The author claims, apparently confidently, that the Romans used this method but had no idea why it worked. It's not clear how he knows this.
I dare say most people who used it didn't know why it works, but then probably most people today don't know why long multiplication works. Barring good evidence to the contrary, I bet that whoever thought the method up had a pretty good idea of why it worked.
Of course it's possible that the Romans just stole the idea from the Egyptians. Or that someone thought up the method and then went to his grave without ever explaining it to anyone else, and that no one else tried to understand it. But I don't believe the author knows.
Incidentally, you can understand the method without having heard of the binary system. For instance, here's how I might explain it to an ancient Roman who happened to speak English. I might start by putting together a rectangle with, say, 10 rows of 4 stones.
If you double one number and halve the other, without any remainder, then the product of the numbers remains the same. (Rearrange the rectangle to be 5 rows of 8.)
On the other hand, if the number you're halving is odd, there is a remainder and the product of the numbers has decreased -- by exactly the value of the other number. (Rearrange the rectangle again: 2 rows of 16, with a "half-row" of 8 left over.)
As you keep doing this, the product stays the same every time you halve an even number (rearrange again: 1 row of 32) but when you halve an odd number you always lose some, as we did a moment ago. At the very end you end up with 1 times something, and of course that product is easy to do.
Now the product of the original numbers is just your final 1-times-something product, plus the bits you lost along the way. We're done.
(It's more elegant mathematically to make zero-times-something the base case, but probably harder to explain.)
Agreed. I have often noticed a tendency among professionals who should know better, to presume that ancient peoples were somehow less intelligent than we are, and less capable of applying their minds to complicated and even abstract problems. It is just plain not true, and there is so much anthropological evidence to demonstrate this fallacy that it is idiotic that it is still repeated.
In this case, it is an especially bad assumption, given that he is talking about a system of mathematics that significantly post-dates the great Greek mathematicians. If Archimedes could invent a method to solve problems today solved by integral calculus, 2000 years before Newton, and implemented Riemann sums in his methods, I hardly think that this little trick confounded explanation at the time it was used, particularly since it works with ANY system of notation.
I don't think the question you're answering ("why were Roman numerals replaced by Arabic?") is the same as the one that was being asked ("why did people using Roman numerals switch from IIII to IV?").
I think the answer to that latter question is unknown.
Ah. I completely misunderstood the question, having just finished reading The Man Who Loved Only Numbers, which had a brief digression on Fibonnaci's evangelism of Arabic numerals to the Pisan mercantile class. Mea culpa.
Contrary to the claim of the article, my understanding is that the Romans actually did their calculations using the abacus. Furthermore the fall of the abacus went hand in hand with the adoption of Arabic numerals.
On a slightly different note, someone noticed that this same article was posted to reddit four years ago and today. Browse the comments and draw your own conclusions about the broadening of a community.
This article is flatly wrong. Some early Romans may well have used so-called 'Roman numbers,' which retained some common use among the masses throughout the years; but any Roman of real education, particularly a Roman geometer or mathematician, of an era after around 100 BC would've used the more convenient and refined Greek numerals, which were base-10 just like our own, and which were in fact a bit more complex than our numbers in that they used separate characters for the tens and hundreds slots. In fact, the Greek system of numerals predates the Roman by hundreds of years. [See here: http://en.wikipedia.org/wiki/Greek_numerals]
This belief that Arabic numerals were the first base-ten system of numbers, that our own modern methods are vastly superior to the methods of the past, or that, as the author of this article puts it, "the Romans did not know anything about the binary system," is silly misinformation. And for what it's worth, Apollonius of Perga's treatise "On Conic Sections" is probably the most complex and beautiful mathematical treatise I know of before or since - and yes, that's counting everything from the modern era.
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[ 10.6 ms ] story [ 45.1 ms ] threadI dare say most people who used it didn't know why it works, but then probably most people today don't know why long multiplication works. Barring good evidence to the contrary, I bet that whoever thought the method up had a pretty good idea of why it worked.
Of course it's possible that the Romans just stole the idea from the Egyptians. Or that someone thought up the method and then went to his grave without ever explaining it to anyone else, and that no one else tried to understand it. But I don't believe the author knows.
Incidentally, you can understand the method without having heard of the binary system. For instance, here's how I might explain it to an ancient Roman who happened to speak English. I might start by putting together a rectangle with, say, 10 rows of 4 stones.
If you double one number and halve the other, without any remainder, then the product of the numbers remains the same. (Rearrange the rectangle to be 5 rows of 8.)
On the other hand, if the number you're halving is odd, there is a remainder and the product of the numbers has decreased -- by exactly the value of the other number. (Rearrange the rectangle again: 2 rows of 16, with a "half-row" of 8 left over.)
As you keep doing this, the product stays the same every time you halve an even number (rearrange again: 1 row of 32) but when you halve an odd number you always lose some, as we did a moment ago. At the very end you end up with 1 times something, and of course that product is easy to do.
Now the product of the original numbers is just your final 1-times-something product, plus the bits you lost along the way. We're done.
(It's more elegant mathematically to make zero-times-something the base case, but probably harder to explain.)
In this case, it is an especially bad assumption, given that he is talking about a system of mathematics that significantly post-dates the great Greek mathematicians. If Archimedes could invent a method to solve problems today solved by integral calculus, 2000 years before Newton, and implemented Riemann sums in his methods, I hardly think that this little trick confounded explanation at the time it was used, particularly since it works with ANY system of notation.
I think the answer to that latter question is unknown.
http://www.reddit.com/r/reddit.com/duplicates/78ud/learn_to_...
This belief that Arabic numerals were the first base-ten system of numbers, that our own modern methods are vastly superior to the methods of the past, or that, as the author of this article puts it, "the Romans did not know anything about the binary system," is silly misinformation. And for what it's worth, Apollonius of Perga's treatise "On Conic Sections" is probably the most complex and beautiful mathematical treatise I know of before or since - and yes, that's counting everything from the modern era.