Q&A #4130

European long division

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From: Pat Ballew (for Teacher2Teacher Service)
Date: Jun 16, 2000 at 18:30:50
Subject: Re: European long division

Here are two additional responses I got to my search for your answer..
The first is from Fernado Gouvea of Colby College and is similar to the
letter from Flo I sent a few days ago.  The second is from Martin Flashman
who provides comments and a link to a page about Stevin, who perhaps can be
called the father of decimal arithmetic..  :

I grew up in Brazil, but attended an American school there, so one of the
more interesting mathematical experiences I had as a young boy was to sort
out why the long division algorithm I learned in school looked so different
from the way my mother did it... In the end, the differences are merely in
the layout of the numbers, I think.

Let's take a specific computation: 133 divided by 12 is 11 with remainder
5. To see this right, you'll need to be using a monospaced font.

The Americans taught me to do it like this:

                12 ) 137

In Brazil, you'd typically write this:

                        137 |12
                         17 ----
                          5  11

In both cases I've approximated the actual symbol used. In the Brazilian
division, the divisor is inside an L-shaped symbol. In the American setup,
the dividend is inside the symbol that looks like a denatured square root
symbol. (In fact, when Brazilian friends saw me write that, they often
assumed I was computing a square root!)

One curious effect of these arrangements is linguistic. In Portuguese, you
always read it as "137 divided by 12". In English, that's allowed, but
"dividing 12 into 137", or simply "12 into 137" or "12 goes into 137" are
more common. Some people have even suggested that the name of the symbol
used for division is "guzinta" (read to sound like "goes into").

Apart from the change in position of dividend and divisor, the main
difference is that the subtractions are typically done mentally rather than
written out in full (though I've seen people do it that way too). The
algorithm is really exactly the same.


Fernando Q. Gouvea
Department of Mathematics               Editor, FOCUS and MAA Online
Colby College                           http://www.maa.org
Waterville, ME 04901

This recent discussion reminded me of the organization of the work for
division by Stevin in his brief treatise on the use of decimal notation
for fractions. You can see this (in part) on the web at
http://home.wxs.nl/~hopfam/Dime.html .

I think you may find this of interest on the issue of how the organization
of the work has changed.
BTW, IMHO Stevin's development of decimal arithmetic was one of the most
important developments in mathematics and western science and culture.

from sunny Humboldt County, California

Hope these help.

 -Pat Ballew, for the Teacher2Teacher service

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