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Yes and no. The format of numbers is important to how you end up manipulating them. The way we do long division with 3 or more digit numbers cannot translate to

Yes and no. The format of numbers is important to how you end up manipulating them. The way we do long division with 3 or more digit numbers cannot translate to roman numerals. A different method is required. This is part of my point. No one goes into how to do complex maths with Roman numerals, Cunieform or anything else. They're just "look at how you can display a count". Once you learn to add, substract, multiply or divide 2 digit numbers on paper with Arabic-Hindu numerals, 30 digit numbers are trivial, only time consuming. Having a system that uses entirely different characters to represent hundreds, five-hundreds, thousands and more, that changes the math system. There was a British show I watched as a kid that played in the US which made a light hearted joke regarding this. I think it was called How 2 or something like that. But it was a Welsh woman that showed how 12/2 can equal 7. XII, when you erase the bottom half, it gives you VII. While a joke, it always bothered me that no one ever talked about how the Roman ACTUALLY did math with their number system. How in the hell do you XII/II on paper by showing your work? Because our 12/2 long division method can translate easily to 1212/2. How do you do the same with MCCXII/II? I find it hard to believe there isn't one. Just, no one talks about it.

fao_ 6y ago
It's why arithmetical calculation was a role one had. They used an abacus for the calculation of complex sums. https://thatsmaths.com/2019/07/04/what-did-the-romans-ever-do-for-maths/ https://thatsmaths.com/2019/07/04/what-did-the-romans-ever-d...
- NoOneNew 6y ago
  So wait... math was never really done long hand in ancient times, but calculators (the abacus or any other form) were used instead? In some respects, that makes sense. Paper like materials were never cheap and long-hand math uses a whole lot of paper. Bead calculators makes sense. I'd be curious to see if there's a correlation to "cheap" paper being plentiful right before the invention of complex math like Algebra. With a quick 5 minute peek, "cheap" paper came from China to the middle east in the ~8th century. Algebra was recorded in the 9th in the middle east. So... eh, maybe. Requires more reading, but it's interesting. Sounds like an interesting infrastructure like correlation to technology.
  - thaumasiotes 6y ago
    > math was never really done long hand in ancient times, but calculators (the abacus or any other form) were used instead? No. I provided an example of peasant multiplication in my response sidethread. (Well, "yes" if you mean using paper. Paper is hard to get. But you can multiply without needing a special tool; the algorithm is commonly executed by using piles of rocks and pits in the ground.) The abacus is a more advanced technology that took time to develop. (It does appear early in some form; see https://en.wikipedia.org/wiki/Abacus#History https://en.wikipedia.org/wiki/Abacus#History )
  - jacobolus 6y ago
    Yes this is largely correct. Calculation was not done with Roman (or Sumerian, Egyptian, Greek, ...) numerals per se, but mentally, with fingers, with piles of tokens, or with a counting board. But we are talking about moving tokens around on some markers drawn on the ground (or on a piece of cloth or animal skin or wooden board or whatever), not beads on rods like a Suanpan/Soroban used in East Asia today. My speculation is that many ancient game boards (e.g. backgammon, mancala) were also (or originally) counting boards. (The Roman phrase 'calculos ponere' literally means 'place pebbles', as on a counting board; this is where our word 'calculate' comes from. To settle accounts with someone was to 'vocare aliquem ad calculos' or 'call them to the pebbles') You should think of written numbers as a historical serialization and storage format. Something like the XML or JSON of the ancient world. And just like a modern computer, ancient computers first deserialized data to a more convenient internal representation for intermediate processing. Arithmetic really took off in Europe with the advent of printed books (easier to teach written arithmetic tricks via books, whereas counting-board tricks were largely passed down orally and by demonstration) and of course with cheaper paper and widespread literacy. The big advantages written arithmetic has vs. counting boards (despite being slower and having a steeper learning curve) is that all of the scratch work remains visible, which makes it easier to check for mistakes. Roman numerals are a more direct representation of the state of a counting board which takes less practice to learn, especially for illiterates.
    - jackcosgrove 6y ago
      That's interesting that the root meaning of calculate is pebbles. I'm sure it's unintentional, but pebbles as a mnemonic for calculus is pretty good. Take the counting stones and pulverize them into dust and you get calculus.
thaumasiotes 6y ago
> The format of numbers is important to how you end up manipulating them. I just pointed out: >> Before the introduction of Arabic numerals to Europe, this manipulation was already common; that's what an abacus was for. The abacus doesn't represent numbers in Roman numerals. It represents them positionally. This poses no problems because conversion between representations is trivial. > How in the hell do you XII/II on paper by showing your work? That's as easy as it gets; half of X is V and half of II is I. This is like asking "how in the hell do you do 7 - 2 on paper by showing your work?". Dividing by 2 is a primitive that is always easy to do in Roman numerals, and is relied on heavily in the method you'd use to multiply them, which you can in fact read about. I'll do 47x9: XXXXVII * VIIII XXIII * XVIII XI * XXXVI V * LXXII II _ CXXXXIIII I * CCLXXXVIII CCLXXXVIII + LXXII + XXXVI + XVIII + VIIII = = CC LL XXXXXXXXX VVVV IIIIIIIIIIIII = = CC C LXXXX XX XIII = = CCCCXXIII = 423
- darkwater 6y ago
  I thought 9 was IX
  - thaumasiotes 6y ago
    You're free to write it that way if you want, but that just adds an unnecessary step to doing math with it.
  - dhosek 6y ago
    The prefix = subtraction convention came late in the history of Roman numerals. I think it might even postdate the introduction of Arabic numerals, but I can't say for sure since I'm going from memory here. If I were less lazy, I'd look at wikipedia which doubtless can give a more definitive answer.
    - thaumasiotes 6y ago
      Wikipedia has plenty of citations to classical Rome. Colosseum gate 44: XLIIII (not, for whatever reason, XLIV, or XXXXIIII) 18th legion: XIIX 22nd legion: IIXX There was evidently not a strict standard. (Note also that the symbols we use for high values, D and M, weren't really what the Romans used. You can see Roman abaci with columns labeled with place values: https://en.wikipedia.org/wiki/File:RomanAbacusRecon.jpg https://en.wikipedia.org/wiki/File:RomanAbacusRecon.jpg https://thatsmaths.com/2019/07/04/what-did-the-romans-ever-do-for-maths/ https://thatsmaths.com/2019/07/04/what-did-the-romans-ever-d... Those look the same; I don't know whether the one is a replica of the other or whether the form was just that standardized. But the columns are plainly labeled as, in ascending order, ϴ, I, X, C, ∞, ((|)), (((|))), and |X|. I speculate that ∞ is a fancy way of writing (|); I believe our M descends from Roman (|) and our D from Roman |). )

fao_ 6y ago

It's why arithmetical calculation was a role one had. They used an abacus for the calculation of complex sums. https://thatsmaths.com/2019/07/04/what-did-the-romans-ever-do-for-maths/ https://thatsmaths.com/2019/07/04/what-did-the-romans-ever-d...

NoOneNew 6y ago
So wait... math was never really done long hand in ancient times, but calculators (the abacus or any other form) were used instead? In some respects, that makes sense. Paper like materials were never cheap and long-hand math uses a whole lot of paper. Bead calculators makes sense. I'd be curious to see if there's a correlation to "cheap" paper being plentiful right before the invention of complex math like Algebra. With a quick 5 minute peek, "cheap" paper came from China to the middle east in the ~8th century. Algebra was recorded in the 9th in the middle east. So... eh, maybe. Requires more reading, but it's interesting. Sounds like an interesting infrastructure like correlation to technology.
- thaumasiotes 6y ago
  > math was never really done long hand in ancient times, but calculators (the abacus or any other form) were used instead? No. I provided an example of peasant multiplication in my response sidethread. (Well, "yes" if you mean using paper. Paper is hard to get. But you can multiply without needing a special tool; the algorithm is commonly executed by using piles of rocks and pits in the ground.) The abacus is a more advanced technology that took time to develop. (It does appear early in some form; see https://en.wikipedia.org/wiki/Abacus#History https://en.wikipedia.org/wiki/Abacus#History )
- jacobolus 6y ago
  Yes this is largely correct. Calculation was not done with Roman (or Sumerian, Egyptian, Greek, ...) numerals per se, but mentally, with fingers, with piles of tokens, or with a counting board. But we are talking about moving tokens around on some markers drawn on the ground (or on a piece of cloth or animal skin or wooden board or whatever), not beads on rods like a Suanpan/Soroban used in East Asia today. My speculation is that many ancient game boards (e.g. backgammon, mancala) were also (or originally) counting boards. (The Roman phrase 'calculos ponere' literally means 'place pebbles', as on a counting board; this is where our word 'calculate' comes from. To settle accounts with someone was to 'vocare aliquem ad calculos' or 'call them to the pebbles') You should think of written numbers as a historical serialization and storage format. Something like the XML or JSON of the ancient world. And just like a modern computer, ancient computers first deserialized data to a more convenient internal representation for intermediate processing. Arithmetic really took off in Europe with the advent of printed books (easier to teach written arithmetic tricks via books, whereas counting-board tricks were largely passed down orally and by demonstration) and of course with cheaper paper and widespread literacy. The big advantages written arithmetic has vs. counting boards (despite being slower and having a steeper learning curve) is that all of the scratch work remains visible, which makes it easier to check for mistakes. Roman numerals are a more direct representation of the state of a counting board which takes less practice to learn, especially for illiterates.
  - jackcosgrove 6y ago
    That's interesting that the root meaning of calculate is pebbles. I'm sure it's unintentional, but pebbles as a mnemonic for calculus is pretty good. Take the counting stones and pulverize them into dust and you get calculus.

thaumasiotes 6y ago

> The format of numbers is important to how you end up manipulating them. I just pointed out: >> Before the introduction of Arabic numerals to Europe, this manipulation was already common; that's what an abacus was for. The abacus doesn't represent numbers in Roman numerals. It represents them positionally. This poses no problems because conversion between representations is trivial. > How in the hell do you XII/II on paper by showing your work? That's as easy as it gets; half of X is V and half of II is I. This is like asking "how in the hell do you do 7 - 2 on paper by showing your work?". Dividing by 2 is a primitive that is always easy to do in Roman numerals, and is relied on heavily in the method you'd use to multiply them, which you can in fact read about. I'll do 47x9: XXXXVII * VIIII XXIII * XVIII XI * XXXVI V * LXXII II _ CXXXXIIII I * CCLXXXVIII CCLXXXVIII + LXXII + XXXVI + XVIII + VIIII = = CC LL XXXXXXXXX VVVV IIIIIIIIIIIII = = CC C LXXXX XX XIII = = CCCCXXIII = 423

darkwater 6y ago
I thought 9 was IX
- thaumasiotes 6y ago
  You're free to write it that way if you want, but that just adds an unnecessary step to doing math with it.
- dhosek 6y ago
  The prefix = subtraction convention came late in the history of Roman numerals. I think it might even postdate the introduction of Arabic numerals, but I can't say for sure since I'm going from memory here. If I were less lazy, I'd look at wikipedia which doubtless can give a more definitive answer.
  - thaumasiotes 6y ago
    Wikipedia has plenty of citations to classical Rome. Colosseum gate 44: XLIIII (not, for whatever reason, XLIV, or XXXXIIII) 18th legion: XIIX 22nd legion: IIXX There was evidently not a strict standard. (Note also that the symbols we use for high values, D and M, weren't really what the Romans used. You can see Roman abaci with columns labeled with place values: https://en.wikipedia.org/wiki/File:RomanAbacusRecon.jpg https://en.wikipedia.org/wiki/File:RomanAbacusRecon.jpg https://thatsmaths.com/2019/07/04/what-did-the-romans-ever-do-for-maths/ https://thatsmaths.com/2019/07/04/what-did-the-romans-ever-d... Those look the same; I don't know whether the one is a replica of the other or whether the form was just that standardized. But the columns are plainly labeled as, in ascending order, ϴ, I, X, C, ∞, ((|)), (((|))), and |X|. I speculate that ∞ is a fancy way of writing (|); I believe our M descends from Roman (|) and our D from Roman |). )