Adding and Subtracting Roman Numerals

Date: 10/26/1999 at 08:51:49
From: Greg Brennan
Subject: Roman Numerals

I would like to know how to add and subtract Roman Numerals without 
converting them to regular numbers. I tried looking for patterns in 
different problems like XXX + X = XXXX, which I know is written XL. A 
problem I can't find a pattern for is LX - XIV = XLVI. What did the 
Romans do to solve this kind of problem?

Date: 10/26/1999 at 13:28:30
From: Doctor Rick
Subject: Re: Roman Numerals

Hi, Greg.

You might want to look at the Roman Numerals page in our Dr. Math FAQ:

   http://mathforum.org/dr.math/faq/faq.roman.html   

I have read that Europeans didn't switch to Hindu-Arabic numerals for 
a long time because they didn't see a reason to do arithmetic on 
paper, using written numerals. They would set up the numbers on an 
abacus, do the math, then write down the answer. (It's a little like 
using a calculator and not bothering to learn to do arithmetic by 
hand.) Roman numerals are closely related to the abacus -- that was 
one reason they liked them.

If you want to add and subtract Roman numerals, I suggest you do a 
little bit of conversion first: get rid of the subtraction rule. That 
is, rewrite XIV as XIIII. Then you can subtract like this:

       LX      =    XXXXX V IIIII
     -  XIIII  =  -     X    IIII
     --------       -------------
                     XXXX V     I

Now you can resume using the subtraction rule, so that the answer is 
XLVI. I "borrowed" or "regrouped," sort of the way we do it in regular 
subtraction: I changed L into XXXXX so there would be enough X's, and 
I changed X into VV and one of the V's into IIIII so there would be 
enough I's. You can make shortcuts by remembering that V - IIII = I, 
for instance.

This method is much like using an abacus. The abacus does not use the 
subtraction rule; 4 (IV) is represented by four beads (like IIII). An 
abacus uses the same principle of changing a bead in one column into 
two "5" beads in the column to the right, then changing a "5" bead 
into five "1" beads in the same column.

When it comes to multiplication, forget it! The Egyptians had a 
similar kind of numeral system (without the subtraction principle), 
and they used a completely different method of multiplication from 
ours -- one that works something like the way computers do it. You can 
find information on Egyptian multiplication by searching our Dr. Math 
Web site.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/