#### Let's learn by an example:

## DCCCLXVIII - CXX - CCLVI - XL - X = ?

#### 868 - 120 - 256 - 40 - 10 = ? the Romans did not have the Arabic numbers

### 1. Simplify the operation right away by crossing out common symbols in the numerals of different signs.

#### Matching numerals / symbols are colored the same.

#### DCCCLXVIII - CXX - CCLVI - XL - X =

#### DII - X - XL - X

#### For example, the numerals XI and IV:

#### XI is an additive group - largest symbol on the left, down to the smallest on the right - to calculate the value add up the symbols: XI = X + I = 10 + 1 = 11;

#### IV is a subtractive group - a smaller symbol precedes a larger one - to calculate the value subtract the first symbol from the second: IV = V - I = 5 - 1 = 4.

#### When subtracting the two numerals, XI - IV = (uncompact the subtractive) XI - IIII = X - III = VIIIII - III = VII (= 7), CORRECT. But if you cross out the common symbol regardless of the fact that it is part of a subtractive group: XI - IV = X - V = VV - V = V (= 5), WRONG.

### 2. Substitute for any subtractives in the roman numerals; that is, "uncompact" them using only the additive notation.

#### Subtractive notation = a group of two numerals, one of a lower value preceding another larger one, the only allowed ones are: IV, IX, XL, XC, CD, CM, M(V), M(X), (X)(L), (X)(C), (C)(D), (C)(M) - to calculate the value subtract the first symbol from the second.

#### Additive notation = a group of two or more numerals, equal or sorted in descending order from high to low, largest symbol on the left, down to the smallest on the right - to calculate the value add up the symbols.

#### XL:

#### XL = L - X = XXXXX - X = XXXX

#### XL = XXXX

### 3. Put the Roman numerals together, catenate them.

#### Catenate all the negative numerals together.

#### X +

#### XXXX +

#### X =

#### XXXXXX

### 4. Starting with the right end (smaller values), combine groups of the same symbols that can make a "larger" one and substitute the single larger one.

#### Combine the repetitive groups of symbols in the negative numeral.

#### XXXXXX =

#### XXXXX X =

#### LX

### 5. Using only the additive notation, convert larger value symbols to some smaller ones, a necessary step in order to cross out all the common symbols.

#### Replacements used, in additive notation:

#### D = CCCCLL; L = XXXXX;

#### DII -

#### LX =

#### CCCCLXXXXXII -

#### LX =

#### CCCCXXXXII

### 6. Rewrite the symbols written in excessive additive notation by using the substractive notation.

#### The numeral C should not repeat itself more than 3 times in a row, rewrite:

#### CCCCXXXXII =

#### CDXXXXII

#### The numeral X should not repeat itself more than 3 times in a row, rewrite:

#### CDXXXXII =

#### CDXLII

## Final answer: