The Base of Roman Numerals

Date: 02/19/2004 at 03:06:50
From: Emma
Subject: Roman Numerals

What base does the Roman Numeral system use?  It appears to have two
bases.

Date: 02/19/2004 at 12:07:28
From: Doctor Peterson
Subject: Re: Roman Numerals

Hi, Emma.

This question is discussed here:

  Base of Roman Numerals
    http://mathforum.org/library/drmath/view/52587.html 

The system is essentially base 10, since a numeral can always be 
broken into parts for each power of ten:

  M CM LX VII
  1  9  6  7

It can be described as a combination of bases 2 and 5, since the 
values of the symbols involved are either 2 or 5 times the value of 
the previous symbol:

  I    V    X    L   C    D     M
  1    5   10   50  100  500  1000
   *5   *2   *5   *2   *5   *2

But that doesn't really make it base 2 or base 5, and since it is not 
a place-value system, the role of 2 and 5 is not very significant.  No 
powers of 2 or 5 are involved, only powers of 10 times 1 or 5.  That's 
why I prefer to think of it as a modified base-10 system influenced 
by base 5.

It's interesting, though, that the abacus (which IS a place-value 
system) uses the same trick of splitting each decimal digit into two 
parts, one base 2 (two beads representing fives, only one of which is 
actually needed) and one base 5 (five beads representing ones).  Roman 
numerals, apart from subtractive notation (as in IV for 4), represent 
well the state of such an abacus, with the digits corresponding to 
each power of ten showing how many 1's and how many 5's there are in 
that "digit".

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

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