An interesting multiplication algorithm from
*Capitalism and Arithmetic*
is known as *gelosia* or* graticola*.

It got that name from the diagonally latticed windows in Venice, from which cloistered women
watched the world. Those windows are also the root word for jealousy, and in contemporary America are called jalousy windows. The alternative name, *graticola*, is Italian for grill.

I'm not sure why they don't still teach this technique in elementary schools - it's pretty cool once you get the hang of it. However, it does not lend itself to HTML, but I'll do my best.

Suppose you want to multiply 934 by 314 [pp. 208-9]. You would write out the 934 in a row, and the 4 and then the 1 and then the 3 (of the 314) going down the right hand side. Like this:

I intentionally left a little space at the left of each row. (I found a somewhat different layout on the internet in this photo).

With imagination, you can envision this is as a window; add diagonals and you have the Venetian gelosia.

OK. Now, multiply the top term in the right hand column times the right term in the top row:4 times 4 is 16.

The trick is to write the answer in a position that corresponds to those two numbers, but with the units digit to the "northeast", and the tens digit to the "southwest".

Now, continue doing that for all the other numbers along the right and top.

When you're done it should look like this:

I put the zeroes in as placeholders, but they are not necessary. If you look carefully, you can can see that there are some diagonals - envision them running from "northwest" to "southeast".

Anyway, the technique is easy now. Just add up the numbers along the diagonals.

Starting at the "northeast" the first diagonal only contains a 6. Write this where you want your answer. The second diagonal contains 2, 1, and 4, and sums to 7. I marked this diagonal in red. Write the 7 to the left of the 6, and continue to the third diagonal. This contains a 6, 1, 3, 0, and 2. I marked these in green on the figure.

Now we have to carry (which is a problem in all methods of multiplication).

These green values sum to 12. Write the 2 to the left of the 7. Carry the 1 from the tens digit of 12 to the top left of the next diagonal (marked in blue), which now sums to 23. Write the 3 next to the 7 in the answer, carry the 2 and add up the next (cyan) diagonal to get 9. Write that next to the 3 in the answer, and do the last diagonal (in magenta). This sums to 2 which is the leftmost term in the answer. You should have 293,276.

Even though this method isn't taught any more, it has a huge advantage, and two smaller ones. The huge advantage is that in doing multiplication the conventional way, it is very easy to make mistakes when carrying: most of us make a tick mark, and we have to remember not to count it more than once. This method does not have that problem, since the carries don't get lined up on top of each other. One smaller advantage is that the method we learn in school requires us to multiply at each step and then remember to add anything we carried, whereas this method completely separates the multiplication and addition. The other smaller advantage is that when we get a result that requires carrying in the conventional method, we have to write part of it below "the line" and part of it above the terms being multiplied, instead of close by each other. This makes checking more difficult.

This is difficult to see because you are used to it. Here's the conventional method:

First multiply the 4 times the 4 to get 16. Write it like this:

That shows the third advantage: the ones and tens digit have to be separated in this method.

Note that I've included all the space and lines we'll need; one reason that *gelosia* seems intimidating is that we don't realize how much advance planning is required to lay out conventional multiplication. In *gelosia*, I did it first because I had to, but we are more cavalier with the conventional method we know so well.

Continuing on, multiply the 4 times the 3 and add the tick mark, which is written like this:

That step shows the third advantage once again. However, it also shows the second and
first advantages as well. The second advantage is that you have to multiply *and* add
at this step, something that never comes up in the *gelosia* method. The first
advantage is more subtle: I had to remember to strikethrough the first tick mark after I used it,
and then put a new one in a different place.

When all is said and done, this method doesn't look that easy either.

It turns out the *gelosia* method is also more efficient. Both techniques require you
to do 9 multiplications. *Gelosia* requires you to do more additions: 14 of them, as
opposed to 10 in the conventional method. However, to save those 4 addition operations, the
conventional method requires you to do 6 carries, 3 strikethroughs, and you have to remember
to shift two rows to the left (and the top row of carries too) to get your columns in the right spots. *Gelosia* only
requires 2 carries. In sum, *gelosia* requires 25 operations of 3 different types, while the
conventional method requires 31 operations drawn from 5 different types. The conventional
method requires you to write 17 digits in the intermediate stage before you get an answer
(and that is not counting superfluous zeroes). The *gelosia* only requires you to write
16 intermediate numbers.

It's also easier to check *gelosia*: all of your products are written out in obvious
spots with the tens and units digits close by each other, whereas the conventional method
sometimes combines multiplication and addition to
yield a single answer which is then split into parts that are no longer contiguous.

As a final check on this, I timed myself doing some multiplication of 2 random 3 digit
numbers. I made a point of doing them the formal way for both methods. After 35 years of
experience, I can do the traditional method in 27 seconds, and having only done the
*gelosia* about a dozen times, it took me 39. However, 14 of those 39 seconds were
spent drawing the *gelosia* matrix. The actual calculation time was smaller. I didn't realize until I was done that I am used to doing the conventional method without vertical rules to line everything up, but if I'd had to draw them the time saved would probably be dissipated.

I wonder if computer programmers know about this algorithm? How much money could be saved worldwide if multiplication operations were done in this more elegant manner?

N.B. This is the post that crashed Typepad on Thursday morning, and made most of my blog disappear. It was very easy to write this the first time around using HTML tables instead of images, but apparently it was more than Typepad could handle.

UPDATE: Craig Depken of Heavy Lifting and UT-Arlington pointed me to this video showing a different technique, and from there I found this video of *gelosia*:

http://www.otsu.ed.jp/ktt-e/bbs/bbs.cgi/bbs.cgi%20%20%20%20%20%20%20%20%20%20%3C/b%3E%20%3C/bbs.cgi

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