This final reflection will take the form of a few experiments in using Roman numerals to do arithmetic. I was intrigued by a paper by W. French Anderson (Anderson, 1956) in which he tries to show how arithmetic might have been done with Roman numerals. Anderson was reacting to a common idea that arithmetic would have been quite difficult for the Romans, since they didn’t have a number system that allowed for ease of use. He wanted to show that arithmetic in that system isn’t as complicated as we might imagine.

The Roman system uses seven digits to write numbers. They are, with modern Hindu-Arabic notation:

I | 1 |

V | 5 |

X | 10 |

L | 50 |

C | 100 |

D | 500 |

M | 1000 |

It will be noticed that this system is a mixture of fives and tens, multiplied by multiples of ten. The fives seem to be used to shorten the written length of numbers–VII is much shorter to right than IIIIIII or IIIX. They seem to act as ‘tally’ marks in an otherwise base-ten system.

Numbers in this system are made by simply concatenating digits until thedesired sum is reached. Therefore, 20 is 10+10 = XX. Six hundred and eventy-three is 500+100+50+10+10+1+1+1 = DCXXIII.

The digits are always listed in descending numerical order. However, it is possible to place a lower digit before a higher digit to *subtract *it’s value (this is called a subtractive):

89=XXCIX. The two Xs are subtracted from the C, and the I is subtracted from the X.

Since this system is based on simple addition and subtraction, addition and subtractions are quite straightforward, especially if there are no subtractives. All one has to do is take all the digits from both numbers, rearrange them into one number, and simplify like terms, taking care to deal with subtractives properly. Anderson uses this example:

MCDLXIX + DCCCXVII = MDDCCLXXXVI = MMCCLXXXVI

Subtraction is simply the reverse process, cancelling out like terms and renaming digits in smaller terms until only one number is left.

Multiplication is a bit more involved. However, like our modern system, it relies on knowing basic multiplication facts (this table is a slightly modified version of one published by Turner, 2007):

times | I | V | X | L | C | D |

I | I | V | X | L | C | D |

V | V | XXV | L | CCL | D | MMD |

X | X | L | C | D | M | MMMMM |

L | L | CCL | D | MMD | MMMMM | |

C | C | D | M | MMMMM | ||

D | D | MMD | MMMMM |

As in the modern system each element in one number must be multiplied by each element in the other. Subtractives should be written in long form. Once the multiplication has been performed, terms should be gathered and simplified as previously.

One of the major limitiations of the Roman system is evident here: long numbers (larger than a few thousand) become unwieldy very quickly. Because the largest digit is M, large numbers can only be written by stringing Ms together. One million, a number often used in our mathematics and engineering, would require a thousand Ms to write down. Different ways of extending the system have been devised over the years, but they rely on an ever expanding set of symbols or introducing new notation (such as a vinculum over a digit to represent ‘times one thousand’ — Turner, 2007).

Division can be done in a similar way to our modern method. It feels unwieldy, but there is a surprising simplification. The system of long division we use requires us to know ‘exactly how many times the dividend can be divided by the divisor in each operation’ (Anderson, 1956, p. 146). For example, if you are dividing 485 (CDLXXV) by 19 (IXX) it is necessary to know how many times 19 goes into 480 (25, with a remainder of 10). With the Roman system, it doesn’t really matter, so long as the number isn’t too big. In this case, it will do to guess XX. IXX (19) times XX (20) is CCCLXXX (380), which leaves a remainder of CV (105). We can repeat the operation with a guess of V (5): IXX times V is VC (95). VC + CCCLXXX = CDLXXV (475). This leaves a remainder of X, which is less than IXX, so the quotient is XX + V = XXV (with a reminder of ten), and our problem is solved.

Anderson also considers whether it is possible to attempt more advanced operations with Roman numerals (and he concedes that the Romans might never have tried to perform these operations.) I won’t go into the details here, but it turns out that it’s quite possible to do what he calls involution and evolution, which are the algorithms used to calculate exponents and roots. It transpires that it is quite possible to do both of these, although the level of accuracy of calculated roots is quite low. It must be said that calculating roots in Hindu-Arabic notation without a calculator is also quite difficult to do with any accuracy.

I was a bit surprised at how fun Anderson’s paper and trying my own examples were. I was quite surprised at how straightforward arithmetic in the Roman system is. There are certainly drawbacks to the system–the unwieldiness of large numbers is a principal one, and the need to deal with subtractives is also a problem. With that said, the whole process was more than satisfying, especially when two unwieldy numbers resolved to a number with only a few digits.

I could imagine giving this as an extension activity to more advanced students, even in primary school. There is nothing difficult about the work. The main issue in this case would be the familiarity of the number system (I found myself converting backward and forwards between the two systems to speed things along sometimes!) It would be a good way to get students to consider why we do things the way we do, and why we replaced an older system that served us well for over a millennia.

#### Bibliography

Anderson, W. F. (1956). Arithmetical computations in Roman numerals. *Classical philology*, *51*(3), 145-150.

Turner, L.E. (2007). Roman Arithmetic. http://turner.faculty.swau.edu/mathematics/materialslibrary/roman/