The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

Roman Numerals: Division

Date: 11/05/2001 at 11:57:06
From: Chalina Mendoza
Subject: Arithmetic of Roman Numerals (specifically division and 

Dr. Math -

I am in a math class and our assignment is to find out how to add, 
subtract, multiply, and divide Roman numerals. I have found 
information on how to add and multiply, but have not found any 
information on how to divide or subtract. Your information would 
greatly appreciated. Thanks. 
Chalina Mendoza

Date: 11/05/2001 at 13:17:47
From: Doctor Sarah
Subject: Re: Arithmetic of Roman Numerals (specifically division and 

Hi Chalina - thanks for writing to Dr. Math.

In the Dr. Math FAQ on Roman Numerals you'll find rules for 
subtracting, with some examples. Jim Loy's page on division is 
temporarily unavailable on the Web; here is a cached version:

Division with Roman Numerals
Copyright MCMXCVII, Jim Loy

Roman numeral division can be very easy or fairly difficult, depending 
on the two numbers. Sometimes you can just write down the two numbers 
and do some division of pieces of the numerator:

XCV / V=XIX (- II + XX + I)

MMCCCXXII / CCX=XI + XII/CCX (the remainder was XII)

In the first example, we had to deal with a borrow. But, that wasn't 
difficult. In the second example, MMC is X x CCX, so MMCCCX=XI x CCX. 
And we had to deal with a remainder.

General method:

The above method works fairly often, but is pure torture in general. 
Try CXLV / VI, if you need to be convinced that it is difficult. We 
need a general method.

Division is repeated subtractions. Our method of long division, which 
we learn in grade school, is just a formalized way of using repeated 
subtractions. In the above example (CXLV / VI) we could just subtract 
VI over and over, keeping track of how many times we subtracted. But, 
there is a short cut, we can subtract LX (which is X x VI) a few 
times, first:



CXLV / VI=LX / VI + LX / VI + XXV / VI


Now we can repeatedly subtract VI from XXV:

XXV=VI + VI + VI + VI + I



The solution, then, is:


Here, we had a remainder of I. This method works fine for all division 
problems. You may have to multiply the denominator by C or even M, 
just as we multiplied by X. Multiplying the denominator by V or L 
sometimes can help. But, usually, that only complicates the process.

We see here that Roman numerals are not a lot more difficult to divide 
than dividing our Arabic numerals. There are more steps, and most of 
us are not as familiar with the process, so it is more tedious for us.

- Doctor Sarah, The Math Forum   
Associated Topics:
High School History/Biography
Middle School Division
Middle School History/Biography

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.