# Roman numerals

Roman numerals are a numeral system that originated in ancient Rome. They are still used today on clock faces, as book chapters, for numbering events or sequels, in the names of monarchs (e.g. Queen Elizabeth II) and popes, to indicate names across generations (e.g. Michael Smith IV), and more.

## What are Roman numerals

Roman numerals are a numeral system that use letters as symbols, each of which represents a certain value.

### Roman numeral symbols

The symbols used in the Roman numeral system are as follows:

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

Like any numeral system, Roman numerals are guided by a set of rules that enable us to express all manner of numbers using a given set of symbols.

## Roman numeral rules

- When consecutive numerals are equal, or the left-most numeral is greater than that of the following numerals (e.g. XX, VIII), their values are added.
- When a numeral of lesser value is followed by a numeral of greater value (e.g. IV), the lesser value is subtracted from the greater one.
- A bar written over a numeral (called a vinculum) indicates multiplication by 1000. For example, M = 1000 × 1000 = 1,000,000.
- No more than 3 of the same numeral can be written consecutively. For example, IIII does not exist. Instead, the subtractive principle (described below) is used.
- Numerals V, L, and D are never repeated; they also are never written before a larger symbol (so are never subtracted).
- The numeral I is only ever written before V or X, meaning that these are the only numerals it can be subtracted from. Similarly, X is only ever written before L, C, or M.

Roman numerals use the subtractive principle, where writing a letter with a smaller value before one with a larger value indicates that the smaller value is subtracted from the larger one (Rule 4). This rule simplifies the process of reading and writing Roman numerals; the more of the same letter we have in a row, the more difficult it becomes to quickly read how many there are. For example, consider if 8 were written as IIIIIIII rather than VIII. Notice how much easier it is to identify that there are 3 "I" compared to the relative difficulty of identifying 8 "I".

The subtractive principle allows us to ensure that we don't use more than 3 of the same letters in a row. Using the number 4 as an example, we know from above that the value of V is 5 and the value of I is 1, so writing IV means 5 - 1 = 4.

## How to read Roman numerals

To read Roman numerals, start reading from left to right and use the list and rules above. If all consecutive numerals are the same or smaller than the left-most numeral, simply add the values of each numeral. If a larger numeral is preceded by a smaller one, subtract the smaller numeral from the larger one. Using these rules, find the sum of the collection of numerals; this is the value of the Roman numeral. Refer to the examples of reading Roman numerals below.

Examples

Find the values (in the decimal system) of the Roman numerals below:

1: DXXXII

Referencing the list above, each consecutive numeral is equal or smaller, so we can just sum the individual values of all the numerals:

D (500) + XXX (30) + II (2) = 532

Note that we could break the Roman numeral up more if we wanted, like counting each individual X as 10 rather than grouping the 3 to form 30, but grouping in the way we did above makes reading Roman numerals more efficient.

2: CMXCIX

This example uses the subtractive principle for each term:

CM (1000 - 100) + XC (100 - 10) + IX (10 - 1) = 900 + 90 + 9 = 999

Since each preceding numeral in each term is less than the following numeral, we subtract the smaller numeral from the larger one. When reading from left to right, we must always separate terms where the following term is larger than the preceding one in this way to be able to determine their values. For example, we cannot break it up as CMX + CIX. If we were to do this, we would be left with CMX (900 + 10) + CIX (100 + 9) = 1019, which is incorrect.

The key takeaway from the examples above is to pay attention to the relative values of each numeral in the Roman numeral you are trying to read. The simplest case is when every subsequent numeral is smaller than the left-most numeral; in this case, just add the value of each numeral. The sum is the value of the Roman numeral. Otherwise, we need to pay attention to numerals where the preceding numeral is smaller than the subsequent one; in these cases, as we read from left to right, we need to subtract smaller numerals from larger subsequent numerals before considering any numerals further to the right. In example 2 above, every term required us to subtract, but consider the example of 989 instead. The Roman numeral for 989 is CMLXXXIX. We would separate this Roman numeral as:

CM (1000 - 100) + LXXX (50 + 30) + IX (10 - 1) = 989

In this case, we needed to subtract numerals, add numerals, then subtract again. The trick to reading Roman numerals is understanding the rules so that we can appropriately separate the terms while summing them. Below is one more example that makes use of the vinculum, which is a bar written over a numeral to indicate multiplication by 1000. For example, X = 10 × 1,000 = 10,000. The vinculum adds one more level of complication to Roman numerals, but is overall relatively straightforward if we understand the rules.

Example

Find the value of XIICCCXLV.

We read XII as 12 × 1000, which is 12000, and add CCCXLV = 300 + 45 = 345. The sum is therefore 12,345. Note that for larger numbers there is not an exact convention, since the Romans did not really use numbers larger than 1000 at the time. The rules above still stay the the same however. Instead of writing the Roman numeral as we did above, we could have also written:

12,345 = XIICCCXLV = XMMCCCXLV.

### How to write Roman numerals

To write Roman numerals, think of each number in expanded notation, where we write a number as the sum of the value of each of its digits. For example, the number 1,987 in expanded notation is 1000 + 900 + 80 + 7. Then, we just use the rules above to write each of these values from left to right. In this case, we know that M = 1000, 900 is written as 1000 - 100 = CM, 80 is written as LXXX, and 7 is written as VII, so 1,987 in Roman numerals is:

1,987 = MCMLXXXVII

For larger numbers, we need to use the vinculum, while noting that there is not an exact convention, since the Romans did not really use numbers much larger than 1000. For example, to write 19,087, first convert it to expanded form:

19000 + 80 + 7

Then, using the vinculum, we would write this in Roman numerals as:

XIXLXXXVII

The above in expanded form is: 19 × 1000 + 80 + 7 = 19,087

## How to write large Roman numerals

For large Roman numerals, we use a vinculum, which is a bar written over the numeral that indicates multiplication by 1,000, similar to the "k" used when expressing thousands (e.g. 5k = 5,000).

- I = 1,000
- V = 5,000
- X = 10,000
- L = 50,000
- C = 100,000
- D = 500,000
- M = 1000,000

## Roman numerals chart

Below is a chart of showing various Roman numerals.

Decimal system | Roman numerals |
---|---|

1 | I |

2 | II |

3 | III |

4 | IV |

5 | V |

6 | VI |

7 | VII |

8 | VIII |

9 | IX |

10 | X |

11 | XI |

12 | XII |

13 | XIII |

14 | XIV |

15 | XV |

16 | XVI |

17 | XVII |

18 | XVIII |

19 | XIX |

20 | XX |

30 | XXX |

40 | XL |

50 | L |

60 | LX |

70 | LXX |

80 | LXXX |

90 | XC |

100 | C |

200 | CC |

300 | CCC |

400 | CD |

500 | D |

600 | DC |

700 | DCC |

888 | DCCCLXXXVIII |

900 | CM |

1,000 | M |

2,200 | MMCC |

4,444 | IVCDXLIV |

9,000 | MX |

80,000 | LXXX |

100,000 | C |

1,536,424 | MDXXXVICDXXIV |

## Years in Roman numerals

The table below shows some years in Roman numerals:

476 | CDLXXVI |
---|---|

730 | DCCXXX |

1215 | MCCXV |

1455 | MCDLV |

1492 | MCDXCII |

1687 | MDCLXXXVII |

1776 | MDCCLXXVI |

1789 | MDCCLXXXIX |

1859 | MDCCCLIX |

1893 | MDCCCXCIII |

1905 | MCMV |

1918 | MCMXVIII |

1945 | MCMXLV |

1959 | MCMLIX |

2001 | MMI |

2008 | MMVIII |

2019 | MMXIX |

2023 | MMXXIII |