[CJ Hinke comments: If this gives you any trouble, listen to Tom Lehrer’s math songs: http://www.youtube.com/watch?v=BRS1H2TBRcU (1997) but note that these pre-date the ‘new math’: https://www.youtube.com/watch?v=UIKGV2cTgqA (1965) .]
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, The Math Dude
February 1, 2013
Do you think math is fun? I absolutely think that it should be. After all, math problems are really just puzzles. And puzzles are fun, right? I realize that not all math problems are created equally—sometimes you have to use math to get real world work done. And that certainly isn’t always tons of fun. But math should be fun most of the time. The key to this is approaching problems with the right attitude. For example, do you think today’s topic—adding and subtracting Roman numerals—sounds like fun? For many of you, the answer is “No!” And you may be wondering how this could possibly be useful?
While those are fair questions, the truth is that math doesn’t always have to be serious and practical. Even though you may never need to use the math we talk about today, it’s still fun to puzzle out how ancient Romans could have added and subtracted their rather cryptic numerals. No, this isn’t a skill you need to survive in the modern world. But learning to see the many challenges you’ll face in life as puzzles that can be solved is a skill that will help you survive, thrive, and have fun. So, with that in mind, let’s spend today having a bit of fun puzzling out how the Romans managed to do arithmetic.
Recap: What Are Roman Numerals?
Before we begin figuring out how to add and subtract Roman style, let’s recap how the Roman numeral system that we learned about last time works. In this system, the letter “I” represents 1, “V” represents 5, “X” represents 10, “L” represents 50, “C” represents 100, “D” represents 500, and “M” represents 1,000. To write numbers other than these we combine various symbols together. If a symbol that represents a smaller number is written to the right of a symbol representing an equal or larger number, we add the values together. In common usage, symbols are repeated no more than 3 times in a row.
For example, the number 1,272 is written MCCLXXII. Since each symbol represents a number that’s equal to or smaller in size than the number represented by the symbol to its immediate left, all of the values represented by the various symbols here are added together. So the “M” represents 1,000, the two symbols “CC” represent 100 + 100 = 200, the “L” represents 50, the two “X”s represent 10 + 10 = 20, and the two “I”s represent 1 + 1 = 2—for a grand total of 1,000 + 200 + 50 + 20 + 2 = 1,272.
If, on the other hand, a symbol representing a smaller value is written just before a symbol representing a larger value, we have to subtract the smaller value from the larger. For example, “IX” represents the number 10 – 1 = 9, and “CM” represents 1,000 – 100 = 900. “I” is allowed to be subtracted from “V” and “X,” “X” is allowed to be subtracted from “L” and “C,” and “C” is allowed to be subtracted from “D” and “M,” but nothing else can be subtracted from anything else. And that’s it! Once you know these rules, you know everything necessary to read and write Roman numerals.
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.
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 firstname.lastname@example.org.
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!