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### Adding and Subtracting Roman Numerals

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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

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/
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