Question

Write 14 in roman numeral form.\[\]
A. IV\[\]
B.XIV\[\]
C.XVI\[\]
D.XV\[\]

Answer 1

Hint: We use the Roman numeral conversion that $d\left( AB \right)=d\left( A \right)+d\left( B \right),\text{if }d\left( A \right)\ge d\left( B \right)$ and $d\left( AB \right)=d\left( B \right)-d\left( A \right),\text{if }d\left( A \right) < d\left( B \right)$ where $A$ and $B$ are symbols from roman numerals. We first find $d\left( A \right)$ where $A$ is the Roman numeral with highest decimal equivalent and then the next largest $d\left( B \right)$ the decimal equivalent of $B$. We add or subtract decimal values accordingly as $B$ is on left or on right of $A$.\[\]

Complete step-by-step solution
We know that roman numerals have symbols I, V, X, L etc. The decimal equivalents of the roman numerals are
\[I=1,V=5,X=10,L=50\]
Whenever a number is represented in roman numerals with more than one symbol say $A$ and $B$ where $A,B$ roman numerals are. Let us denote the decimal equivalent of $A$ be $d\left( A \right)$ and the decimal equivalent of $B$ be $d\left( B \right)$ and. Then the decimal equivalent of $AB$ be $d\left( AB \right)$. We have
\[\begin{align}
  & d\left( AB \right)=d\left( A \right)+d\left( B \right),\text{if }d\left( A \right)\ge d\left( B \right) \\
 & d\left( AB \right)=d\left( B \right)-d\left( A \right),\text{if }d\left( A \right) < d\left( B \right) \\
\end{align}\]
We can extend it for more than two symbols and we compare inequalities with numerals having the largest decimal equivalent.

A. Here the Roman numeral is IV where the decimal equivalent of V is $d\left( V \right)=5$ and the decimal equivalent of I is $d\left( I \right)=1$.Here we have $d\left( V \right)>d\left( I \right)$ and I is on the left of V. we have the decimal equivalent of IV as
\[d\left( IV \right)=d\left( V \right)-d\left( I \right)=5-1=4\]
So we have $d\left( XI \right)\ne 14$ and option A is not correct. Let us check option B.
B. Here the Roman numeral is XIV. We see that decimal equivalent of all symbols after the first symbol X is less than decimal equivalent of first symbol X and then the largest symbol is V and on the left side of I lies V. So we have
\[\begin{align}
  & d\left( XIV \right)=d\left( X \right)+\left( -d\left( I \right)+d\left( V \right) \right) \\
 & \Rightarrow d\left( XXXII \right)=10+\left( 5-1 \right)=10+4=14 \\
\end{align}\]
So option B is correct and as numbers are uniquely represented in Roman numeral we do not need to check option C and D.

Note: We can alternatively by using the algorithm for decimal conversion where in the first step find the highest decimal value $v$ is less than or equal to the given decimal number $x$ and write its corresponding numeral symbol $A$ and in second step $x=x-v$. We repeat step-1 and step-2 until we get $x=0$. We note that 0 does not exist in the Roman numeral.