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For many cases one can ignore division, but yes, as the article said that generally required an abacus. Multiplication doesn't in general require an abacus if

For many cases one can ignore division, but yes, as the article said that generally required an abacus. Multiplication doesn't in general require an abacus if you're trained in roman numerals. Let's take your "more complicated numbers", 42 * 13. Expand XLII * XIII = X * XLII + XLII + XLII + XLII, the first requires you to know a times table to see that it's CDXX, the rest you'd add mentally, LLL - XXX = CXX, so it's CDXX + CXXVI = DXLVI. Five hundred forty six. Like, I'm not saying it's as clean as 42 * 13 = 420 + 126 = 546, it's not. But it's not educationally prohibitive either.

kbenson 9y ago
I think we can all agree addition is fairly easy. Multiplication, since it can be explained in terms of addition, it also not really that hard. Subtraction and division are harder, as I originally noted, and by extension division, are harder, as there is not one simple rule for conversion before subtraction that won't require additional conversions unless you go to the simplest form, which is unwieldy (along string of I's). From my example, which was meant for subtraction, not addition or multiplications, we get the following: XLII - XIII so we convert to an optimized form for arithmetic (which is, as I've noted a second format you need to know and keep track of, since it's not valid Roman numeral format XXXXII - XIII then we note that we have to borrow from the X "place", so let's do that XXXVIIIIIII - XIII now we can actually subtract (XXXX - X) + V + (IIIIIII - III) = XXX + V + IIII = XXXVIIII Now we have to convert back to valid Roman numerals XXXVIV = XXXIX This is using common current arithmetic techniques. I'm not sure whether this is the method they would have employed, or whether it would have been a different method that could have been easier or harder depending on how it combines with the attributes of this number system. Important things to note about the method shown are a pre and post conversion step (or steps, if you don't get every sub-conversion right initially or don't go to a lowest common format), and then the arithmetic itself requires more borrows.
- mywittyname 9y ago
  Well, we can eliminate notations like XL, IV, IX, etc because, as noted in the article, those were not in common usage until much later. Even if we don't, then the first step is to expand these values, much like you would with fractions (i.e., convert to lowest common denominator). Subtraction is as easy as addition (unless you go into negative numbers). The stategy is to walk the subtrahend left-to-right, eliminating from both as you go. If you run into a value that you don't have, then you expand the minuend from the right until you do. For your "difficult" example of 42 - 13: XXXXII - XIII 1 Eliminate an X from both. Now you have XXXII - III. 2 Eliminate I from both Now you have XXXI - II. Repeating (2) gives you XXX - I. 3 Since there is no I to eliminate, expand the minuend from the right to create one. Now you have XXVV - I, repeat since you still don't have a I, repeat (3) which gives you XXVIIIII - I Eliminate the from I both, giving the final result: XXVIIII. Optional, you can "reduce" this to XXVIV, like you would if you had a fraction, like 3/9, but as stated in the article, this notation was not common place in ancient Rome.
- jacobolus 9y ago
  XLII – XIII = XXLIX, of course ;)

kbenson 9y ago

I think we can all agree addition is fairly easy. Multiplication, since it can be explained in terms of addition, it also not really that hard. Subtraction and division are harder, as I originally noted, and by extension division, are harder, as there is not one simple rule for conversion before subtraction that won't require additional conversions unless you go to the simplest form, which is unwieldy (along string of I's). From my example, which was meant for subtraction, not addition or multiplications, we get the following: XLII - XIII so we convert to an optimized form for arithmetic (which is, as I've noted a second format you need to know and keep track of, since it's not valid Roman numeral format XXXXII - XIII then we note that we have to borrow from the X "place", so let's do that XXXVIIIIIII - XIII now we can actually subtract (XXXX - X) + V + (IIIIIII - III) = XXX + V + IIII = XXXVIIII Now we have to convert back to valid Roman numerals XXXVIV = XXXIX This is using common current arithmetic techniques. I'm not sure whether this is the method they would have employed, or whether it would have been a different method that could have been easier or harder depending on how it combines with the attributes of this number system. Important things to note about the method shown are a pre and post conversion step (or steps, if you don't get every sub-conversion right initially or don't go to a lowest common format), and then the arithmetic itself requires more borrows.

mywittyname 9y ago
Well, we can eliminate notations like XL, IV, IX, etc because, as noted in the article, those were not in common usage until much later. Even if we don't, then the first step is to expand these values, much like you would with fractions (i.e., convert to lowest common denominator). Subtraction is as easy as addition (unless you go into negative numbers). The stategy is to walk the subtrahend left-to-right, eliminating from both as you go. If you run into a value that you don't have, then you expand the minuend from the right until you do. For your "difficult" example of 42 - 13: XXXXII - XIII 1 Eliminate an X from both. Now you have XXXII - III. 2 Eliminate I from both Now you have XXXI - II. Repeating (2) gives you XXX - I. 3 Since there is no I to eliminate, expand the minuend from the right to create one. Now you have XXVV - I, repeat since you still don't have a I, repeat (3) which gives you XXVIIIII - I Eliminate the from I both, giving the final result: XXVIIII. Optional, you can "reduce" this to XXVIV, like you would if you had a fraction, like 3/9, but as stated in the article, this notation was not common place in ancient Rome.
jacobolus 9y ago
XLII – XIII = XXLIX, of course ;)