Roman numerals

Roman numerals are a number system that was developed in ancient Rome. They were used throughout Europe for more than 2000 years until they were replaced by the Arabic numerals we use today.

Roman numerals are elegant but difficult to calculate with, since the system is not based on place value and has no zero.

The seven symbols

Roman numerals are composed of 7 letters, each with a specific value:

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

The largest number that can normally be written with standard rules is:

$$ MMMCMXCIX = 3999 $$

Rules for use

If a smaller number is written before a larger one, it must be subtracted, e.g. $IV = 5-1=4$.
If a smaller number is written after a larger one, it must be added, e.g. $VI = 5+1=6$.
The largest numbers should stand to the left, except when subtraction is applied.
$I, X, C$ and $M$ can be repeated up to three times in a row, e.g. $III = 3$ or $XXX = 30$.
$I, X$ and $C$ can only be subtracted once and only in specific combinations:
- $I$ only before $V$ and $X$ → $IV=4, IX=9$.
- $X$ only before $L$ and $C$ → $XL=40, XC=90$.
- $C$ only before $D$ and $M$ → $CD=400, CM=900$.
$V, L$ and $D$ must not be repeated.

Extension to large numbers

If there is a bar over a symbol, it means the value is multiplied by 1000.

$$ \overline{V} = 5000, \quad \overline{X} = 10000, \quad \overline{M} = 1000000 $$

This makes it possible to write numbers larger than 3999, for example:

$$ \overline{IV} = 4000, \quad \overline{V}CMXCIX = 5999 $$

Examples

$III = 3$
$IV = 4$ → $I$ stands to the left of $V$, so 1 is subtracted from 5.
$XII = 12$ → $II$ stands to the right of $X$, so 2 is added to 10.
$XL = 40$ → $X$ before $L$, so 10 is subtracted from 50.
$XC = 90$ → $X$ before $C$, so 10 is subtracted from 100.
$CM = 900$ → $C$ before $M$, so 100 is subtracted from 1000.

A year in Roman numerals

Let us translate 2016:

$MMXVI = 1000+1000+10+5+1 = 2016$

Another example: $MCMLXXXIX = 1989$

$M = 1000$
$CM = 900$
$LXXX = 80$
$IX = 9$

Together $1000+900+80+9=1989$.

Modern use of Roman numerals

Clocks – often with “IIII” instead of “IV” on dials
Kings and popes, e.g. Christian IV or John Paul II
Books and films in series, e.g. Star Wars Episode IV
Buildings and monuments, often with years carved in stone

Summary

Roman numerals are based on 7 symbols and a few simple rules.

The system has no zero and no place value, which makes it impractical for calculation.

On the other hand, it is a beautiful and historically important number system that is still used today in special contexts.