CCCXLII + MMDCXCI - MCMLXXXVIII + MMXXXI: How to Add and Subtract Roman Numerals. Calculate Without Converting to Arabic Numerals. Roman Numeral Calculator For Adding and Subtracting

Roman numerals addition and subtraction calculator:
» CCCXLII + » MMDCXCI - » MCMLXXXVIII + » MMXXXI = ?

Do we use Arabic numerals below?

  • The Romans did not have the numbers we use everyday, the so called Arabic numbers.
  • They were using Roman numerals for making addition and subtraction calculations.
  • Add and subtract the Roman numerals without converting to Hindu-Arabic numbers
  • Steps and explanations below, without using any Arabic numbers.

  • » Roman numerals, reading rules


1. Simplify the operation.

Cross out the common symbols in the numerals of different signs.

The matching numerals / symbols are colored the same.


CCCXLII + MMDCXCI - MCMLXXXVIII + MMXXXI =

CCCXL + MDCXC - CMLV + MMI

» When crossing out, do not mix the symbols in the subtractive and additive groups. See why.


2. Substitute for subtractives.

Substitute for any subtractive groups of symbols in the roman numerals; that is, 'uncompact' them using only the additive notation.

» The subtractive and the additive notation: Find out more info on this topic.


CCCXL:

XL = L - X = XXXXX - X = XXXX

CCCXL = CCCXXXX


MDCXC:

XC = C - X = L XXXXX - X = L XXXX = LXXXX

MDCXC = MDCLXXXX


CMLV:

CM = M - C = D CCCCC - C = D CCCC = DCCCC

CMLV = DCCCCLV


3. Put the Roman numerals of the same sign together.

Catenate the positive numerals together:

CCCXXXX +

MDCLXXXX +

MMI =

CCCXXXXMDCLXXXXMMI


4. Sort out the symbols in descending order.

Sort out the symbols in descending order from left-to-right with the largest symbol on the left, down to the smallest on the right.


Sort out the symbols of the positive numerals:

CCCXXXXMDCLXXXXMMI =

MMMDCCCCLXXXXXXXXI


5. Cross out (remove) any symbols occurring in both strings.

MMMDCCCCLXXXXXXXXI -

DCCCCLV =

MMMXXXXXXXXI -

V


6. Get rid of all the symbols in either string.

Using the additive notation, convert larger symbols to smaller ones and then cross out the common symbols that occur in both numbers.

» The subtractive and the additive notation: Find out more info on this topic.


The larger symbols that are being converted, written in additive notation:

X = VV;


MMMXXXXXXXXI -
V =
MMMXXXXXXXVVI -
V =
MMMXXXXXXXVI

7. Merge the repeating symbols together.

Starting on the right end of the number (starting with the smaller values), merge the repeating symbols into larger ones.


MMMXXXXXXXVI =

MMMXXXXX XXVI =

MMMLXXVI


CCCXLII + MMDCXCI - MCMLXXXVIII + MMXXXI =

MMMLXXVI

The operation written with Arabic numerals:

342 + 2,691 - 1,988 + 2,031 = 3,076

The Romans did not have the numbers we use everyday, the so called Arabic numbers.

The Romans also did not have the concept of negativity with their numeral system.


» Monthly Addition and Subtraction Calculations With Roman Numerals

» Month 01, 2026 [January]: Addition and Subtraction Calculations With Roman Numerals


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How to add and subtract Roman numerals the right way, like the Romans calculated, without the Hindu-Arabic numbers: steps and explanations

  • 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.

1. Cancel the numerals having the same value but different signs.

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

2. Cross out the common symbols in the numerals of different signs.

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

When crossing out,
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.

3. Replace the groups in subtractive notation.

  • 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;

4. Put the Roman numerals of the same sign together.

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

5. Cross out (remove) any symbols occurring in both strings.

  • DCCCCLXVIIII - XXXXV = DCCCCLIIII - XXX;

6. Using the additive notation, convert larger symbols to smaller ones and cross out the common symbols.

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

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

  • 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

The final answer:

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

  • 90 + 1,953 + 219 - 1,248 - 90 = 924
    The Romans did not have the Hindu-Arabic numbers.