[BITList] Doing maths with Roman numerals

[John Feltham]

     ROMAN NUMERALS  
                1997       
 
Europe adopted the Arabic way of presenting numbers; 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. (Some say the concept really came from India).
 
Just think how cumbersome it would be if we had to do math in Roman Numerals. The Roman system consists of seven numbers represented by seven capital letters:
 
I = 1; V = 5; X = 10; L = 50; C = 100; D = 500; M = 1000.
 
Most Roman numbers use the principle of addition, for instance XI = 10 + 1 = 11, but 4s and 9s use subtraction, so IV = 5 - 1 = 4,  IX = 10 - 1 = 9.  So the numbers from 1 to 20 are:
 
I    = 1
II   = 2
III  = 3
IV   = 4
V    = 5
VI   = 6
VII  = 7
VII  = 8
IX   = 9
X    = 10
XI   = 11
XII  = 12
XIII = 13
XIV  = 14
XV   = 15
XVI  = 16
XVII = 17
XVIII= 18
XIX  = 19
XX   = 20

 
This works the same way with the higher numbers:
 
XL   = 40  (50  - 10)
LX   = 60  (50  + 10)
XC   = 90  (100 - 10)
CX   = 110 (100 + 10)
 
The Roman system really gets complicated when you want to write down irregular numbers;
 
3,625 would be: MMMDCXXV
 
How did I do that?
 
Start with MMM (3 times 1000)
 
Add DC  (500 + 100) and then
 
XXV  (10 + 10 + 5)
 
The Romans knew that their system was hard. They finally devised a way to write large numbers by using symbols. One of the most important was the vinculum.
The vinculum is a line drawn over the Roman Numerals you are working with and the line or vinculum means "multiply the number under the vinculum by 1000":
 
 
            _________________
                MMMMM           OR   5,000,000
 
 The Romans built fantastic bridges, aqueducts, the Apian Way, fine buildings, and ships; all using Roman Numerals to do their engineering mathematics. We need to give them credit. They ruled the known world for 1000 years and we owe them even more for keeping the Christian Church alive. Some sample problems:

 
            XVII                     XVII
            + XI                       XI
 
          ______                   ______
              XI                      X)XX
             x I
 
 
 
 
            XLIV                          CIX
              CV                        - XLII
            + IX
 
 
 
             XLV                 _________
         x   CIX              LIV)MCMXXXIV
 

So to add, without converting, is fairly simple but tedious since there is no concept of base:
To add, one expands to the basics number, e.g. iv become iiii, xl becomes xxxx, etc.
Then strings all the characters together in decreasing order, the collects lower numbers into higher numbers.
Subtraction is done the same way. 
Multiplication and division with Roman numerals is a nightmare if you don't convert.

Really makes one appreciate numbers systems where position indicates exponent.

xvii + xi = xxviii

xliv + cv + ix = xxxxiiii + cv + viiii = cxxxxvviiiiiiii = clviii

Subtracting is done the same way

xxv - viii = xx - iii = xviiiii-iii = xvii

Multiplication

xvi X iii means add xvi together iii times
xvi + xvi + xvi = xxxvvviii = xxxxviii = xlviii 
(i)     (ii)    (iii)

Division is more nightmarish

xix / iv means subtract iv from xix until there is nothing left or what is left is less than iv and keep track of the number of times that the subtraction is done.

i. xix - iv = xv
ii. xv - iv = xi
iii. xi - iv = viiiiii - iiii = vii
iv. vii - iv = iiiiiii - iiii = iii
since iii < iv, then answer is iv remainder iii

OUCH!!!!

-------------- next part --------------
An HTML attachment was scrubbed...
URL: http://lists.bcn.mythic-beasts.com/pipermail/bitlist/attachments/20100509/f4ce856f/attachment-0001.shtml