## Number Systems

So far, we have worked with the Base $10$ number system. In this lesson, we will learn about other number systems.

### 1. Roman Numerals

This numeric system originated in ancient Rome over $2000$ years ago. Instead of using separate symbols (like in our decimal system) to represent numbers, they utilised the following letters of the Latin alphabet to represent these values.

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

Numbers were expressed by writing these symbols together. The following algorithm/rules are applied to compute the value of the number.
1) The value of a symbol is subtracted, if it is written to the left of a higher-valued symbol.
2) If not, the value of the symbol is added.

For example,
VIII $= 5 + 1 + 1 + 1 = 8$
IX $= -1 + 10 = 9$
XLIV $= -10 + 50 -1 + 5 = 44$
CDLVII $= -100 + 500 + 50 + 5 + 1 + 1 = 457$

Note that $49$ is written as XLIX and not IL. This is because of the following additional rules.
1) I can be written before V or X only
2) X can be written before L or C only.
3) C can be written before D or M only.

In Roman Numerals, $0$ or negative numbers cannot be written and decimals cannot be represented. Large numbers are difficult to write. For instance, the distance between sun and the earth is written as $150,000,000$ km in the decimal system. To write this in Roman Numerals, the symbol M has to be written $150,000$ times.

Upon the advent of the decimal system, Roman Numerals for mathematical operations ceased. Today we use Roman Numerals to indicate smaller numbers, such as the hours in a clock, the class in school, movie sequel numbers, book volumes, etc.

### Example 1

Which of the following equals the sum of the roman numbers CDLVI and CCXXXIX? ($1$) CDXCVI           ($2$) DCCXV           ($3$) DCXCV           ($4$) DCCXXI

### Solution

CDLVI $= -100 + 500 + 50 + 5 + 1 = 456$
CCXXXIX $= 100 + 100 + 10 + 10 + 10 - 1 + 10 = 239$

$456 + 239 = \textbf{695}$

We do not consider options $1$ and $4$ as their last digits are $6$ and $1$ respectively.

Option 2: DCCXV $= 500 + 100 + 100 + 10 + 5 = 715$

Option 3: DCXCV $= 500 + 100 - 10 + 100 + 5 = \textbf{695}$