I terms are repeated more than one time, but no more than three times. This leads us to our first rule.

Rule 1 – Repetition

The M, C, X, and I numerals are the only numerals that can be repeated, consecutively, up to three times.

Observation 2

In any grouping of numerals, if a larger numeral is followed by a smaller numeral, or series of smaller numerals, the individual numerals are added. This leads us to our second rule, the addition rule.

Rule 2 – Addition

If a larger numeral is followed by a smaller numeral, or series of smaller numerals (from left to right), add the numerals.

Observation 3

Notice that the C, X, and I numerals are the only numerals that precede any other numerals. This observations leads us to Rule 3, the subtraction rule.

Rule 3 – Subtraction

The subtraction rule states the following:

• C, X, and I are the only numerals that can, individually, precede another numeral.
  • C can only be placed before M or D.
  • X can only be placed before C or L.
  • I can only be placed before X and V.

The Forming Rules

Our final set of rules are based on the forming of Roman numerals outside of what is in the table above. The forming rules will allow us to write all Roman numerals, or at least the ones between 1 and 3999.

Rule 4 – Order

All numerals must be written in descending order, from left to right, by place value, e.g., by thousands, hundreds, tens, and ones.

Rule 5 – Concatenating

Only one entry of each column from the table can be used to form a Roman numeral. Refer to the example below.

• 3,456 is written in Roman numerals by executing the following operations:
  1. Write the decimal number in expanded form:
     • 3,456 = 3000+400+50+6
  2. Convert each term of the expanded decimal number into a Roman numeral.
     • 3000 = MMM
     • 400 = CD
     • 50 = L
     • 6 = VI
  3. Concatenate the terms in descending order, from left to right.
     1. 3,456 = MMMCDLVI
     2. 1004 = MIV
     3. 813 = DCCCXIII
     4. 91 = XCI

Final Observations

Below are some final observations to keep in mind as you begin writing Roman numerals.

• Notice that D, L, and V are only ever used once in a numeral.
• The largest Roman numeral that can be formed with the basic numerals is 3999 = MMMCMXCIX.
• Anything that falls outside of these rules are considered invalid numbers.

Algorithms for Converting Between Systems

MATLAB and C++ algorithms have been developed to simplify the processes of converting between number systems.

Roman Numeral to Decimal Conversion

https://eemaginations.com/roman-numeral-to-decimal-conversion/

Decimal to Roman Numeral Conversion

https://eemaginations.com/decimal-to-roman-numeral-conversion/