Chapter 1 Abacus: An Early Computer - Computer Science Home Page

15 Pages·2011·0.32 MB·English

Chapter 1
Abacus: An Early Computer
An abacus is a device, usually made of wood, or
plastic in recent times, which has a frame that
holds rods with freely-sliding beads mounted
on them.
An abacus usually consists of 13 columns of
rods, separated into two parts: the upper part
contains two rows of rods, each representing
a 5, and the lower part contains 5 rows, each
representing a 1.
The following figure shows a typical abacus.
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Where can it be used?
Abacus can be used to carry out normal oper-
ations such as addition, subtraction, multipli-
cation, and division.
Each column in an abacus represents a place,
e.g., the rightmost column represents one’s
place, the next the ten’s place, etc..
In each column, when counting by 1, you push
up a rod in the lower part, until all the five
rods are pushed up; when you move down the
bottom rod in the upper part and restore the
five rods in the lower part.
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How is it used?
A significant part of using abacus is that there
is a system of “rule”s, which you just keep in
your mind after a lot of practice. Most don’t
know anything about the “why” part, they only
know “what” to do.
In other words, an intelligent activity is con-
verted to a mechanical one.
This is just like what we do with computers,
where we incorporate our knowledge into pro-
grams, with the difference being a computer
will carry them out, without knowing anything
about this knowledge.
Let’s check out a video clip....
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How to add a 1 to a column?
There could be three cases, depending on the
current situation:
•
“One: Up 1”, if, in the column, at most
three rods have been move up; or
•
“One: Down 5 and remove 4”, if all the
four rods have been moved up, (4+1=5);
or
•
“One: Remove 9 and forward 1”, if not
only all the four rods in the lower part are
moved up, but one of the rod in the upper
part is also moved down (9+1=10).
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Other rules
Such rules also exist for adding 2, 3, ..., 9,
which we have to apply according to the situa-
tion, as well as for subtraction, multiplication,
division, find out the square root of a number,
etc.
For details, please check out the included ref-
erence material, such as Abacus: The art of
computing with beads, available at
http://www.ee.ryerson.ca/ elf/abacus/.
They might look intimidating at first, but, with
enough practice, it can be done very quickly,
particularly combined with mental mathemat-
ics.
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An example
As a demonstration, let’s try to sum up 1
through 10, i.e., by using a real abacus or,
more easily, the computer simulator, e.g., the
one in
http://www.tux.org/~bagleyd/java/AbacusAppC.html
Its link can also be found in the course page.
To calculate
.
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10
we go through the following steps:
1. Start with 1, just apply “One: Up 1”.
2. To add 2, the rule is simply “Two: Up 2”.
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3. To add 3, as three rods in the lower part
have been pushed up, the rule in this case is
“Three: Down 5 Away 2”, i.e., 3=5-2. We will
move down two rods in the lower part, while
moving down one rod in the upper part of col-
umn 1, with the combination of one rod move
down in the upper part and one rod up in the
lower part of column 1 representing 6=5+1.
4. To add 4, as one rod in the lower part is
moved up, and one rod in the upper part is
moved down, the rule is “Four: Remove 6 and
forward 1”, i.e., 4=10-6. We thus restore both
rods in column 1 and push up one rod in the
lower part of column 2.
5. To add 5, as column 1 is now empty, we
just apply “Five: Up 5 ” by moving down a
rod in the upper part of column 1. We now
have 15.
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6. To add 6, since a rod is down in the upper
part, the rule is “Six: Up 1, remove 5 and
forward 1”, i.e., 6=11-5. We thus move up
one rod in the lower part of column 1, while
restoring the rod in the upper part of column 1
and move up another rod in the lower part of
column 2. We now have 21, two rods and one
rod in column 2 and 1, respectively.
7. To add 7, it is easy(?). The rule is simply
“Seven: Up 7”. We now have 28.
8. To add 8, we have to apply “Eight: Away
2 and forward 1”, i.e., 8=10-2. We will move
down two rods in the lower part of column 1,
and move up another rod in the lower part of
column 2. We now have 36 in the abacus.
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9. To add 9, we have to apply the rule of
“Nine: Away 1 and forward 1”, i.e, 9=10-
1. More specifically, we restore the rod in the
lower part of column 1 and move up another
rod in the lower part of column 2, as 9+1=10.
We now have 45.
10. Finally, to add a 10, in the current situa-
tion, the rule used here is “Ten: Down 5 and
Remove 4”, as 1=5-4. We restore the four
rods in the lower part of column 2 and move
down one rod in the upper part of column 2
to get a 55, which is the correct answer of
· · · .
1 + 2 + + 10
We can certainly continue....
Rules for other operations such as subtraction,
multiplication and division also exist. They are
much more complicated to grasp, but once you
have, it will take much less time to get the
results.
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Abacus vs. calculator
The following site contains some interesting
data, comparing the time it took a Japanese
using an abacus to carry out various operations
vs what it took a calculator to do the same
stuff.
http://www.ee.ryerson.ca:8080/~elf/abacus/abacus-contest.html.
Although it is rather old, similar comparative
results have been reported in recent time.
It is rather apparent that abacus is much faster
in terms of addition and subtraction, but not
so in terms of multiplication and division.
We indeed saw last time how Arabian kids used
abacus to do calculation and actually beat an
calculator.
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