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Comment by philwelch | Hacker News Reader

Adding Roman numerals, at least, isn’t hard. If it was I doubt Roman numerals would have ever lasted. Arabic is still, in my opinion, easier to add—from the perspective of a lifetime spent exclusively doing arithmetic in Arabic numerals—but it’s not much easier until you’ve memorized all 50 unique sums of one-digit numbers. Multiplication, though, that’s the real difference maker.

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> Adding Roman numerals, at least, isn’t hard.

I find it insanely difficult - but as you note, a lifetime of arabic numerals, and a lack of skill in appropriate tools (eg. an abacus) will skew that comparison.

> ... but it’s not much easier until you’ve memorized all 50 unique sums of one-digit numbers.

This doesn't feel right.

I don't think I have memorised the sum of all pairs of 1-digit numbers - but contemplating this now, it's impossible for me to be sure. I'm not sure where multiplication kicks in for breaking down larger numbers into quotient and divider for me, let alone 'most people'.

Given the maximum value (sum) of two x 1-digit numbers is 18, a naive assumption is the permutations don't really number 50 (I get 45) - and given that roman numerals didn't have a zero, a fairer comparison would be [1-9][1-9] (36 unique combinations)

Either way, in any counting system there's presumably a similar 'memorisation' gate you have to pass for the fundamental set. With roman numerals there was historically 7 I think - I, V, X, L, C, D, M - and summing those wouldn't be anywhere near as straightforward as summing sets of single-digits, so I don't think the comparison of this requirement is as skewed against arabic numerals as you suggest.

I was going off the rough estimate that 0-9 is 10 digits and 10x10 is 100 but that reduces to 50 thanks to commutativity. I think 45 if you don’t count zeroes—I don’t know where you get 36 from.

I also use the term “memorize” pretty loosely—I remember that memorizing times tables was a thing but not so much for plus—but addition is simple enough that most people can kind of intuit what 7+4 heuristically if they’re sat down and forced to do arithmetic as small children for long enough. (Also I’ve never had the patience for memorization; I just rely on my brain to cache things that I use frequently and it ended up working for times tables. Also other things.)

But I do want to acknowledge that Arabic numerals make multiplication nearly as easy as addition, which is a staggering achievement over Roman numerals.

Though I will say, on the other hand, that the Romans weren’t that stupid and neither were their medieval successors prior to the adoption of Arabic numerals. They could add things up and we’ve discussed in a parallel thread how that algorithm works. The Roman numeral system isn’t as optimized informationally—let’s not underestimate the sheer awesomeness of seamlessly expressing numbers as large as 108730026190037365462849562635965—but that would be useless to most cultures that used Roman numerals.

I would even question one more thing. To someone who doesn’t know a numeral system to begin with, do Arabic numerals actually make addition harder? I mean, very small children (and programming languages like JavaScript if you accidentally express one number as a string) sometimes make the mistake of thinking 11+8=118, but in Roman numerals that’s just like saying XI + VIII = XIVIII, which is also wrong, but not as wrong as 118. A Roman child could easily be taught no, that’s XVIIII since V’s go before I’s, and then maybe reduce to XVIV. A child today is like, “wait wtf are places?” Roman numeral users never have to learn the concepts of places, carry, or borrow, which honestly sounds like a good trade off for a civilization that doesn’t have to do multiplication and division that easily.

I think I'd calculated combinations, but dubiously ignored where both were identical - that's where I ended up with 36.

OTOH do we memorise any of the x+1 combinations? I hope we don't, but perhaps we do. I genuinely can't say at this point. I was trying to work out how I processed sums such as 8+7 earlier, and concluded that so far I can tell, I work out the difference of one of those numbers from 10, subtract it from the other, then it's a very simple addition - ie that becomes 10+5. But I'm now unsure if that's what I do as a general rule, and am even less sure what other people may do.

Times table I vaguely recall learning by rote in formative school, but that's an awfully long time ago, and trying to self-analyse my mechanisms for multiplications is highly challenging. It feels like I try to move those back to multiples of 10 or 100, again, too.

I recall reading aeons ago that the only intuitive interface is the nipple - beyond that, everything is learned. So what makes for an intuitive or sensible mathematical representation of things is probably so arbitrary as to be pointless arguing about. It feels that the kinds of things we do, day to day, with numbers, that base-10 arabic number system is optimum, but that may simply be the lack of exposure to a better system.

As a parent of young children and former math teacher, yes, we do absolutely memorize those sums. It usually happens at an early enough age that the process of doing so evaporates early in life and becomes part of your base mental code. I spent a significant part of my teaching life teaching high school math for college students (and occasionally grade school math for college students) and there are very much people who never managed to get that memorization step completed. What's really fascinating is that it's an orthogonal skill to higher mathematics. I've seen students who needed to use a calculator to do 6+5 and yet managed to be able to solve algebraic problems. This is, I must add, uncommon, but it's less because one skill depends on the other but rather because the failure to gain the basic math skill leads to an unwillingness to try to gain the more abstract math skill.

I agree that Arabic numerals are better, but I did want to make the point that Roman numerals aren’t quite as bad as you might think having already learned Arabic numerals.

Your method for summing 8+7 is what I think I do for things like 7+4 (since I can visualize 7 as “three less than 10” and 4 as “one more than three” all in the same thought to reach 11), but for 7+8 my brain noticeably spits out 15 immediately and only a moment later does it actually do the processing you mention.

This is only tangentially related, but until now I never really thought about how I sum single digit numbers, but it's not by having memorized all combinations. I'd say I have all combinations that sum to 10 or less memorized, which adds up to 25 unique combos (plus the rule that number + 0 = number).

If I can tell it sums to more than 10, I break it up mentally into [larger number] + [smaller number] = [larger number] + ([smaller number] - remainder) + remainder, where [larger number] + ([smaller number] - remainder) = 10

You could break it down further by just memorizing what each digit less than 10 is when you add 1. Then you can do addition like 5 + 4 = 1 + 1 + 1 + ... = 2 + 1 + ... = 8 + 1 = 9. Then you'd only have to memorize 9 things (10 if you include 0). I guess this assumes that you know the order numbers go in though, whereas memorizing all of the combinations doesn't require that.

Honestly I feel like I do the same thing when I think about it, but it’s so well cached that whenever my brain sees 5+7 or 7+5, I simultaneously think “12”, forget the specific order, and imagine 5+5=10 with 2 overflow all at once and I can’t honestly tell you which concept arrives first.

And yeah, you can get by with less memorization and more counting, but eh...either you’re taught not to do that or you do enough arithmetic that your brain just caches the whole table eventually anyway.

Please explain the easy method for adding, for example MCMLXVII and LXV. I mean that seriously, I'm curious what the trick is.

I think originally Roman numerals didn't have shorthands like IV instead of IIII. In that case, to add Roman numerals, you just write the letters all together, sort them, and combine smaller digits into larger digits as necessary.

  MCMLXVII + LXV
   = MCCCCCCCCCLXVII + LXV (canonicalize)
   = MCCCCCCCCCLXVIILXV    (concatenate)
   = MCCCCCCCCCLLXXVVII    (sort)
   = MCCCCCCCCCLLXXXII     (combine, VV => X)
   = MCCCCCCCCCCXXXII      (... keep combining, LL => C)
   = MMXXXII               (... C{10} => M, nothing left to combine)
   = MMXXXII               (optionally, look for ways to re-write with the subtraction rule)

I think the practical way of thinking would be:

- seeing L + L = C and converting CM to M ( basically striking L L and C )

- then V + V = X, ( striking Vs and add an X)

Then write whats left:

MMXXXII

Then try to re-write.

I originally thought canonicalize was an important step, but it actually isn’t for humans. For humans, CCCCCCCCC requires a lot more tedious counting than CM or even DCCCC, leading to more errors than simply allowing the human to notice that CM+C=M.

MCMLXVII+LXV = MCMLLXXVVII = MMXXXII

Addition in Roman numerals is dead simple: you do it by just bunching symbols together. The only difficulty is subtractive notation, which wasn’t really used in Ancient Rome.

Tokenize, concatenate, sort, reduce.

MCMLXVII + LXV

M CM L X V I I + L X V

M CM L X V I I L X V

M CM L L X X V V I I

V+V=X, L+L=C, C+CM=M so MMXXXII. Convert everything to decimal if you wanna check my work.

Note that converting CM to DCCCC is actually pretty unnecessary since you can just combine CM and C to make M instead of having to count up lots of C’s. A computer algorithm would be simpler by reducing CM to DCCC but adjusting for human fallibility, allowing for CM + C = M makes things a little easier.

Also note that this method scales to any number of sums, not just adding two numbers together.

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