This has been attributed to so many places. I actually spent some time coming up with an algebraic proof that it worked when I was in college. It turns out that it is perfectly clear what is happening if you write both numbers in binary as another poster has pointed out.
However for historical purposes, the question is why? I think there is a very simple reason. Before you have based number systems, most number systems are usually derived from tally sticks. This makes some functional sense, and once you think about it from a tally stick, roman numerals make perfect sense. For example consider the number XXIV. It represents a position on a tally stick:
Count to the second X and then one before the V. IXL is the same. Count backwards one X and then one I from the L. So while our system of numbers today is based on position of digits, many earlier systems were based on visualized markings on tally sticks. So you have numbers without a base system. This divide and add approach works very well when you don't have a number system. You can take your tally stick, figure out the number, and the same on the other side. I cannot think of another way to do multiplication on a tally stick.
Interestingly tally sticks of a slightly different sort were in use up until the early 19th century in Britain as tax receipts and in 1836 the attempt of the British government to get rid of these humble staves leveled both houses of parliament. Before you have widespread numeracy, tally sticks are how you track amounts for everything from debts to receipts. It isn't surprising that this would be a widespread method.
The interesting part is that Romans wouldn't have written 24 as XXIV. They'd have written it as XXIIII, which makes it much simpler because you only ever go left to right..
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This has been attributed to so many places. I actually spent some time coming up with an algebraic proof that it worked when I was in college. It turns out that it is perfectly clear what is happening if you write both numbers in binary as another poster has pointed out.
However for historical purposes, the question is why? I think there is a very simple reason. Before you have based number systems, most number systems are usually derived from tally sticks. This makes some functional sense, and once you think about it from a tally stick, roman numerals make perfect sense. For example consider the number XXIV. It represents a position on a tally stick:
IIIIVIIIIXIIIIVIIIIXIIIIVIIIIXIIIIVIIIIXIIIIVIIIIL
Count to the second X and then one before the V. IXL is the same. Count backwards one X and then one I from the L. So while our system of numbers today is based on position of digits, many earlier systems were based on visualized markings on tally sticks. So you have numbers without a base system. This divide and add approach works very well when you don't have a number system. You can take your tally stick, figure out the number, and the same on the other side. I cannot think of another way to do multiplication on a tally stick.
Interestingly tally sticks of a slightly different sort were in use up until the early 19th century in Britain as tax receipts and in 1836 the attempt of the British government to get rid of these humble staves leveled both houses of parliament. Before you have widespread numeracy, tally sticks are how you track amounts for everything from debts to receipts. It isn't surprising that this would be a widespread method.
The interesting part is that Romans wouldn't have written 24 as XXIV. They'd have written it as XXIIII, which makes it much simpler because you only ever go left to right..
the Romans, IIRC were extremely inconsistent. Sometimes, for example, you see IIXX for 18. However either way it represents a place on a tally stick.