Division with Roman Numerals

Date: 03/02/2004 at 15:38:58
From: Trish
Subject: Division using Greek, Roman or Egyptian

I'm having real problems trying to divide using a non place system.  I
can think in the Hindu place system, but the concept of using letters
rather than numbers is confusing me.  Can you please give me an
example of a division calculation?

Date: 03/02/2004 at 16:55:13
From: Doctor Peterson
Subject: Re: Division using Greek, Roman or Egyptian

Hi, Trish.

There is a demonstration of addition and subtraction in our FAQ:

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

That also points out that the Romans and others probably wouldn't have 
bothered doing arithmetic with their numerals in the sense we think of 
doing it; they would put the number on an abacus and use that.

The Egyptians did multiplication and division using a method of 
doubling and halving, which is described here:

  Egyptian Method of Multiplication
    http://mathforum.org/library/drmath/view/57542.html 

  Egyptian Division
    http://mathforum.org/library/drmath/view/57574.html 

  Egyptian Division, Shelley Walsh
    http://faculty.ed.umuc.edu/~swalsh/Math%20Articles/EgyptDivide.html 

That method didn't require memorizing multiplication tables (which 
would have to be different from one place to another).  Actually, the 
idea that one could memorize such tables was considered preposterous, 
at least for most people, long after Hindu-Arabic numerals became 
popular.

Each operation can in principle be done in any numeral system by 
thinking in terms of place value.  For example, to add CIV and CCXCII, 
you would arrange each number in columns (as on an abacus, or in a 
place-value system) and add the columns:

     C    IV
  + CC XC II
  ----------
   CCC XC VI

I can see why division would be the hardest to do this way; estimating 
a quotient requires some sort of familiarity with multiplication 
tables.  Probably anything reasonable you do will amount to either 
translating (bit by bit) into Hindu-Arabic, or thinking in terms of an 
abacus, or using the Egyptian method.  We could try to demonstrate 
several such techniques, but you get the idea: if you want to do as 
the Romans did, you won't bother!

I'll just try the Egyptian division using Roman numerals, since you 
asked for a sample.  Here is 153 / 9, as in the link above:

      I  |  IX   *      C  L III       X  VI
     II  |  XVIII      -C XL IV       +    I
     IV  |  XXXVI      ---------      ------
   VIII  |  LXXII            IX        X VII
    XVI  |  CXLIV*

I doubled starting at I and at IX until the next doubling would take 
me past CLIII; then subtracted each number in the second column that 
I could, marking those I used with *; then added the corresponding 
numbers in the first column to get my answer, XVII.

If you have any further questions, feel free to write back.

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