

A Different Kind of Multiplication - oscardelben
http://www.phy6.org/outreach/edu/roman.htm

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gjm11
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.)

~~~
teilo
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.

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Jermey128
This is also known as Russian peasant multiplication.

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raganwald
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.

[http://www.reddit.com/r/reddit.com/duplicates/78ud/learn_to_...](http://www.reddit.com/r/reddit.com/duplicates/78ud/learn_to_multiply_the_ancient_roman_way/)

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btilly
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.

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koeselitz
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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pcestrada
Here's another way, called Mayan Multiplication:
<http://www.youtube.com/watch?v=wA0VLbPGorI>

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tjmaxal
Why did Roman numeral notation change in the middle ages?

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Daemmerung
Fibonnaci and his _Liber abaci_ , written after he encountered the superior
Indo-Arabic system. See <http://pass.maths.org.uk/issue3/fibonacci/index.html>

~~~
gjm11
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.

~~~
Daemmerung
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.

