As was correctly answered in the comments to the previous post, I am now 40 years old (XL) and the next time that my age in Roman numerals will be the same length as my age in Western (Hindu-Arabic) numerals will be when I am 51 (LI). 49 is not a correct answer in this case because the Romans did not habitually use subtraction in this way; irregular formations like IL (49) and XM (1900) do sometimes occur irregularly, but normally one cannot ‘skip’ a power. I can only be subtracted from V and X; X can only be subtracted from L and C; and C can only be subtracted from D and M.
As any schoolchild can tell you, one of the purported disadvantages of Roman numerals is that their numeral-phrases are long and cumbersome (e.g., 37 vs. XXXVII). And of course, for many numbers that is true. But for many other numbers (e.g., 2000 vs. MM) the Roman numeral is equal in length or shorter than its Western numeral counterpart. Note, in particular, that round numbers tend to be those that are shorter in Roman numerals; this is because, since the Roman numerals don’t have a 0, that numerals that in Western notation would have a 0 have nothing in their Roman counterpart.
Among numbers whose Roman numeral and Western numeral notations are exactly the same length, there doesn’t initially appear to be much of a pattern:
1, 5, 11, 15, 20, 40, 51, 55, 60, 90, 102, 104, 106, 109 …
but then we see a new sequence emerge –
111, 115, 120, 140, 151, 155, 160, 190 …
which are just the numbers in the sequence from 11-90 with an added C on the front, which makes sense, since you’re just adding a 1 in front of them, similarly.
You could be forgiven in thinking that these numerals come up frequently, since we’re in the midst of a giant cluster of years with this property –
… 2002, 2004, 2006, 2009, 2011, 2015, 2020 …
but then after that, it’s another 20 years before 2040.
Unsurprisingly, numerals containing 3 rarely have equal-length Roman equivalents and numerals containing 8 never do. So once you get past 3000, these numbers become extremely rare. Roman numerals don’t have a standard additive representation for 4000 and higher; you can write 4000 as MMMM, but normally one would expect a subtractive expression with M (1000) subtracted from 5000. There are Roman numerals for 5000, 10000, 50000, 100000, etc., but they are extraordinarily rare, and the Romans during the Empire instead tended to place a bar (or vinculum) above an ordinary Roman numeral to indicate multiplication by 1000; thus, IV=4000. The addition of this feature creates a real conundrum: does the vinculum count as a sign or not? If it does, then IVI=4001 has four signs; if not, then IVII=4002 does.
I’ll leave this aside and stop here to save all of our brains. Thanks for playing!