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The History of the Abacus

Kevin Samoly, Miami University

The abacus is a counting tool that has been used for thousands of years. Throughout history, calculating larger
numbers has been problematic, especially for the common uneducated merchant. Out of this necessity, the
idea of the abacus was born. Solving problems on an abacus is a quick mechanical process rivaling that of
modern-day four-function calculators. After first addressing basic counting procedures and memorizing a few
simple rules, students can use the abacus to solve a variety of problems. The abacus is a timeless computing
tool that is still applicable in today’s classrooms.

Introduction
Throughout history, keeping track of
numbers has been problematic. How would
you keep track of numbers without pen or
Counting paper? What if you couldn’t read or write?
tables have Without a written language, how would
over 2000 you add, subtract, multiply, or divide large
Fig 2 Bead frame abacus
numbers accurately and efficiently? All
years of of these problems can be solved with an Both computing tools calculate in a similar
documented abacus. fashion.
use, dating An abacus is a computing tool used for Counting tables have over 2000 years
back to addition, subtraction, multiplication, and of documented use, dating back to Greeks
division. The abacus does not require pen and Romans. The earliest documented
Greeks and or paper and works for any base number counting boards were simple stone slabs
Romans. system. There are two basic forms of the with parallel and horizontal lines serving as
The earliest abacus, the first being a counting table, place value indicators. The “normal method
document which is a specially marked flat surface that of calculation in Ancient Greece and Rome
uses small stones or beans as markers, as was done by moving counters on a smooth
counting shown in Figure 1. The second is a bead board or table suitably marked with lines or
boards were frame abacus, which is a frame with beads symbols to show the ‘places’” (Pullan, 1968,
simple stone strung on wires or rods, shown in Figure 2. p. 18). The counting board’s development
slabs with then stagnated for hundreds of years with
the only noticeable change being the move
parallel and to vertical place-values lines.
horizontal The origin of the portable bead frame
lines abacus is not well-known. It was thought
serving as to have originated out of necessity for
traveling merchants. Some historians credit
place value the Chinese as the inventors of bead frame
indicators. abacus, while others believe the Romans
introduced the abacus to the Chinese
Fig 1 Counting table representing 3874 through trade (Moon, 1971). From there,

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Russia and Japan developed their own fourth century BC (Pullan, 1968). It’s made
versions of the abacus. of white marble and measures 149 × 75 ×
Today the abacus lives on in rural parts 4.5 cm. On its top face is a set of eleven
of Asia and Africa and has proven to be parallel lines accompanied with Greek
a timeless computing tool. Its operation numerals. Metal counters were placed
offers everyone the ability to compute large on the board between the lines, and the
numbers without any electrical devices. As number of counters on each line represents
seen in the next section, the simplicity of different values. The vertical spaces were
counting on the abacus provides endless used to indicate base units, tens, hundreds,
computing possibilities to people of all ages etc., and intermediate spaces allowed the
around the world. operator to mark larger values, such as
fives, fifties, etc. Figure 3 depicts this idea.
History Throughout the Ancient Tables could easily be arranged to compute To unravel
Worlds addition and subtraction problems. As time the history
It is certain that mechanical aids have passed and cross-cultural exchange took of the abacus
been used as calculation devices by ancient place, the abacus evolved into a tailored
civilizations. Merchants may have even tool that reflected the operator’s individual
we look next
used a crude form of the counting table needs. at classical
during prehistoric times. Ever since the paintings and
Phoenicians, Egyptians, and Greeks were literature;
involved in Mediterranean trade, the abacus
has become a world-wide computing tool
however, these
(Moon, 1971). historical
Since the abacus’ history is not clear, the artifacts
origin of the tool can only be speculated. do not
Counting boards are thought to be the
predecessor of the beaded abacus. Counting
portray the
boards were likely crudely constructed lower class
wooden tables, lost to decomposition like merchants and
the wooden huts in which they were used businessmen
(Pullan, 1968). To unravel the history of the
abacus we look next at classical paintings
who
and literature; however, these historical frequently
artifacts do not portray the lower class used counting
merchants and businessmen who frequently boards.
used counting boards. “One could hardly
expect the aristocratic Plato, for instance,
to write a treatise on a device used by slaves
and petty tradesmen” (Moon, 1971, 21).
For this reason, the abacus is often left out
of Greek and Roman history.
Salamis Counting Board
The oldest existing counting board was
discovered on the island of Salamis about a
century ago (see Fig 3). It has been estimated
that the counting board dates back to the Fig 3 Salamis Counting Board

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Medieval Counting Boards Other Abaci

Counting boards were used throughout The bead frame abacus and a counting
Europe during the Middle Ages until the board have many features in common but
16th century when they were replaced by are unique to the cultures that use them.
pencil-and-paper calculations. Arithmetic The counting boards are thought to be more
books depicting counting boards became flexible than the bead frame abacus because
the first documentation with the advent of the operator has an unlimited supply of
more modern publication methods (Moon, markers, but the bead frame abacus is
1971). The illustrations often display more compact, portable, and is capable of
two mathematicians solving equations, more rapid computing. In the following
one using a counting table and the other sections we will explore the unique features
calculating with pencil and paper. One of different forms of beaded abaci as they
such painting is shown in Figure 4. evolved around the world.
Roman Abacus
Arithmetic The Roman abacus features sliding
counters permanently attached to the
books depicting device. It consists of a metal plate with
counting fixed counters sliding in grooves. Each
boards . lower grove holds four markers, except for
. . often the right-most groove which holds five.
Similarly, each of the upper grooves holds
display two one marker. Beads in the upper slots have
mathematicians a value five times that of the marker below.
solving Suan-pan
equations, “Some authorities believe that the Roman
abacus was introduced into China early in
one using a the Christian era by traveling merchants”
counting table, (Moon, 1971, p. 30). The earliest mention
and the other of the abacus in Chinese literature did not
calculating occur until the 12th century. The Suan-
Fig 4 Mathematicians solving equations with a pan is widely used throughout China and
with pen and counting table and pen and paper other parts of Asia. The Suan-pan features
paper. five unit beads on each lower rod and two
Medieval counting boards were arranged
‘five-beads’ on each upper rod, as shown in
vertically with the numbers increasing
Figure 5.
as they moved away from the operator.
Soroban
Counters were placed on lines or grooves
The modern Japanese abacus, known as
signifying a specific place-value while larger
a Soroban, was developed from the Chinese
values could be marked between the lines.

Fig 5 Other abaci

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Fig 6 Soroban Abacus

Suan-pan. The Sorban features four unit Each heaven bead in the upper deck has a
beads on the lower rods, and one ‘five bead’ value of 5, each earth bead in the lower deck The Soroban
on the upper rods as depicited in Figure 6. has a value of 1, as seen in Figure 6. The
abacus is
The Sorban also features large sharp-edge beads are considered counted when they
beads that are easily manipulated, and are moved towards the divider separating ideal for a
the distance through which they move is the upper and lower decks. base-ten
relatively short, allowing for high-speed Once it is understood how to count numbering
operation. using an abacus, any integer can easily be
system, in
Schoty found. The general rule is that the value
The Russian abacus, the Schoty, features of any given bead is ten times that of the which each
ten beads per rod and no dividing bar, bead to its immediate right, or one tenth rod acts as
as depicted in Figure 5. This requires the the value of the bead to its immediate left a placeholder
movement of a larger number of beads over (Kojima, 1954). Decimals are represented
and can
a relatively far distance, but is said to be right of the unit rod, also seen in Figure 6.
easier to learn than instruments utilizing There are two general rules for quickly and represent
markers of value five. While all the previous easily solving any addition and subtraction values 0
devices were operated horizontally, the problem with the Soroban abacus. First, through 9.
Russian instrument can also be operated the operator should always solve problems
vertically (Pullan, 1968). from left to right (Kojima, 1954). This may
be confusing at first but will drastically
Abacus Operation increase the speed of operation, and with
In general, all abaci work in a similar a little practice can make computing an
fashion. In this section we explore how easy-to-learn mechanical process. Second,
to operate the popular Soroban abacus. the operator must be familiar with how to
The Soroban abacus is ideal for a base-ten find complementary numbers, specifically,
numbering system, in which each rod acts always with respect to 10 (Kojima, 1954).
as a placeholder and can represent values A number’s complement is found through
0 through 9. On each rod, the Soroban a simple addition problem. The value
has one bead in the upper deck, know as added to the original number to make 10
a heaven bead, and four beads in the lower is the number’s complement. For example,
deck, know as earth beads (Kojima, 1954). the complement of 7, with respect to 10,

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is 3 and the complement of 6, with respect This leaves us with 1 bead registered on
to 10, is 4. rod G (the tens rod) and 2 beads on rod H
(the unit rod), as shown in Figure 9. The
Addition result shows that 8+4 is 12.
When solving addition problems with This rule remains the same regardless of
the Soroban abacus, subtraction is required the numbers used. Consider, for instance
when the addends sum to a value greater the addition problem 356+472. First
than 9. In such a case, the complement is register 356 on rods F, G, and H, with H as
subtracted and 1 bead is added to the next the unit rod, as shown in Figure 10.
highest place value (Heffelfinger and Flom,
2011). For example, consider adding 8 and
This lesson 4. The process begins by registering 4 on
built upon the unit rod H, as shown in Figure 7.
the concept
of measuring
lines using
iterated Fig 10 356 on rods FGH
lengths, and Recall that when performing operations,
successfully we work from left to right. Next, add 4
built a bridge Fig 7 4 on rod H
hundreds to rod F (adding 5 and subtracting
1). The abacus now reads 756, as shown in
from this Figure 11.
concept to Because the sum of the two numbers is
greater than 9, subtraction must be used.
the more We subtract the complement of 8 - namely
abstract 2 - from 4 on rod H and add 1 bead to tens
ruler. rod G. This is illustrated in Figure 8.

Fig 11 756 on rods FGH

Now add 7 tens to rod G. Because

5+7 > 9, the complement of 7 - namely
3 - is subtracted from rod G, and 1 bead
Fig 8 Subtracting 2 and adding 10 added to rod F. The resulting value, 826, is
illustrated in Figure 12.

Fig 9 Final result showing 4+8=12 Fig 12 826 on rods FGH

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The final step is to add 2 to the unit

rod H. This yields the final solution 828
illustrated in Figure 13.

Fig 15 6432 on rods EFGH

Fig 13 828 on rods FGH

Subtraction
As we all know, subtraction is the
opposite operation of addition. Thus,
when subtracting with the Soroban abacus,
we add the complement and subtract 1
Fig 16 1432 on rods EFGH
bead from the next highest place value. For
example, when subtracting 8 from 12, start
by registering 1 on the tens rod G and 2 on
the units rods H. No tens are subtracted,
so we begin by subtracting 7 from rod H.
Because 2 − 8 < 0, the complement of 8
- namely 2 - is added to the units rod H,
and 1 bead is subtracted from the tens rod
G. This is shown in Figure 14. This leaves
the answer 4. Fig 17 1132 on rods FGH

When subtracting 6 tens from rod G, the

complement of 6 - namely 4 - is added to
rod G, and 1 bead is subtracted from rod
F. The abacus now reads 1072 as shown in
Figure 18. The final step is to subtract 1
from the unit rod H. The abacus depicts
the final solution, 1071 (left to the reader).
Fig 14 Adding 2 and subtracting 10
Next, we examine a more challenging
problem, 6432-5361. First register 6432
on rods E, F, G, and H, with H as the
units rod (Fig 15). Subtract 5 thousands
from rod E, leaving 1432 (Fig 16). Next,
subtract 3 hundreds from rod F. As Figure
17 illustrates, the abacus now reads 1132. Fig 18 1072 on rods FGH

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Multiplication
Multiplication problems, although
potentially more difficult than addition
and subtraction, can be easily computed
with the Soroban abacus. Before students
can successfully complete multiplication
problems, they must first be familiar with
multiplication tables through 9 × 9.
Registering the multiplicand and the Fig 19 Registering 36 x 4
multiplier is the most critical step in the
process. This ensures the ones value of the
product falls neatly on the unit rod.
As an example, let’s consider the
multiplication problem 36 × 4, with
multiplicand 36 and multiplier 4. We
begin by placing our finger on unit rod H
and count left one rod for every digit in
the multiplier (1 position to rod G) and Fig 20 Partial product 24 on GH with F reset to 0
one rod for each digit in the multiplicand
(2 positions to rod E) (Heffelfinger and
Flom, 2011). Next, register 36 on rods E
and F. Then place 4 on rod B. This leaves
enough space to help students distinguish
the multiplicand from the multiplier, as
suggested in Figure 19.
Performing multiplication on the abacus
involves nothing more than the addition
Fig 21 Final product 144 on FGH with D reset to 0
of partial products. Our first step is
multiplying 6 by 4, and adding the partial
and-pencil calculations. Furthermore, the
product on the two rods, GH, to the right
process utilizes addition, subtraction, and
of the multiplicand. Since we’ve accounted
multiplication we’ve discussed in previous
for the 6, we reset rod F to zero. These steps
sections of this paper. As such, we explore
are illustrated in Figure 20.
the following example, 19 ÷ 5, by means
A similar process is followed to multiply
of a series of visual steps that connect our
30 by 4. Its product, 120, is added to rods
abacus work to the traditional paper-and-
EFG. Since we’ve accounted for the 30 in
pencil procedure familiar to many teachers.
our calculation, we reset rod D to zero. This
As Figures 22-25 suggest, division
leaves the final product, 144, on rods FGH
on the abacus provides students with
as shown in Figure 21. Once addition
an alternative to paper-and-pencil
is mastered, the reader is encouraged to
calculation that provides a more hands-
try multiplication problems that involve
on experience for young learners. Our
carrying, such as 36 × 9.
example is rudimentary. For a more in-
depth treatment of the topic, readers are
Division
encouraged to explore resources provided
Solving division problems on the
in the reference list of this paper.
Soroban abacus mirrors familiar paper-

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Fig 22 Setting up 19 divided by 5

Fig 23 Determing the number of 5’s in 19

Fig 24 Multiplying divisor by quotient and subtracting

from dividend

Fig 25 Final solution, 19 ÷ 5 = 3R4

Summary
The abacus has played a vital role in mathematics that can still be seen today. Its
portability made it a popular tool used by merchants and allowed it to be introduced to
all parts of the world. Its flexibility allowed the abacus to evolve into a tool that could be
used across cultures. Its mechanical operation allows addition, subtraction, multiplication,
and division to be done accurately and efficiently without pencil and paper, rivaling that

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of a four-function calculator. The abacus Kojima, T. (1954). The Japanese abacus.

is window into the past which allows Retrieved from www.omerique.
users to process operations in the same net/twiki/pub/Recursos/ElAbaco/
fashion of that done for thousands of years. takashikojima1.pdf
Moreover, the device provides students Moon, P., (1971). The abacus: its history,
in today’s classrooms with alternatives to its design, its possibilities in the modern
paper-and-pencil procedures that let them world. New York, New York: Gordon
explore calculations in a more hands-on & Breach Science Publishers
manner. Pullan, J.M., (1968). The History of the
Abacus. London, England: Hutchin &
References Co LTD.
Heffelfinger, T. & Flom, G., (2011). The
Bead Unbaffled - An Abacus Manual. Sources of Images
Abacus: Mystery of the Bead. Retrieved Figure 1: http://homepage.mac.com/
September 27, 2011, from http:// shelleywalsh/MathArt/CBAddition.html
webhome.idirect.com/~totton/abacus/ Figure 2: http://encyclopedia2.
thefreedictionary.com/Bead+frame
KEVIN SAMOLY, Figure 3: http://www.historyofinforma-
samolykm@ tion.com/index.php?id=1664
muohio.edu, is a Figure 4: http://www.mlahanas.de/
senior at Miami Greeks/Counting.htm
University. He Figure 5: http://www.ucmas.ca/our-
enjoys traveling, programs/how-does-it-work/what-is-an-
cycling, and abacus/
playing sports. His areas of interest Figure 6: http://webhome.idirect.
include middle school mathematics com/~totton/abacus/
and social studies education. He is Figures 7-25: Drawn by the author using
currently looking for employment Adobe Illustrator
opportunities in Ohio.

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