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### Roman Numerals: Division

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

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 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
subtraction)

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

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 - LX - LX=XXV

So,

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

CXLV / VI=XX + XXV / VI

Now we can repeatedly subtract VI from XXV:

XXV=VI + VI + VI + VI + I

So,

XXV / VI=IV + I/VI

The solution, then, is:

CXLV / VI=XXIV + I/VI

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
http://mathforum.org/dr.math/
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Associated Topics:
High School History/Biography
Middle School Division
Middle School History/Biography

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