Your task is to write a function that takes two roman numerals (character strings) as input and returns their sum as a roman numeral as output. Be sure that input can be given in either the additive or subtractive forms of Roman numerals; give output using the subtractive form. What is add-roman("CCCLXIX", "CDXLVIII")?
Roman numerals are a natural development from tallying and finger counting. Some examples:- 1 = I = extend one finger on right hand
- 2 = II = two fingers
- 3 = III = three fingers
- 4 = IIII = four fingers
- 5 = V = right thumb
- 6 = VI = right thumb and one finger
- 9 = VIIII = right thumb and four fingers
- 10 = X = one finger on left hand
- 15 = XV = one finger on left hand and thumb on right hand
- 50 = L = thumb on left hand
- 77 = LXXVII = left thumb and two fingers, right thumb and two fingers
- 99 = LXXXXVIIII = extend all fingers on both hands
- 100 = C = out of fingers!
- 500 = D
- 1000 = M
These examples all take the additive form of Roman numerals, which I call "old style". A later innovation, driven by desire for compactness is to use position and subtraction to avoid repetition of 4 symbols in a row.
Key cases:
- 4 = IIII = IV
- 9 = VIIII = IX
- 40 = XXXX = XL
- 90 = LXXXX = XC
- 400 = CCCC = CD
- 900 = DCCCC = CM
Please note that this more compact form is a step backwards in terms of computation: The beauty of tallying (and additive notations) is that you can add by simple collecting like symbols and then reducing. For example:
LXXXX + LXXXX
= LL XXXXX XXX (collect the like symbols)
= C L XXX (since LL reduces to C and XXXXX reduces to L)
In other words, addition with (additive) Roman numerals is simpler than conventional arabic addition:
90 + 90 = 180 (in this case)
There's less to remember, and no carrying to worry about. The full reduction rules form an easy pattern:
- IIIII -> V
- VV -> X
- XXXXX -> L
- LL -> C
- CCCCC -> D
- DD -> M
Easy, and forming a logical pattern. Of course, ultimately, the arabic place-value system is far superior for an array of reasons, but for adding and subtracting small positive numbers, tallying and its extensions are arguably better.
By contrast, I know of no simple, direct way to perform addition with subtractive roman numerals, and would normally convert to another representation -- arabic or additive -- do the computation there, and then convert back.
Here's some Python code for inter-converting between both styles of Roman numerals and the more familiar arabic representation, plus addition of additive Roman numerals using reduction rules -- the more authentic method -- and addition of subtractive Roman numerals via arabic form.
Notes on the code:
- I use a rule-based form so that the same algorithm (with different rule-sets for additive and subtractive conventions) converts Roman numerals <-> arabic
- The practice of iteratively reducing a number to zero or chomping through a string until there is nothing left while building up the result is a common pattern
- There's no error-checking yet for ill-formed roman numerals: we're avoiding that pain for the moment.
Ideas for extensions and explorations:
- Subtraction and multiplication of additive Roman numerals via reduction rules
- Devise an augmented Roman numeral system with six more symbols for five thousand, ten thousand, fifty thousand, one hundred thousand, five hundred thousand, and one million; extend the code and do some sums.
- Explore the connections between Roman numerals and the (Roman) abacus
