Roman Numerals - Rules for Forming Roman Numerals

The Roman numeral system is built upon seven fundamental symbols: $I = 1$, $V = 5$, $X = 10$, $L = 50$, $C = 100$, $D = 500$, and $M = 1000$. There is no symbol for zero, which distinguishes it from our decimal system.

Rule of Repetition: When symbols $I, X, C,$ or $M$ are repeated, their values are added (e.g., $II = 2$, $XXX = 30$). A symbol can be repeated up to a maximum of three times. Symbols $V, L,$ and $D$ are unique because they are never repeated.

Rule of Addition: If a symbol of lower value is placed to the right of a symbol of higher value, the values are added together. For example, in $LX$, the $X$ ($10$) is after $L$ ($50$), resulting in a sum of $50 + 10 = 60$.

Rule of Subtraction: If a symbol of lower value is placed to the left of a symbol of higher value, the lower value is subtracted from the higher value. Visually, this 'prefixing' indicates a reduction, such as $XC$ representing $100 - 10 = 90$. Note that $V, L,$ and $D$ are never subtracted.

Subtraction Limitations: To maintain consistency, $I$ can only be subtracted from $V$ and $X$; $X$ can only be subtracted from $L$ and $C$; and $C$ can only be subtracted from $D$ and $M$. Subtractions like $LC$ or $IC$ are not mathematically valid in this system.

Small Symbol Between Two Larger Symbols: When a symbol of smaller value is placed between two symbols of greater value, it is always subtracted from the symbol immediately to its right. For example, in $XIX$, the $I$ is between two $X$s, so it is calculated as $10 + (10 - 1) = 19$.

Vinculum (Bar Notation): For representing numbers greater than $3,999$, a horizontal line called a vinculum is drawn over a Roman numeral to multiply its value by $1000$. A symbol like $\bar{V}$ signifies $5,000$, and $\bar{X}$ signifies $10,000$.