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How to Add and Subtract Roman Numerals

What’s 51 + 12? Easy, right? How about LI + XII? Not so easy…or is it? Keep on reading to learn all about adding and subtracting Roman style.

By
Jason Marshall, PhD
5-minute read
Episode #140

How to Add Roman Numerals

But how in the world could you possibly do arithmetic with these numerals? It seems impossibly cumbersome and confusing to add Roman numerals like MCLXXIV + CXXXIX (aka 1,174 + 139), but it’s really not! You just have to think a little outside the decimal-system-lined box that we’re accustomed to living in. If you want to have some fun—of the puzzling type that I talked about at the outset—I encourage you to stop for a few minutes and try to come up with a method for adding Roman numerals. Like any good puzzle, it’s always best to try and solve it on your own instead of peeking at the answer.

Let’s start by thinking about the example problem from before, MCLXXIV + CXXXIX. The only tricky part about adding Roman numerals is what you should do about subtractive symbols like “IV” and “IX” that have smaller numbers to the left of larger numbers. As it turns out—and as should be clear by the time we’re finished—the best thing to do is to rewrite these subtractive symbols using additive symbols (even if you have to violate the rule about having no more than 3 of the same symbol in a row). Which means we need to rewrite “IV” as “IIII” and “IX” as “VIIII.”

So the problem we’re now trying to solve is: MCLXXIIII + CXXXVIIII. The next step is to gather up and rewrite the symbols in order from largest to smallest. In this case, that’s MCCLXXXXXVIIIIIIII. Now all you have to do is combine symbols together wherever you can. For example, we can combine “XXXXX” into the symbol “L,” and we can combine “IIIIIIII” into the symbol “VIII.” This gives us MCCLLVVIII. But we’re still not done because we can now combine “LL” into the symbol “C” and “VV” into the symbol “X.” When we do that, we get an answer of MCCCXIII or 1,313. Addition isn’t so bad, right?

How to Subtract Roman Numerals

But what about subtraction? How does that work? Well, let’s see by working out the problem MCLXXIV – CXXXIX (aka 1,174 – 139). Again, I encourage you to stop and think about how you could work this puzzle out. As with addition, the best way to start is by turning all of those subtractive symbols like “IV” and “IX” into additive symbols like “IIII” and “VIIII.” That leaves us with the problem: MCLXXIIII – CXXXVIIII. Now the fun part: Cross out pairs of symbols that appear on both sides of the problem (since that’s just subtracting the same amount from each side). That means we can get rid of the “C,” two of the “X”s, and all four of the “I”s from both sides, leaving us with the much simpler problem: ML – XV.

Now, start with the largest value on the right side and find the value on the left side that’s bigger than it. In this case, that’s the “X” on the right and the “L” on the left. Then rewrite the larger symbol on the left in terms of the smaller symbol on the right and carry out the subtraction. In other words, ML – XV = MXXXXX – XV. As before, cross out symbols that appear on both sides, and then repeat this process as many times as necessary until you’re all done. In this case, the final answer is MXXXV or 1,035.

Wrap Up

Okay, that’s all the mathematical puzzling and un-puzzling about ancient Roman arithmetic that we have time for today. I hope you enjoyed it! Remember to become a fan of the Math Dude on Facebook where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too. Finally, please send your math questions my way via Facebook, Twitter, or email at mathdude@quickanddirtytips.com.

Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!

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About the Author

Jason Marshall, PhD

Jason Marshall is the author of The Math Dude's Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.