I have seen a quote purporting to be fifteenth century (or so) advice to a father about his son. The advice was that a German university would be adequate if the son only desired to learn to add and subtract, but if the lad was to multiply and divide, he should go to Rome.
I would love to believe it, but I fear that this quote is a bit too good to be real.
By the fifteenth century, they wouldn’t have been using Roman numerals. And even when Roman numerals were used, multiplication and division wasn’t so much something that you learned, as you inherited closely-guarded tables of multiplication and division, and sold your services of using those tables.
This article suggests that the Greeks at least had a method for multiplication for their letter-based calculation system. So, I wouldn’t be surprised if the Romans had also. This doesn’t help find your quote, but I just happened to be reading this yesterday, so… there it is.
Maybe I’m being whooshed here, but such a quote, if real, would have had nothing to do with Roman numerals. It’s just that following the Quattrocento, Italy in the 14th and 15th century was going through a famous Renaissance period (mostly artistic I guess, though in the sciences there was Leonardo da Vinci and Galileo). The man is supposed to be telling his son “for advanced learning, go to Italy.”
I don’t know when they stopped using roman numerals in northern Europe, but Hindu-Arabic numerals were brought to Europe by Leonardo Bonacci, generally known as Fibonacci for some reason. (I had always assumed that he was the son of Bonacci, but Wiki says his father was Bill, see http://en.wikipedia.org/wiki/Fibonacci). His father was a businessman with business in North Africa and he saw how superior, at least for business reckoning, that system was. In 1202, he published a book (Liber Abaci, which means, I imagine, Book of Counting) advocating the new system. There was, as you might imagine, tremendous to this new system, considered very hard to understand. Sometimes it seems it still is. I once asked a student in a history of math course, if she could explain the connection between 36 and numerals 3 and 6. She had no idea.
I recall reading once about some 18th century notable who, at age 40, engaged a tutor to teach him long division and he found it an impossible struggle. So even if our numbers were around, it might have been not a subject that any but accountants and navigators learned. Of course, ignorance of multiplication and division is now near-universal among those born after, say 1980.
Back when I was perhaps in Junior High or thereabouts, my Mom showed me the nasty mental and procedural gymnastics one had to go through to multiply a couple of relatively large (our equivalent of 3-digit) Roman numerals, or to divide using Roman numerals.
Man alive. It’s one thing to have folks tell you all your life what a fantastic development the ZERO was, but you don’t really get it just counting or adding or subtracting. I never complained about “carry the two” ever again!
In all seriousness, no it isn’t. My son was born in the late 1990s and could expertly perform both multiplication and long division with pencil and paper by 4th or 5th grade.
I was born in the late 1960s and of course also learned to do multiplication and long division by hand. I also learned (at the age of 8 years in 3rd grade), that if you had a bunch of long division problems to do for homework, the whole trial-and-error part of long division could be avoided if you used a calculator to pre-determine what the answer was going to be. 
(I then showed all the rest of the work, as well as to figure out what the remainder was.) I did this when there was just one calculator in the house, an expensive Texas Instruments model with the red LEDs.
Despite the ubiquity of calculators today, my son apparently never picked up on this labor saving technique.
I spent just one afternoon studying what was known about Greek arithmatic, and I’m prepared to say that the first thing you need to know about it is that you don’t need to know anything about it. Okay, I can see the appeal. I myself gave it an honest try. But consider this: Have you already mastered the use of the abacus? The slide-rule? Have you tried doing subtraction in binary using 2’s compliment addition? How about in decimal using 10’s complement? No? Then there are a lot of fun and rewarding nerdy challenges for you before you have to resort to Greco-Roman arithmetic.
Eh, we had our multiplication tables pretty well drilled into us in elementary school (I was born in '85, fwiw), and if you know those, there’s not really much trial and error in long division.
What if the divisor has more than one digit? I don’t know about you, but I only learned my multiplication tables up through 12 x 12.
If you are dividing a five digit number by a three digit number (for example), there is generally some trial-and-error involved. YMMV.
Sounds like the SR-50.
I went to college in 1973 and I started learning the slide rule for fun in high school. I never did become a whiz at it like my dad. (There’s a trick for estimating the 3rd digit when doing large-number multipilcation) Calculators were available by then - I bought one with 4-functions for $100 in spring 1974 and a full scientific for $29 only 3 years later.
The key to a lot of the advanced math is memeorizing the math tables. Forced rote memorization seems to be something modern teaching methods have trouble with, but it’s something you cannot avoid to be proficient.
What’s really needed is a class on estimating: “Joe has $137,980 and offers you 16% - how much roughly is that?” Your brain should say “That’s about $14,000 for 10%, 5% more is another $7,000 for $21,000. Plus 1% is $1,400 and I rounded up; so about $22,000.”
If your geekitude comes into play, you can estimate 16% of $140,000-137,980 =$2,020 and subtract that from $22,400.(16x2=32, so about $320 plus $3.20; etc.) And amaze your friends with your mental math before they finish getting their cellphone out, switching to calculator, and keying it in.
So many people use a calculator, make a typo (or multiply instead of dividing) and come up with a number that’s wildly off base, and don’t realize that an answer around $2,000 or $180,000 could not possibly be right.
Oh, some, certainly, but not that much. I mean, even when you get into big numbers, it’s not like you’re shooting blind. For instance, I tried dividing 19,204 by 353:
Ok, so 1920 divided by 353, well, 353 * 3 is gonna be a little over 1000, so call it 3 + 2: 5 * 353 = 1765. 19204 - 17560 = 1554 - well, 353 * 5 is gonna be a little more than that, so call it 4 - that’s 1412, and you get 142 left over. Well, we already know 353 * 4 is 1412, which gives you a remainder of 8. It’s pretty damn close to 54.4.
Basically, estimation rules. Once you get up into the multi-digit problems, sure, it’s a little more complicated, but you can fudge a lot by paying attention to the highest digits and guessing. Maybe you’ll have a little trial-and-error, but for that problem, all I needed was trial-and-success.
When you are eight years old and you have a dozen or two long division problems to work for homework, it’s even easier to use a calculator. 
Even at that age, I didn’t have a lot of patience for what I believed to be repetitious busy work. I understood the concept immediately in school. I didn’t need to work dozens of problems at home. Everybody is different, though. Some folks need the repetition, so I understand why it was assigned that way.
Yeah, but estimating: 20000 divided by 400 = 50; 1900/300=60 and a bit. So your calculator better give you an answer between those two or you made a typo. Estimating…
Dividing by 353? Well, 5400 =2000; 5300=1500; So 5350 should be half-way, 1750. and 5353=1750+15. Of course, this is a lot easier to do if I can see it instead of in my head…
The whole issue of numeracy came up in the media about 10 or 15 years ago. We fuss a lot about literacy, but a lot less about numeracy. Perhaps the answer to the original post was, other than adding a whole sum of numbers - in which case you use an abacus - the Romans didn’t use a lot of higher math. I haven’t heard about any fancier math work they did.
I’m surprised we had to wait for the arabs to give us digits. You would think using an abacus, the concept of a repetitive set of digits would eventually become obvious pretty fast. Zero for a number position would mean “all beads to the right” or whatever.
What have the Romans ever done for us?
I felt the same way. We had computer programs that got around this by teaching us only if we failed a pretest, and I thought that was going to be the way of teaching in the future–yet it still hasn’t happened.
Anyways, the problem with using the calculator is that every teacher I’ve ever had actually checked to see if we’d actually done the work. Only in high school did it become somewhat optional: if you didn’t show your work, and made a mistake, you couldn’t get partial credit. But that was only because there was no way we could just enter the problem directly in to the calculator (though you could get close.)
Actually, the Romans used Calculus for calculating addition, subtraction, multiplication, and division. Well, what they called Calculus. Calculus means stones (like Calcium). Romans used to have a tablet with grooves in it (or they simply dug grooves in a dirt floor), and they used small round stones to represent the number much the way an abacus does.
In fact, most civilizations at the time of the ancient Greeks had devices like abacuses to do their calculations. What amazes me is that it took almost 1000 years for someone to look at an abacus and say “Wait a second, if I make digits represent each column of stones, I could do my calculations without carrying around this heavy stone tablet and this pouch of stones!”. Maybe it took someone to figure out how to represent an empty column (i.e. invent the zero).
My guess is that it really took the invention of paper as a commodity. It doesn’t do you any good to create a base 10 numeric system which would allow you to do your calculations with a pen and paper if you paper was too expensive to use for scratch.
Or, maybe it took double entry bookkeeping which is what Fibonacci really demonstrated in Liber Abaci. Liber Abaci said, “Look what you can do if you use Indian Method of calculating!” Before that, you couldn’t do a running total of accounts like you’d do in a checkbook. Thanks to decimal based arithmetic, you could more easily keep track of how much each account owns you.
Just because Arabic numerals were already common does not mean that the use of Roman numerals had ceased. Roman numerals would continue to be widely used, especially for accounting purposes, for another couple of centuries.