LXXV + DCCLVIII + MMCMXLVII - MMCCCXLVI =

LXX + DLVII + CMVII - C

LXX + DLVII + CMVII - C

the symbols in the subtractive and additive groups:

for example ...

75 + 758 + 2,947 - 2,346 = 1,434

- Let's learn with an example:
#### XC + MCMLIII + CCXIX - MCCXLVIII - XC = ?

- 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.

- The matching numerals / symbols are colored the same.
- XC + MCMLIII + CCXIX - MCCXLVIII - XC = MCMLIII + CCXIX - MCCXLVIII;

- The matching numerals / symbols are colored the same.
- MCMLIII + CCXIX - MCCXLVIII = CML + XIX - XLV;

do not mix the symbols in subtractive and additive groups.

- For example, the numerals XI and IV:
- XI is a group in additive notation - the largest symbol is to the left, down to the smallest to the right - to calculate the value of the group simply add up the values of the symbols in the group: XI = X + I = 10 + 1 = 11
- IV is a group in subtractive notation - a smaller symbol precedes a larger one - to calculate the value of the group subtract the value of the first symbol from the value of 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.

- Replace any groups 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.
- CML: CM = M - C = D CCCCC - C = D CCCC = DCCCC; CML = DCCCCL;
- XIX: IX = X - I = V IIIII - I = V IIII = VIIII; XIX = XVIIII;
- XLV: XL = L - X = XXXXX - X = XXXX; XLV = XXXXV;

- Catenate the positive numerals together.
- DCCCCL + XVIIII = DCCCCLXVIIII;

- DCCCCLXVIIII - XXXXV = DCCCCLIIII - XXX;

- The larger symbol that will be converted to additive notation: L = XXXXX;
- DCCCCLIIII - XXX = DCCCCXXXXXIIII - XXX = DCCCCXXIIII;

- The numeral C should not repeat itself more than 3 times in a row, rewrite:
- DCCCCXXIIII = CMXXIIII;
- The numeral I should not repeat itself more than 3 times in a row, rewrite:
- CMXXIIII = CMXXIV

#### XC + MCMLIII + CCXIX - MCCXLVIII - XC = CMXXIV ( = 924).

- 90 + 1,953 + 219 - 1,248 - 90 = 924

The Romans did not have the Hindu-Arabic numbers.