Longest Roman Numeral

Date: 05/03/2007 at 11:59:22
From: Vince
Subject: trying to find greatest number

Which year in our recorded history has had the greatest number of 
Roman numerals in it?

I haven't used Roman numerals for many years.  This question came up 
in a trivia session and no one could come up with an answer.

Date: 05/03/2007 at 13:53:01
From: Doctor Ian
Subject: Longest Roman numeral for what year? 

Hi Vince,

Well, note that there are multiple ways to write a given number using
Roman numerals.  For example, '4' is usually written as 'IV', but if
you look at lots of watches and clocks with Roman numerals on their
dials, you'll frequently see it written as 'IIII', because that has
more symmetry with the 8, which is written 'VIII'.

So you could write the current year as

  IIIIIIIIIII...III

which would give you 2007 numerals.  And if you decide to use the
Hebrew calendar, the current year is 5767, so that's quite a lot 
bigger.  And I wouldn't be surprised to find that there are other
calendars that I don't know about, with even larger years.  

Probably the person who formulated the question meant to ask:

  Using the minimal Roman numeral representation, what 
  year since 1 AD, using the standard civil calendar, uses 
  the greatest number of digits to express? 

But if he didn't say that, then you can just start slapping I's
together.  :^D

Note that Excel has a 'roman' function, so you can enter something like 

  =roman(2007)

into a cell, and get back something like 'MMVII', and 

  = len(roman(2007))

to get the number of digits (5).   

So for fun, I set up a spreadsheet to do this for all the integers
from 1 to 2007, and it turns out that the longest string is for 1888,

  1888 = MDCCCLXXXVIII

which requires 13 digits.  

As soon as you see it, it makes sense that this would be the winner...
but I would have had trouble figuring that out ahead of time!

Does this help? 

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