Adding and Subtracting Roman Numerals

Date: 10/07/97 at 15:25:36
From: meghan
Subject: Adding and subtracting Roman numerals

Dear Dr. Math,

I am a student of St. Johns University in Queens, New York. I don't 
have a specific question, but do you have any suggestions for how to 
teach adding and subtracting of Roman numerals?

It would be great if you could help!

Thanks!
Meg

Date: 10/10/97 at 11:47:54
From: Doctor Luis
Subject: Re: Adding and subtracting Roman numerals

The basic roman numerals are (i,v,x,L,C,D,M,....) which correspond to 
groups of the basic numeral i, and, in our decimal notation, to
(1,5,10,50,100,500,1000,...)

The Roman numerals, defined by the successor property, are the 
following: 

   i      --> i
   ii     --> i+i
   iii    --> ii+i  --> i+i+i
   iiii   --> iv    --> v-i
   iiiii  --> iv+i  --> v-i+i --> v
   vi     --> v+i   
   vii    --> vi+i  --> v+i+i
   viii   --> vii+i --> ..... --> v+i+i+i
   viiii  --> viii+i--> ix 
   viiiii --> ix+i  --> x-i+i --> x
   xi     --> x+i
   xii    --> xi+i
           .
           .
           .


Let a be a basic numeral. Then, in general, the juxtaposition or 
concatenation of a with itself (or another numeral) is taken to 
signify addition:

   a     = a 
   aa    = a+a
   aaa   = a+a+a
   aaaa  = a+a+a+a 
   aaaaa = a+a+a+a+a

This last number is generally represented by another Roman numeral b, 
defined by

   b = aaaaa

and bb is represented by yet another Roman numeral

   c = bb = aaaaaaaaaa

One can clearly see that b is a^5, and that c is a^10

Such usual group definitions are

  i          --> i  (identity numeral or basic pure numeral)        
  iiiii      --> v
  iiiiiiiiii --> vv --> x
  xxxxx      --> L
  xxxxxxxxxx --> LL --> C
  CCCCC      --> D
  CCCCCCCCCC --> DD --> M

Similarly, one can define higher groups of (pure) numerals.

Composite numerals are of the form ba, ab, abb, or abbb. (where b<a)
and could be defined as

   ba   = a-b (in other words, ba is the number you add to b to get a)
   ab   = a+b
   abb  = a+b+b
   abbb = a+b+b+b

   a and b, of course are pure numerals.

Usually, groups of the same four pure numerals are written in the 
composite form
  
    bbbb --> bbbbb-b --> a-b --> ba


Now, Roman numerals, in general, are written as a concatenation of 
either pure or composite groups of numerals, and writing the greater 
groups first:

  as in 1567 = MDLXVII  (1000+500+50+10+5+2)


Here I have some examples which you might want to examine. I have 
included the equivalent operation in decimal numbers to the right 
(don't you love place-value notation?) (notice that I'm using the 
definition of the numerals to evaluate the products, sums, and 
subtractions)

  ii*iv = ii*(v-i)      2*4 = 2*(5-1)
        = ii*v - ii*i       = 2*5-2*1
        = vv - ii           = 5+5-2
        = v+v-ii            = 5+5-2
        = v+iiiii-ii        = 5+(1+1+1+1+1)-(1+1)
        = v+iii             = 5+(1+1+1) = 5 + 3
        = viii              = 8

  xxi+xxxiv = xxi+xxxiiii       21 + 34 =  (20+1)+(30+4)
            = xxxxxiiiii                =  (20+30)+(1+4)
            = LV                        =  50 + 5 = 55

  Civ - xLvii = LLiv - xxxxvii      104 - 47 = (50+50+4) - (40+5+2) 
              = (LL+iv)-(L-x+v+ii)           = (50+50+4) - (50-10+5+2)
              = L+L+iv - L+x-v-ii            =  50+50+4  - 50+10-5-2
              = (L+L-L)+(x+iv-v-ii)          =  50+50-50 + 10+4-5-2
              = L + (x+v-i-v-ii)             =  50       + 10+5-1-5-2
              = L + (x-i-ii)                 =  50       + 10-1-2
              = L + (ix-ii)                  =  50       + 9-2  
              = L + (viiii-ii)               =  50       + 5+4-2   
              = L + vii                      =  50       + 5+2
              = Lvii                         =  57 
              = LVII

If you have any more questions, feel free to ask again :-)

-Doctor Luis,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/