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

## CIX + CV + CCCXL = ?

#### The Romans did not have the Hindu-Arabic numbers. So we will solve out this operation exactly the way the Romans were calculating it, without using the Hindu-Arabic numbers.

### 1. Replace the groups written in subtractive notation.

#### Replace any groups written in subtractive notation in the roman numerals; that is, "uncompact" them using only the additive notation.

#### A group in subtractive notation = a group of two numerals, one of a lower value preceding another of larger value, 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 of the group subtract the value of the first symbol from the value of the second.

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

#### CIX:

#### IX = X - I = V IIIII - I = V IIII = VIIII

#### CIX = CVIIII

#### CCCXL:

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

#### CCCXL = CCCXXXX

### 2. Put the Roman numerals together.

#### CVIIII +

#### CV +

#### CCCXXXX =

#### CVIIIICVCCCXXXX

### 3. Sort the symbols in descending order of their values.

#### Sort the symbols in descending order of their values, from left-to-right, with the largest symbol to the left, down to the smallest to the right.

#### CVIIIICVCCCXXXX =

#### CCCCCXXXXVVIIII

### 4. Combine the repeating symbols together, in groups.

#### Starting on the right end (smaller values), combine groups of the same symbols into larger ones.

#### CCCCCXXXXVVIIII =

#### CCCCCXXXXXIIII

#### CCCCCXXXXXIIII =

#### CCCCCLIIII

#### CCCCCLIIII =

#### DLIIII

### 5. Rewrite the repeating symbols.

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

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

#### DLIIII =

#### DLIV

## The final answer: