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### Adding and Subtracting Roman Numerals

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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?
```

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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/
```
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