# How did the Romans do math?

This has bugged me for years, and I’ve never been able to come up with an answer. We know that the Romans built a lot of great structures, aqueducts, etc., so their engineering skills must have been pretty good, but how in the world did they make those complex calculations using Roman numerals?

Maybe I’m just not able to grasp it, since I’ve been working with numbers that have fixed place notation since grade school, but even adding and subtracting Roman numerals gives me a headache. I always have to convert first and then do the math. Long division? Forget about it. Don’t even mention algebra, trigonometry, etc. For that matter, do Roman numerals even have any notation for non-whole numbers?

Was there some method of calculation that lent itself to working with this kind of notation? Would someone who grew up in Rome be able to do calculations in their head, but be totally stumped by our notation?

Any insights would be appreciated.

They used counting boards and abaci.

You might want to take a gander at this book, as its written by a Roman architect.

Romans were taught in school how to calculate with Roman numerals just as you were taught to calculate with Arabic numerals. They didn’t find them at all strange or confusing. They thought them completely natural.

So would you if you were brought up that way.

Certainly not “just as we were taught”, since the same methods don’t work. So what the OP is asking is what the methods the Romans used would have been.

And from what I understand, most Romans weren’t taught in school how to do calculations, beyond perhaps addition and subtraction. There were actually professionals called calculatores whom you would hire if you really needed arithmetic done, and they mostly did their work by using very large tables of values which they jealously guarded. Yes, this is very inefficient, which is precisely why Arabic numerals took over from the Roman ones with time.

So if you ever find yourself transported through time to ancient Rome, you now know how you can support yourself.

This is veering from the subject, but what numeral system would Pythagoras have used? He was around well before Greece came under Roman rule, and the Greeks didn’t use the same alphabet as the Romans so I don’t think they used the same numeral system.

The Greek numbering system used their entire alphabet. The first 9 letters stood for the units 1 through 9, the next nine letters stood for the tens 10 through 90, and the next nine letters for the hundreds 100 through 900. They retained three obsolete letters for this since the Greek alphabet has only 24 letters. You can see how they did higher numbers plus other details at the above link.

As I understand it, the Hebrews and Phoenicians used essentially the same system, although they used their own letters, of course.

Astronomers like Ptolemy used the Babylonian base-60 numbers for their calculations. Geometers such as Euclid did the math in their heads. Pythagoras was a wild and weird cultist… I don’t know what he used.

It is easy to consider Roman numerals as a ridiculously ineffective and quaint way of doing math, and I have no desire to use. But they have their merits. For example, when you read, damaged characters are generally obvious, and usually don’t make the sentence look correct but convey a different meaning. The same is true of Roman numerals. With Arabic numerals, changing one of the characters always produces a different but correct-looking statement.

Regious Romanus has 4 gold coins; Billy Brittus has 10 ( stolen off a passing viking ). RR arrives with his legionaire mates, takes all BB’s coins and anyone else’s… who needs to count them?*

• Apart from at the level of “One for you, one for you, two for me.”

Someone once told me, long ago in the mists of time, that if you chop a C in half horizontally, you get an L. And if you chop an X in half horizontally, you get a V.

I wonder if that was based on any real evidence, or just a handy mnemonic-type device.

That’s not true. Words have a generally accepted spelling; if someone types ‘haces’, it’s a good bet that they misspelled ‘haves’. Numbers generally do not: if someone types ‘334,556’, that could just as easily have been intended to be ‘334,557’. However, if someone types a random string, there could be a typo in there and nobody would know, and if someone types ‘3.14259’, most people are going to see the typo. It’s primarily because the rules for spelling words are far more rigorous than for spelling numbers that this disparity exists, in my opinion.

You seem to be saying the same thing, and yet are writing as though you have opposite positions.

That’s what he’s saying. If Roman numerals were editted, it’d look funny to someone used to reading it. With Arabic numerals, not so much. He’s saying what you’re saying- Roman “math” is more like reading.

Well, it sure didn’t prevent the Romans from using the Quadratic Equation:

**
-b ± √(b[sup]Ⅱ[/sup]-Ⅳac )
x = ———————————————
Ⅱa
**

Actually the Romans used only capital letters so they had to wait until lower case were invented centuries later. Not to mention algebra.

Arabic numbers do not. Roman numerals don’t have ‘spelling’, but they have a formalized method of construction that’s almost as good. For instance, suppose you see DLCIV. This isn’t generally correct - it could be evaluated as being five hundred and fifty-four, (five hundred for the D, 100-50 for the L before C, 5-1 for the I before V,) but the LC doesn’t make sence. You’d write 554 as just L, giving DLIV.

Therefore, you look for a spelling error the same way you would with a word. I would say the most likely correct construction would be DCCIV - seven hundred and four. The first C was mistaken as an L, or the top of the C wasn’t clear so it looked like an L.

Edited to add: Of course, the tradeoff for this ‘spell-check’ is that roman numerals take up more space than comparable arabic numbers and are harder to do multiplication/division with on paper. There’s no free lunch.

And, modern mathematical symbols (like the + sign and the = sign) weren’t in use until many centuries after the ancient Romans. In the olden days, a lot of math was written out in words.

Note that the strict representation rules for Roman numerals as we know them are a modern convention. The Romans used a subtractive rule sometimes, but applied it inconsistently, and sometimes DID use things like “IIXX” to represent 18, as well as “VIIII” for 9 or “XLIIII” for 44.

http://www.web40571.clarahost.co.uk/roman/howtheywork.htm

See some of the historical examples near the bottom of that article.

My old World Book Encyclopedia had three stills from a stop-motion film someone made of Romans using Roman Numerals to perform simple multiplication. It looked interesting, but the encyclopedia never explained exactly how the procedure worked. Too bad – I’ve always wanted to know. And I never did find out who made the film, and haven’t seen it. But there appears to have been a formalism akin to those we use for multiplying two numbers together, or finding square roots, that makes it relatively easy to perform.

Of course, you could also use the abacus, which is what I expect most people did. The formalism looked as if it took up a LOT of space on your tablinum.