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The Japanese Abacus
NOTE
The Publisher
pleased to announce, for the benefit of readers
obtaining abacuses, that Japanese-made
instruments of fine quality in various sizes are now available in
is
who have had difficulty in
the United States as listed below:
Small
(13 rods)
Medium
(15 rods)
Large
(23 rods)
Please inquire from Charles E. Tuttle Company, Rutland, Vermont,
Mail orders promptly filled.
The
JAPANESE ABACUS
Its
Use and Theory
BY
Takashi Kojima
CHARLES
E.
RUTAND VERMONT
TUTTLE COMPANY
TOKYO JAPAN
Representatives
BOXERBOOKS, INC., Zurich
British Isles: PRENTICE-HALL INTERNATIONAL, INC., London
Australasia: PAUL FLESCH & CO., PTY. LTD., Melbourne
Canada: M. G. HURTIG LTD., Edmonton
Continental Europe:
Published by the Charles E. Tuttle Company, Inc.
of Rutland, Vermont
& Tokyo, Japan
Editorial offices:
Suido 1-chome,
2-6,
Bunkyo-ku
Tokyo, Japan
Copyright in Japan, 1954
by Charles E. Tuttle
Co., Inc.
All rights reserved
Library of Congress Catalog Card No. 55-3550
Standard Book No. 8048 0278-5
First edition, April 1954
Twenty-fifth printing, 1970
Printed in Japan
CONTENTS
Foreword by Yoemon Yamazaki
Author's Foreword
Page
7
9
Abacus versus Electric Calculator
11
Brief History of the Abacus
23
III.
Basic Principles of Calculation
26
IV.
Addition and Subtraction
31
Multiplication
53
VI.
Division
64
VII.
Decimals
81
Mental Calculation
87
Exercises
88
I.
II.
V.
VIII.
FOREWORD
It
me
gives
abacus
is
English on the
great pleasure that this book in
ready for publication.
Japanese abacus operators have
long cherished the desire, here finally realised, of introducing the
countries in view of the remarkable
Japanese abacus to other
advances and developments
strument and
its
which have been made in the
in-
use during the past quarter of a century.
The Japanese abacus, simple and primitive looking though it
be, can be operated with greater speed and efficiency than even
a fact proven in numerous
the electric calculating machine
—
tests
and well documented by Mr. Kojima
This
is
parti culary so in
addition
handle figures
abacus can
fast as the electric
credible speed
and subtraction, where the
any number of
of
To
machine.
in^ his first chapter.
and mystifying
explain
efficiency
the
it is
digits twice as
instrument's in-
essential not only
to introduce the
newest improved methods of operation but also
to elucidate the
most advanced theories of rational bases and
of bead manipulation.
In
writing
this
practical
book Mr. Kojima has kept these
two requirements well in mind.
of the
The Abacus Research Institute
Japan Chamber of Commerce and Industry has been
most pleased
to cooperate with
search data and correcting his
him by making
available
its
re-
manuscript in the light of
all
the latest information.
YOEMON YAMAZAKI
Vice-Chairman, Abacus Research Institute
Professor of Economics, Nippon University
Vice-President, All-Japan Federation of Abacus Operators
AUTHOR'S FOREWORD
This book has been written as a guide for those who, though
knowing more about the use and theory of the
in
interested
Japanese abacus, have until
now been unable
explanation in the English
language.
Chapter
most important facts about the speed and
calculation, with special reference to a
and the
electric calculating
of
survey
Chapter
the
history
examples,
three
how
on the abacus.
machine.
chapters
presents
the
efficiency of abacus
Chapter
II gives
a brief
explain
in
and
of abacus calculation.
detail,
with numerous
the four processes of arithmetic are
worked out
Particular attention should be given to Chapter
of bead manipulation.
to give a theoretical
and
as
Many
it
embodies the essential
notes have been included
scientific explanation
fundamental principles as such knowledge
est
I
and development of the abacus,
IV, on addition and subtraction,
rules
full
comparison of the abacus
III introduces the basic principles
The next
any
to find
but will prove of great aid in
the
instrument.
The book concludes with
and mental
calculation,
and a
is
of the rules and
not only of inter-
actual operation of the
short chapters
on decimals
selection of exercises.
Among many who kindly gave me information and suggestions, I am particularly grateful to Mr. Yoemon Yamazaki, who
kindly wrote the foreword and provided me with many valuable
suggestions and a large part of the information in
He
is
Chapter
the Vice-Chairman of the Abacus Research Institute
I.
and
Advisor to the Central Committee of the Federation of Abacus
Workers
(hereafter referred to simply as the
Abacus Committee),
both organizations being under the sponsorship of the Japan
Foreword
10* Author's
Chamber of Commerce and
I also
wish to express
Industry.
my
special
gratitude
to
Mr. Shinji
Ishikawa, President of the Japan Association of Abacus Calcu-
who spared himself no
lation,
the manuscript and
trouble in reading the whole of
much important
furnishing
up-to-date in-
formation.
I also
Arai,
my
extend
acknowledgements
grateful
Mr. Zenji
to
Chairman of the Abacus Research Committee of the Japan
of Abacus
Federation
Education
,
and Mr. Miyokichi Ban, of
They kindly read the
the above-mentioned Abacus Committee.
whole of the manuscript and provided
me
with
many
necessary
and valuable suggestions.
My
grateful acknowledgements are also due to Mr.
Takeo Uno
on the Abacus Committee and Mr. Tadao Yamamoto, who conducts his
in parts
I also
own abacus
They kindly read the manuscript
school.
and gave me valuable suggestions.
wish to thank Mr. Kiyoshi Matsuzaki, of the Savings
Bureau of the Ministry of Postal Administration, who kindly
furnished the table on page 13.
Finally
I
suggestions
must express
my
on
style
English
sincere thanks for
from
Mr.
Commonwealth Public
formerly
of
the
Far
East
Relations
Network
;
;
invaluable
Wells,
Chief
Gosling,
of the
C. G.
Writer for the Far East Network; Mr. Harold
British
many
Mr. Richard D. Lane,
and
above
all
from
Mr. Meredith Weatherby, of the Charles E. Tuttle Company,
without whose painstaking efforts this book could not have be-
come what
it
is
now.
T-K.
ABACUS VERSUS ELECTRIC CALCULATOR
L
The
first
abacus, or soroban as
objects that strongly attract the attention of
When
in Japan.
he buys a few
one of the
called in Japan, is
is
it
the foreigner
trifling articles at
he soon notices that the tradesman
does
some
perplex
not
store,
himself
with mental arithmetic, but instead seizes his soroban, prepares
it
by a
tilt
and a
rattling
manipulation of rapid
sweep of his hand, and after a deft
that the Japanese tradesman
even when the problem
head, but this
is
is
often
off
uses
true
the
price.
his
board and beads
It is
simple enough to be done in one's
only because the use of the abacus has become
a habit with him.
37 and 48
reads
clicks,
If
he
in his head.
tried,
But
could no
he
such
is
the
doubt easily add
force of habit that
he does not try to recognize the simplicity of any
instead, following the line of
soroban for manipulation, and
escaping any need of mental
resistance,
least
begins
problem;
he adjusts his
clicking the beads, thus
effort.
Doubtlessly the Westerner, with
his
belief in the powers of
mental arithmetic and the modern calculating machine, often
mistrusts the efficiency of such a primitive looking instrument.
However, his mistrust of the soroban
into admiration
is
likely to
be transformed
when he gains some knowledge concerning
it.
For the soroban, which can perform in a fraction of time a
difficult
arithmetic
calculation
that
the
Westerner
could
do
means of pencil and paper, possesses distinct
advantages over mental and written arithmetic.
In a competi-
laboriously only by
tion
in
arithmetic
problems, an ordinary Japanese tradesman
The Japanese Abactas
12
with his sgroban would
An
exciting contest
electric calculating
adding machine.
his
between
the
Japanese
abacus
Stripes
and
Army
:
newspaper,
In reporting the contest, the Stars and
Stripes.
remarked
and the
machine was held in Tokyo on November
12, 1946, under the sponsorship of the U. S.
the Stars
and accurate
rapid
a
outstrip
easily
Western accountant even with
" The machine age
took
a
step
backward
yesterday at the Ernie Pyle Theater as the abacus, centuries old,
dealt defeat to the
most up-to-date
machine now being
electric
used by the United States Government
.
.
.
The abacus
victory
was decisive."
The Nippon Times reported
the contest as follows
" Civiliza-
:
tion,
on the threshold of the atomic age, tottered Monday
noon
as the 2, 000-y ear-old abacus
beat the electric
after-
calculating
machine in adding, subtracting, dividing and a problem including
all
UP. Only
three with multiplication thrown in, according to
in multiplication alone did the
machine triumph.
.
.
."
The American representative of the calculating machine was
Pvt. Thomas Nathan Wood of the 240th Finance Disbursing
Section
of
General MacArthur's headquarters,
selected in an arithmetic contest as the
the electric calculator
who had been
most expert operator of
The Japanese representative
champion operator of the abacus
Japan.
in
was Mr. Kiyoshi Matsuzaki, a
in the Savings Bureau of the Ministry of Postal Administration.
As may be seen from
the results tabulated on the following
page, the abacus scored a total of 4 points as against 1 point
for the electric calculator.
Such
results
should convince even
the most skeptical that, at least so far as addition
tion are concerned, the abacus possesses
over the calculating machine.
multiplication
demonstrated:
and
division,
Its
and subtrac-
an indisputable advantage
advantages
however, were
in the fields of
not so decisively
Abacus versus
Electric Calculator
•
13
RESULTS OF CONTEST
MATSUZAKI
Type
of
(Abacus) vs.
Name
Problem
Addition:
50 Matsuzaki
number each consisting of from 3
to 6 digits
Wood
Subtraction:
5
problems,
with Matsuzaki
minuends
1st
Heat
lm.
14.8s.
Wood
each
Multiplication: 5
problems,
each Matsuzaki
containing 5 to 12
digits in multiplier and multipli-
Wood
cand
digits
in
divisor
Wood
and dividend
lm.
All correct
4 correct
No decision
30s.
Defeated
lm. 44.6s.
lm.
-
36s.
4 correct
No decision
lm.
19s.
lm.
All correct
lm.
22s.
4 correct
Defeated
2m. 14.4s.
All correct
3 correct
Defeated
2m. 22s.
Victor
Defeated
lm. 53.6s.
lm.
36.6s.
lm. 20s.
All correct
Defeated
lm. 23.4s.
4 correct
lm. 21s.
All correct
4 correct
Victor
Defeated
lm. 19s.
Victor
All correct
All correct
4 correct
Defeated
Victor
Defeated
48s.
1
Victor
All correct
lm.
1
Victor
4 correct
4 correct
Score
1
lm. 53s.
Defeated
lm. 0.8s.
All correct
3rd Heat
16s.
2m. 0.2s.
Defeated
lm. 0.4s.
Victor
Division: 5 prob- Matsuzaki
lems, each containing 5 to 12
2nd Heat
Victor
lm.
of
to 8 digits
(Electric Calculator)
Victor
Victor
and
subtrahends
from 6
WOOD
lm.
1
26.6s.
Composite problem: 1 problem in
addition of 30 6numbers; 3
lm. 21s.
digit
problems
in sub- Matsuzaki
.
each
traction,
with
c
wo
All correct
Victor
1
6-digit
numbers; 3 problems in multiplication, each with
two
figures containing a total of
6 to 12 digits; 3
problems in division,
each with
Wood
lm. 26.63.
4 correct
Defeated
two figures containing a total of
6 to 12 digits
Matsuzaki
4
Total Score
Wood
1
14
•
The Japanese Abacas
For reliable information on the comparative
merits of the
abacus and the calculating machine, we can do nothing better
then turn to the Abacus Committee
Commerce and
concerning
Industry, which has
the
Committee has acted
made minute
of
the
as judge of
the
potentialities
for abacus operators' licenses
Chamber of
of the Japan
Japanese
investigations
The
abacus.
semi-annual examination
since the
examinations were in-
1931, such licenses being divided into three classes,
itiated in
according to the manipulators' efficiency.
The Committee
says:
"In a contest in addition and subtraction,
a first-grade abacus operator can easily defeat the best operator
of an
latter,
electric
machine,
solving problems twice as fast as the
no matter how many
digits the
numbers do not contain over
numbers contain.
six digits, the abacus
If the
manipulator
can halve the time of the operation by relying in part upon
system peculiar to the abacus, to be des-
mental calculation
(a
cribed hereafter).
In multiplication and division the first-grade
abacus operator can maintain some margin of advantage
over
the electric calculator so long as the problem does not contain
more than a
total
of about ten digits in multiplicand and mul-
tiplier or in divisor
and quotient.
The abacus and
the electric
machine are on a par in a problem which contains a
ten to twelve digits.
With each additional
digit in a
total
of
problem,
the advantage of the electric calculating machine increases."
A
similar view
is
held by Mr. Kiyoshi Matsuzaki,
who made
the following remark concerning the contest described in preced-
ing pages
:
"In
addition and subtraction even the third-grade
abacus worker can hold his
machine.
own
against the electric calculating
In multiplication and division the first-grade abacus
worker may have a good chance to win over the calculating
machine, provided the problem does not have more than a
total
of ten digits in multiplicand and multiplier or in divisor and
Abacus versus
quotient.
than
I
felt
felt the
same, though.
to be able to
make
As examples of
rator,
it
15
nervous at the contest and made more mistakes
My
might have done otherwise.
I
Electric Calculator
A
good
a better
opponent may have
abacus worker ought
first-grade
showing when he
the proficiency required
will be of interest to cite a
is
at ease."
of the abacus ope-
few problems used in the
examination for abacus operators' licences.
A.
No.
ADDITION AND SUBTRACTION
¥
¥40,693,718.52
4
5
6,395,082.74
269.31
541,793.60
82.706.314.95
72,940.18
6
7
8
9
10
3.014,725.86
98,156.02
15,726,408.39
970,285.13
45,963.78
26,073.94
309.861.75
8,714.905.26
346.17
295.130.86
11
12
13
14
15
6.831,750.24
64,371.59
249,168.07
70,593,826.41
4.352.80
94,038,726.51
69,052.74
150,938.42
43,281.65
7.916,403.28
1
2
3
52,687.09
7.180,592.43
1,745.38
63.847.529.10
4
3
2
1
¥
'
160,384.72
83,479,051.26
-21,479.50
9.058,627.13
-3,780.29
27,915.64
40,715,368.92
86,223.41
-504,189.76
-6,037,512.89
924.35
763,815.04
-20,849,136.57
4,102,653.93
95,467.83
¥
5
730.49
6,089,547.31
463,195.28
97,820.56
3,985,271.04
10,476,825.93
54,613.78
218,769.45
3,428.01
82,605,917.34
61,853.20
250.376.19
3,576,904.82
49,021.67
57,316,482.90
¥
352,719.48
84,936.20
92,460,385.71
-718,024.36
45,178.62
8.327.605.94
-19,062.53
-4,085,237.61
25,963,180.47
70,941.28
-6,798.05
-50,824,361.79
953.16
3,107,425.89
639,507.14
tr
B.
No.
1
MULTIPLICATION
759.843x57.941 =
C.
No.
DIVISION
4,768,788,093-^-14,593 =
2
302.162x83.602 =
2
971,337,349-f-51.682 =
3
967.408x70.589 =
3
47,408,509, 168-+49. 201
4
20.359x628.134 =
4
0.3481095257-^0.06457 =
5
84.2697x9.4076 =
5
66,014,150,202-^-92,378 =
6
135.941x46.295 =
6
3,657.6146092+80,914 =
7
0.4271805x0.2513 =
7
166.4719833-^-0.6702 =
8
669.378x0.31908 =
8
0.4537275087^-7.38609 =
9
0.914053x68.037 =
9
328,399.09042-^35.746 =
10
587.216x17.452 =
10
=
24,484,596,290-^28.135 =
16 The Japanese Abacus
•
D.
No.
2
1
¥
1
MENTAL CALCULATION
74.63
¥
3
¥
3.46
4
¥
52.31
5
¥
8.09
90.47
2
2.98
97.98
30.64
2.41
3.51
3
50.41
6.05
-9.28
56.37
-76.29
4
83.72
2.13
14.75
1.52
-1.83
5
1.35
50.79
8.39
70.86
54.02
6
6.84
8.21
-45.05
3.94
8.35
7
95.01
19.64
6.17
96.70
29.14
8
3.27
78.30
-2.83
6.28
-6.50
9
65.10
4.56
7.14
37.19
47.26
JO
4.92
82.07
91.26
48.05
1.83
To
receive a first-grade license an applicant must be able to
work problems similar
to
foregoing
the
accuracy within a time limit of
three groups
— A,
The problem
five
—and
B and C
of
abacus
it
cent
first
it
a few words of
depends directly upon
abacus, and being an integral
of the
technique,
per
minutes each for the
in mental calculation requires
a knowledge of the use
80
one minute for the fourth.
explanation as the method of solving
part
with
is
different
entirely
from any
Western method of calculation. This abacus method of mental
arithmetic
is
is
described in
to say here that the
some
method
detail in
Chapter VIII.
consists in
Suffice
mentally visualizing
an abacus and working the problem out by standard methods
on the imaginary instrument.
The process
is
easier
than
it
sounds and accounts for the incredible and almost mystifying
peaks of efficiency attained by masters of abacus operation.
To
cite
but one example of proficiency in this type of mental
arithmetic,
on May 28, 1952, during the Sixth
Contest, held in Tokyo, a master
All- Japan
Abacus
abacus operator, Mr. Yoshio
Kojima, gave a demonstration of his
skill in
mental arithmetic.
Abacus versus
Electric Calculator
•
17
In one minute and 18.4 seconds he gave correct answer to 50
which contained
division problems, each of
in
and
dividend
its
divisor.
Next,
five to
twinkling
a
in
seconds he added 10 numbers of ten digits each.
two remarkable records
mentally
visualized
added mentally the
A.
¥
5
71,896
306,425
839
50,178
2,941
6
7
8
9
10
567,308
762
82,037
3,694
470,589
11
24,310
165
5,742
904,213
68,951
1
2
3
4
12
13
14
15
— and
This
abacus!
fifteen
with
all
seven digits
of
Thus he
B.
means
numbers given
that
in
he could have
one of the columns
ADDITION AND SUB TRACTION
¥
93,502
8,164
802,635
378
25,910
-8,756
650,481
362
71,049
2,913
134,795
-19,247
-804
-65,798
746,083
¥
130,745
59,280"
4,968
102
701,539
¥
60,374
875,126
23,601
7,284
932
48,075
6,714
429
602,893
27,564
-506,849
-39,256
4,931
93,270
315,687
90,312
856
-96,178
-4,517
1,023
485
187,683
-760
248,951
19,074
MULTIPLICATION
6,742x358 =
C.
No.
¥
9,180
25,634
418,275
54,361
7,903
86,215
903,587
643
71,852
408
640,729
210
92,674
5,096
837,921
DIVISION
435,633-921 =
2
2,681x609 =
2
315,56 + 805 =
3
5,093x176 =
3
18.998+236 =
4
0.825x94.12 =
4
63.162-+0.087 =
5
3,310x803 =
5
223.792 + 394 =
6
9,478x0.645 =
_6
400.026 + 418 =
7
76,506x5.2 =
7
180,096 + 64
8
193.4x4.18
8
0.105118 + 0.753 =
9
0.4052x0.267 =
9
104,249 + 1.709 =
10
9,718x703 =
set
no aid other than the
It
No.
13.6
10
0.21918+5.62 =
18 The Japanese Abacus
•
of Problem A, page 15, in one-fourth of the given time limit of
one minute, and in one-eighth of the time required by the best
operator of an electric calculating machine.
To
receive a third-grade
work problems similar
to
license
an applicant must be able
to those on page 17 with ?0 per cent
accuracy within a time limit of
five
minutes for each group.
The primary advantage of the abacus is its incredible speed
resulting from the mechanization or simplification of calculation,
by means of which the answer
on the board, thus reducing
mechanically
itself naturally or
problem form3
to a given
mental labor to a minimum. The theoretical explanation of
mechanization of calculation
3 to Example
Example
to
9,
Note 3
to
given in Chapter IV
is
Example
10,
ranging generally between
to quote prices
in
U.S.
more does the ordinary
gleaming
of the
3,
Note
4 and 5
20).
Another big advantage of the abacus
ate price,
and Notes
(see
this
is
2S<f
extremely moder-
its
and $2.50 or $3.50
How many
equivalents.
dollar
calculating
machine
cost, to say
times
nothing
machines which abound in Western
electric
business houses?
Among many
overlook
its
other
merits
handy construction,
operational methods,
its
of the
its
one
abacus
should
not
and the ease of
portability,
which are nothing more than simpli-
fications of the four processes of arithmetic.
The most
in addition
instead of
metic,
peculiar advantage of the abacus
and subtraction
from
is
that a
worked out from
right to left as
is
left
problem
to right
the case with written arith-
and thus harmonizes perfectly with the normal way of
reading and writing numbers.
added or subtracted while
the
is
first
number
in the
it
In
is
problem
this
way
being given.
is
a
number can be
For example,
if
753, the operator can enter
7 on the abacus the instant he hears or sees " seven hundred/'
Abacus versus Electric Calculator
and then proceed on next
calculation
in written
whereas
finally the 3,
he would usually have to wait until
were given and then
figures
and
to the 5
19
all
backward from the
start calculating
3 of 753.
The one admitted disadvantage of
instrument produces only a
calculation
But
end.
rapidity
it
this
seeming disadvantage
more than
is
and accuracy which the abacus makes
the length of time and practice required to
abacus
Certainly the
not so great an obstacle as
Some
tific
to
offset
by the
possible.
And
requires
But
it
this
become a
much more
practice
apparent disadvantage
thought
generally
is
skilled
to
be.
experience and practice with this simple but highly scien-
instrument will convince the reader that this Western idea
largely a prejudice.
each day
skill
made, the whole
must be carried through again from beginning
than the calculating machine.
is
is
chief factor which discredits the abacus in Western eyes
operator.
is
any error
the result that counts.
is
The
is
If
the
that
is
without preserving any
final result
record of intermediate steps.
abacus
the
to
proper
with
turn
the
to
A
few weeks of practice for an hour
procedures
abacus
will
instead
give
anyone
sufficient
of pencil and paper for
arithmetical computation.
According
to the
Abacus Committee, average students, who
begin their practice while in their teens, should be able to pass
the examination for third-grade licenses
daily practice of
after
half year
of practice
will
should
of
make him
Generally speak-
enable
abacus operator to obtain a second-grade license
full year
year
a
one hour, and bright students or students with
a mathematical bent after only three months.
ing, another
half
;
a first-grade operator.
a
third-grade
and one more
As
is
ally the case with
any other
to start practising
under right guidance when young.
art or
accomplishment,
it
generis
best
Those
20 The Japanese Abacus
•
who
up
take
their
study of
their teens are never able to
but
it
the
pass
is definitely possible for
and occasionally even
abacus after they are out of
the
them
to the second.
Receat years have shown the abacas
an amazing
to enjoy
in-
1965 nearly one million ap-
crease in popularity in Japan. In
plicants took the examinations, about a fourth of
(roughly 5,000
examination,
first-grade
to attain to the third rank,
out of 55,000
received
whom
first-grade
passed
licenses;
25,000 out of 280,000, second-grade; and 230,000 out of
750,000, third-grade). In the same year about half a million
took the
exam held by
the National Federation of Abacus Oper-
ators, a private association
eration of
Chamber
independent of the
Abacus Operation which
of
All- Japan Fed-
with the Japan
is affiliated
Commerce and Industry. Additionally, another half
exam held by the National Association of
a million took the
Vocation High School Principals.
The abacus has even found
grade schools as
all
there are
its
way
one of the elements of arithmetic, and
now numerous
abacus
schools to meet the needs of
those preparing to go into business.
become such a popular
tically every
How
are
into the curriculum of
favorite that
the abacus has
In short,
it
is
to be
found in prac-
household.
we
to account for the
sudden spurt in the popularity
of the old-fashioned abacus, here in the middle of the mechanized twentieth century ?
lies
in the fact that
its
Undoubtedly the principal explanation
operational methods have recently been
markedly simplified and improved.
As
will
be explained in
Chapter VI, the old method of division required the memorization of a difficult division table,
and was the chief factor which
alienated the average, non-commercial Japanese
Once
this
difficulty
was overcome by the introduction of the
newer method of division
so
much more
from the abacus.
— so
accurate that
it
much
simpler and,
marked a milestone
in
a sense,
in the im-
Abacus versus Electric Calculator 21
•
provement of abacus technique
the universal popularity which
— the instrument rapidly
it
now
attained
enjoys.
But how account for the almost exclusive use of the abacus
in offices
machines
and firms which could well afford
?
electric calculating
According to figures
Let statistics give the answer.
compiled by the Abacus Committe, in the conduct of an average business the four types of arithmetical
about the following proportions
in
subtraction
five
percent,
occur
calculation
addition seventy percent,
:
multiplication
and
percent,
twenty
As previously mentioned, the abacus can
As
faster than the electric calculating machine.
division five percent
add and subtract
for problems in multiplication
more than a
total
and
division, those
which contain
of ten digits in their mutiplicand and multi-
and quotient are exceptional. This means
good operator can work out most mutiplication and
plier or in their divisor
that
a
division problems as fast as or even faster
on an
electric calculation
on an abacus than
much
machine, to say nothing of the
slower non-electric machine. So even today
when Japanese
nesses are fitted with a variety of electronic computers,
surprising that the abacus retains
its
store as well as the giant corporation.
nese businesses 85 per cent of
abacus while most of the remainder
is
it is
not
popularity with the tiny
To
give figures, in Japa-
calculation
all
busi-
is
done on the
done on machines with
a small percentage on calculating table, slide rules, or written or
mental arithmetic.
At the present time various experiments are being undertaken
to
improve
But
still
this little
established
further the operational technique of the abacus.
handbook
methods and
to operate the abacus
Once
will
introduce
only
the
of
best
&q
verified theories, essential for learning
with good
understanding
and
rapidity.
the basic rules have been mastered, the secret of acquiring
skill in
abacus operation
lies in
constant daily practice.
Problems involving the extraction of roots can also be solved
22 The Japanese Abacus
•
on the abacus with great
which
is
rarely used
But the extraction of roots,
rapidity.
everyday and business calculation,
in
is
outside the scope of this book.
The abacus can be
tic
also a great aid in the instruction of arithme-
50%
in grade school. For about
of
problems
all
in
elementary
school textbooks of arithmetic are calculation problems, and the
other
50% are those which can
be reduced
problems
to calculation
by the process of some reasoning. Some adequate training enables
children to calculate with the abacus
traditional
means of
much
faster than with the
pencil and paper. So abacus experts are of
opinion that, taking into consideration the time spent on practice
for learning
abacus operation, the abacus or "soroban," made
good use of
in the course of arithmetic,
and learning arithmetic,
business and everyday
The
final
to say nothing of its
facilitate
immense
teaching
utility in
life.
but most remarkable
enables the blind to calculate
who
may
calculate with pencil
utility
much more
of the abacus
is
that
it
rapidly than the sighted
and paper. In Japanese schools
blind, all kinds of instruments for aid to calculation
for the
by the blind
have, hitherto, been introduced only to be discarded soon for the
abacus, which has proved by far the most effective instrument for
teaching the ideas of numbers and mathematics to blind children.
In Japan an abacus grading examination for blind children was
initiated in
1964 by the Chamber of Commerce and
Today we have reasons
to expect that a world-wide
tion for blind children will be instituted
Industry.
examina-
under the joint spon-
sorship of the Pan -East Asian Abacus Association of Japan,
South Korea, and Taiwan; and the newer abacus associations
the U. S., Europe, Mexico, and other countries.
of
H.
BRIEF HISTORY OF THE ABACUS
The imperfect numerical notation and
the scarcity of suitable
writing materials in ancient times are presumed to have given
rise to the
need for devices of mechanical calculation.
the definite origin of the abacus
son for believing that
its
covered with sand or fine
earliest
dust,
is
obscure, there
is
some
form was a reckoning
which
in
While
rea-
table
were drawn
figures
when necessary. The
English word abacus is etymologically
from
derived
the Greek abax, meaning a
with a stylus, to be erased with the finger
9
ill!
9
reckoning table covered with
dust,
which in turn comes
from
a Semitic
ing
l!
dust
or
In
sand.
Roman Grooved Abacus
lines to
indicate numbers.
were in
common
disks
with dust or
time
this
sand-
were arranged on
Various forms of this Kne abacus
use in Europe until the opening of the seven-
In rather remote times, a third form of abacus
appeared in certain parts of the world.
which loose counters were
sliding
reckoning
dust abacus gave place to a
upon which counters or
teenth century.
a
{
table covered
ruled table
word mean-
laid, the table
up and down grooves.
Instead
of lines
on
had movable counters
24 The Japanese Abacus
•
All three types of abacuses were found at
Rome
in ancient
- the dust abacus,
Out of
grooved abacus.
abacus was developed
a frame.
calculations can be
is
still
or other
abacus,
and the
line
this last type yet a fourth
—one
form,
This
the
some time
form of
the
with beads sliding on rods fixed in
the
bead or rod abacus, with which
made much more quickly than on
paper,
used in China, Japan, and other parts of the world.
In
Europe, after the introduction of Arabic numerals, instrumental
make much
arithmetic ceased to
altogether to the graphical as
progress and finally gave
the supply
way
of writing materials
became gradually abundant.
As
for the Orient, a
form of the counting-rod abacus,
called
cKeou in China and sangi in Japan, had been used since an-
means of
cient times as a
itself
The Chinese abacus
calculation.
seems, according to the best evidence, to have originated
in Central or Western Asia.
There
is
a sixth-century Chinese
reference to an abacus on which counters were rolled in grooves.
The
description of this ancient Chinese abacus
West give us good reason
intercourse between East and
Chinese
lieve that the
The Chinese
write
from
to right.
is
left.
generally
Rut the abacus
called
the
that
that the
use
line,
worked from
is
dialect,
was a
abacus
they
left
was not
till
the fourteenth century.
people of the
Orient,
later
development,
and did not come
probably appearing in the twelfth century,
common
Roman.
The present Chinese bead abacus, which
suan-pan (arithmetic board) in Mandarin
and soo-pan in the southern
into
the
to be-
columns from above downwards.
another indication
indigenous to China.
is
by
suggested
compelled to write in a horizontal
right to
This
was
abacus
write in vertical
If they ever are
and the known
It is
only natural
having retained a system of
numerical notation unsuited for calculation, should have developed the abacus to a high
degree,
and
its
continuous universal
—
Brief History of the Abacus
use even after the introduction of Arabic numerals
testimony to the great efficiency achieved in
The Japanese word
for abacus, soroban,
rendering of the Chinese suan-pan.
is
its
•
25
eloquent
is
development.
probably the Japanese
Although the soroban did
not come into popular use in Japan until the seventeenth century, there is
no doubt
that
it
must have been known
anese merchants at least a couple of centuries earlier.
to Jap-
In
any
of calculation became
case,
once
widely
known in Japan, it was studied extensively and intenby many mathematicians, including Seki Kowa (1640
who discovered a native calculus independent of the
sively
1703),
this
convenient
Newtonian theory.
instrument
As" a result of all this study, the
form and
operational methods of the abacus have undergone one improve-
ment
after
Like the
another.
present-day
the soroban long had two beads above the
Chinese
beam and
But toward the close of the nineteenth century
it
suan-pariy
five
was simplified
by reducing the two beads above the beam to one, and
around 1920
it
acquired
its
beam from
Thus the present form of the soroban
and ingenuity
science.
We
in the field
feel sure
that
finally
present shape by omitting yet an-
other bead, reducing those below the
labor
below.
the
is
of Oriental
soroban,
use in this mechanical age on account of
five to four.
a crystalization of
mathematics and
enjoying
its
widespread
distinct advantages
over the lightning calculating machine, will continue to be used
in the
coming atomic age
as well.
BASIC PRINCIPLES OF CALCULATION
IIL
The abacus
arithmetical calculation.
or board
holding
tions
It
balls, or
performing rapid
vertically
wooden frame
on
arranged rods,
A
counters slide up and down.
across the board divides the rods
into
two
sec-
The most common type of abacus
upper and lower.
:
for
consists of an oblong
number of
a
which wooden beads,
beam running
instrument
simple
a
is
Japan has twenty-one bamboo rods, and
is
in
about twelve inches
But larger types with twenty-seven
long by two inches wide.
or thirty-one rods, and smaller ones with seventeen or thirteen
rods, are also
preceding
As described
used.
and
chapter,
illustration, the
as
at
may be
conclusion
number of beads per rod has been
and
to six,
finally to five.
of the
seen in the accompanying
reduced, in the interests of simplicity
from seven
the
progressively
and ease of operation,
Until recently an abacus
with five beads in the lower section of each rod was in general
But
use.
this type
now been
abacus
has
largely
replaced
one
with
of
four
by a
beads
on each rod below the
Fig.
1.
Modern Japanese Abacus
imvyvywATP^TTyyy
beam.
The abacus
YYYYYYYYyYYyYAYYYYYY^
is
based
on the decimal system.
For convenience in
cal-
Fig.
2.
Qlder-T>tpe Japanese Abacus
Basic Principles of Calculation
mm
beam
dilation the
27
marked
is
with a unit point
•
every
at
These unit points
third rod.
serve to indicate the decimal
and other
point
units
of
For
ex-
decimal measure.
ample, select any rod near
Fig.
which
the
Modern Chinese Abacus
3.
is
marked with a unit
of the problem. Then the
the second
the second
is
and
rod to
first
the hundreds' rod,
is
another unit point)
hand, the
point,
rod,
On
etc.
another point)
the
the
is
the hundredths', the third (likewise
is
rod,
(marked with
rod
rod to the right of the unit rod
first
unit rod
left is the tens'
third
the thousands'
board
the
call this the
its
the
of
center
other
tenths',
marked with
the thousandths' rod, etc.
is
Each of the four beads on the lower section of a rod has
the value of
while the bead on the upper
1,
Each of the
has the value of
5.
obtains
when
its
value
loses its value
tion.
On
beam
obtains
and
when
it
is
it
1-unit beads below the
value
loses its value
The beads
when
beam
moved up toward the beam, and
moved back down to its former posiis
the other hand, each of the
its
section of a rod
when
it
is
it
is
5-unit
beads above the
moved down
moved back
to the
up.
in Figure 1, using the third unit point
right to designate the unit rod,
represent
beam
the
from the
number
1,345,
while Figure 2 shows 46,709.
Before using the abacus,
make
sure that
the neutral position representing zero.
up
all
5-unit beads
and moving down
clearing the abacus for use,
middle finger on
lower edge, and
This
its
move
upper
all
hold the
all
the beads are in
is
done by moving
left
edge and your
beads
down by
beads.
In
end with your
left
1-unit
all
left
thumb on
slanting
the
its
upper
—
28 The Japanese Abacus
•
edge toward your body.
moving the
5-unit beads by
all
After leveling the abacus again, raise
right index finger
from
left to
right along the upper edge of the beam.
When
calculating
on the abacus, use two
Some
index finger and thumb.
finger, but
thumb
Nearly
as well.
index finger to move
more
two
use
use the
efficient to
Use the
fingers.
down and to move
thumb only to move 1-unit
For instance, to place the figure 7 on the abacus
beads up.
with only the index
move down
beads
experts
the index
5 -units beads up and
down, while using the
1-unit beads
first
all
it is
the right
:
use only
operators
experiments show that
fingers
— whereas
two successive motions
finger requires
a 5-unit bead,
and then move up two
1-unit
these motions can be performed simultaneously
with two fingers, with a corresponding increase in efficiency.
Moreover, in our everyday actions we commonly employ two
more
or
fingers, say in
pen, and the hand
always
requires
is
the
picking up something or in holding a
so
made
assistance
that
of the
thumb.
for the proven fact that, in the long run,
to operate the abacus with
finger almost
index
the
it
is
This
much
accounts
less tiring
two fingers than with but one.
Experiments also show that the index finger can move beads
down more
the
other
quickly and accurately
best
pulation
is
and accuracy than the index
and quickest way
to acquire
to use the index finger
with the prescribed rules for bead
finger
the
thumb, while on
hand the thumb can move beads up with
speed, force,
The
than
movements
greater
finger.
skill in
and thumb
abacus mani-
in strict accord
manipulation.
will be indicated in detail for a
The
correct
number of
problems in the next chapter. They should "be carefully heeded
and practised many times
as nimbly
and
until
you can
effortlessly as the fingers
the keys in executing a sonata.
flick
your two fingers
of a pianist glide over
:
Basic Principles of Calculation
Another important secret for acquiring rapid
calculation
always
is
skill in
•
29
abacus
keep your fingers close to the beads.
to
Never raise your fingers high from the beads nor put them deep
your
their ridges just slightly with the tips of
The guiding
hereafter,
down by touching
Glide the beads up and
between the beads.
principles for the
may be summarized
movement of
flngevs.
beads, as followed
thus
General Rules for Bloving Bead3
1.
Move down
a 5-unit bead
beads as the same time.
2.
First
move down one
up a 5-unit bead.
3.
(See
In quick succession
and move up one or more 1-unit
(See
or
more
Example
first
In quick succession
first
and then a 5-unit bead.
5.
In
addition,
operation
on the unit
rod,
up
the tens'
move up one
on
Examples 11, 13, 15,
17 and 19.)
6.
In subtraction, after
subtracting a
1-unit
bead from the
tens'
rod,
operate on the
unit
rod.
amples
12,
(See Ex14,
16,
a
5-unit
bead,
or
more
and
1-unit beads,
(See Examples 8 and 9.)
(See
rod.
and then move
(See Examples 7 and 9.)
move
a 1-unit bead
next chapter.)
6.)
after
finishing
5,
1-unit beads,
move down
then one or more 1-unit beads.
4.
Example
FiG
:
30 The Japanese Abacus
•
18 and 20.)
When working
good posture
will
with the abacus,
have
much
to
up
sit
straight at a desk.
A
do with the speed and accuracy
of your calculations.
Finally, in
studying
the
illustrations
examples given throughout the
which accompany the
rest of the
book, the following
key should be kept in mind
Key
1.
to Illustrations
A
white bead (<>)
is
and has no numerical
2.
A
striped bead
(^^)
one which
is
value.
is
one which has
thereby having either obtained or lost
3.
A
black bead
(^)
in its original position
is
its
just
been moved,
numerical value.
one which obtained numerical value
in a previous step.
4.
| indicates that
beads
are
to
be
moved down with
the
index finger.
5.
f indicates that beads are to be
moved up with
the index
finger.
6.
7.
£
indicates that beads are to be
Figures in parentheses
moved up with
the thumb.
accompanying the foregoing signs
indicate the order in which beads are to be moved.
ADDITION AND SUBTRACTION
IV.
There are four
abacus
principal
arithmetical
and subtraction are basic
you know how
to
addition
The
and
our daily
In
most important of
life
processes, for unless
and business accounts,
first
However, when many large numbers are to be
number
number and
set
generally used for addition
is
is
rod, that
is,
any case a one-
In
left.
the last digit of a
on a unit
of the abacus,
set at the right side
because the working extends to the
ways be
Of
all.
central part of the abacus
subtraction.
digit
division.
used far more frequently than the other processes
is
added, the
on the
add and subtract on the abacus, you cannot
multiply or divide.
is
and
addition, subtraction, multiplication,
:
these, addition
and
calculations
larger
on a
number should
rod
al-
marked with a
unit point.
1.
Adding and Subtracting One-Digit Numbers
Example
Step 1
:
1.
1
+ 2=3
Set the
1-unit bead with the
set 1
number 1 by moving up one
thumb (Fig. 5). See that you
on a unit rod marked with a unit
Step 2
same rod
:
;
,
Add 2
to
two more
1
by
1-unit
z+z
point.
moving up, on the
beads,
using
the
Fig. 5
Fig. 6
:
32 The Japanese Abacus
•
thumb
(Fig. 6).
Note
1
This
:
example
adding one or more 1-unit beads.
procedure applies are
2+1
2+2
1+1
1+2
1+3
Note 2
bead," "
to "
set
The problems
to
which
this
:
3+1
5+1
6+1
5+2
5+3
5+4
6+2
6+3
Hereafter, such phrases as
:
procedure used in
the
illustrates
move down three
1," " move down
7+1
7+2
8+1
" move up one 1-unit
1-unit beads," etc., will be shortened
3," etc.
3-2 = 1
Step 1
Step 2
thumb
:
Set 3 with the
:
Subtract 2 by moving down two
unit beads with the index finger
Note
Fig. 7
Fig. 8
2-1
which
to
2
Step 1
:
Step 2
:
j
this
6-1
Set 2
7-1
7-2
beads.
8-1
8-2
8-3
9-1
9-2
9-3
9-4
5 with the index finger
:
This example illustrates the procedure
used in adding a 5-unit bead.
+5
1-unit
(Fig. 9).
Move down
procedure applies are
1
one or more
procedure applies are
10).
Note
which
(Fig. 8).
+ 5-7
(Fig.
Fig. 9 Fig. 10
this
4-1
4-2
4-3
3-1
3-2
1-
This example illustrates the procedure
:
used in subtracting
The problems
(Fig. 7).
2+5
The problems
:
3+5
4+5
to
.
:
:
Addition asai Subtraction
Example
7—5
4-
—2
Step 1
:
Set 7
Step 2
:
Move up 5
(Fig.
(Fig.
11).
with the index finger
example shows the procedure
This
:
blems to which
procedure are
this
2+
:
Set 2
Step 2
:
Move down
time
(Fig.
Note
(Fig.
with
1
13).
thumb
the
5-unit
to
at the
6.
same
this
Fig. 13
procedure applies are
3+6
2+7
8 — 6=2
:
Set 8
Step 2
:
After moving
(Fig.
15)
down
move up 5 with
the
1
with the in-
same
finger
how
to sub-
16).
Note 1
:
This example shows
both one or more 1-unit beads and a
problems to which
9-9
this
Fig. 15
5-unit bead
procedure applies are
9-8
8-8
Fig. 14
and one or more 1-unit
2+6
Step 1
finger,
bead
which
1+6
1+7
1+8
tract
9-5
14).
The problems
Example
Fig. 12
5 with the index finger
used in adding both a
beads.
8-5
This example illustrates the procedure
:
Fig. 11
6=8
Step 1
and move up
pro-
:
6-5
5-5
Example 5
The
5-unit bead.
used in subtracting a
(Fig.
w
12).
Note
dex
33
9-7
8-7
7-7
9-6
8-6
7-6
6-6
:
S4 The Japanese Abacus
•
Note 2
The above-mentioned procedure
:
moving up 5
and
first
1
next with the index finger.
explained in Note 2 of Example 15, in some
procedure makes
move
difficult to
it
preferable
is
the
cases
the fingers
to
As
is
latter
nimbly,
e. g.,
the addition of 4 to 9.
1=5
4+
1
:
Set 4
Step 2
:
Move down 5
Step
(Fig.
17).
4<i)
,2>
„,
first
and 4
|(
next in close succession with the index
finger
Fig. 18
Fig. 17
Fig. 19
(Fig.
Note
1
:
18).
This example illustrates the
procedure used in setting 5 when the addition of two numbers
makes
The problems
5.
Step
:
which
3+2
4+1
Note 2
to
this
19 illustrates the incorrect way
Fig.
single continuing
rect
way
(2)
move
1+4
2+3
Note that the correct way
2.
procedure applies are:
down
to
perform
18) requires but a
(Fig.
stroke of the finger whereas the incor-
move down 4,
move down 5, resulting
requires three separate movements: (1)
the finger back up,
in a loss of time
Example
8.
and
(3)
and effort
5—
1=4
Step 1
5
Sp
\
:
Set 5
(Fig. 20).
move up 4 with
thumb, and then move up 5 with
Step 2
:
First
the
the
index finger in close succession (Fig. 21).
Fig. 20
Fig. 21
Fig. 22
Flick the
thumb and
the
index
finger
with the idea of performing the two motions at the same time.
Note 1
:
This example illustrates the procedure of subtract-
ing a number from
applies are
5.
The problems
to
which
this
procedure
Addition and Subtraction
5-1,
Note 2
is
As explained
:
from
subtracted
may perform
22
by
or
way
Example
9.
to
and
time, he
perform Step
2.
the incorrect
means
This
that
further,
travel
Note
up stroke
single continuous
attention has to
sulting in a loss of time
same
(Fig. 21).
movements.
your
beginner
the
at
the index finger, whereas
least
at
if the
pushing up four 1-unit
first
way requires but a
separate
three
But
time.
motions
illustrates the incorrect
thumb and
requires
finger
same
each separately
that the correct
of the
the
and then a 5-unit bead
Fig.
beads and one 5-unit bead
1-unit
hard to make the two
it
beads,
Step 2 of this example, when 1
in
at
5-4
6-4
7-4
8-4
5-3,
6-3,
7-3,
four
5,
pushed up
should be
finds
5-2,
6-2,
35
way
your
re-
effort.
4 + 3=7
Step 1
:
Set 4
Step 2
:
First
(Fig. 23).
move down
5,
and then 2
in
close succession with the index finger (Fig. 24).
Note
1
used in
This example illustrates the procedure
:
adding a 5-unit bead
dure applies
:
\
to
which
this proce-
:
4+2
3+2
4+1
Note 2
I
subtracting Fig. 23 Fig. 24
and
The problems
one or more 1-unit beads.
,
Since three
four 1-unit beads, 5
is
excess.
This operation
equation
:
1-unit
4+3
3+3
2+3
beads
added and 2
may be
is
4+ 4
3+4
2+4
1+4
cannot be added to the
subtracted to offset the
represented in the form of the
4+-3=4 + (5 ~~2)=7
36 The Japanese Abacus
•
Note 3
When working
:
Since 3 plus 4 equals 7,
simply remember that 2
the foregoing example, do not thii.k:
must form 7 on the board.
I
the complementary digit with which
is
down 5 and 2
3 makes 5, and by flicking
sum
allow the
7 to form
and 4 and
is
much
~31F
X
jT
digits
Set 7
Step 2
:
In close succession,
the
one or more
index
two motions
1-unit
illustrates
Problems to which applicable
Note 2
the two
:
idea
the
the 5 with
performing
of
same time
(Fig.
26).
the procedure for adding
one
subtracting
5-unit
bead.
:
5-3
6-3
7-3
5-2
6-2
5-4
6-4
7-4
8-4
Since three 1-unit beads cannot be subtracted from
1-unit
operation
with
move up 2
first
move up
then
at the
beads and
5-1
20.
(Fig. 25).
finger
example
This
:
mode
^
:
with the thumb, and
J".?
Example
Step 1
Fig. 25 Fig. 26 the
1
see Note 3 to
digits,
7-3 = 4
Example 10
Note
3 and 2,
For furiher explanation of calculation by means
of complementary
Ss
:
simpler and less liable to error than the ordinary
of calculation.
«F
digits for 5
There
by means of complementary
Operation
1.
in rapid succession,
on the board.
itself naturally
complementary
are only two groups of
Instead,
2
beads,
may be
is
added and 5
is
represented by the equation
subtracted
This
:
7-3=7+2-5=4
Note 3
:
When working
7 leaves 4, so 4 must
remember
makes
board.
5,
that
2
is
fce
the
and allow the
(See Note 3 to
example, do not think: 3 from
this
formed on the board. Instead, simply
complementary
result
to
Example
form
20.)
digit
with
which
itself naturally
3
on the
;
37
Addition and Subtraction
Example 11.
Step 1
Set 3
:
B
shall call
Step 2
3
+ 7 — 10
on a unit
which we
rod,
S
,.
(Fig. 27).
In close succession,
:
move
first
down then 3 on B with the index finger,
and then move up 1 on the tens' rod, here
B
A B
Fig. 27 Fig. 28
Flick
called A, with the thumb (Fig. 28).
the index finger and the thumb in a twisting manner so that
you may perform the two motions
Note 1
sum of two
digits.
1
on the
2+8
+9
tens' rod until
rod B.
If
Problems
10 when
it
is
you follow
which applicable
to
3+7
In working out this
:
to set
time.
the
This procedure requires the subtraction of
one or more 1-unit beads.
Note 2
how
This example shows
:
same
at the
:
4+6
do not move up
example,
1
you have moved down 3 on the unit
this incorrect procedure,
improve in bead calculation.
you
will never
For the theoretical reasons for
the advantages of the correct procedure, see Note 5 (The Order
of Operation) to Example 20.
Example 12.
Step 1
10-7 = 3
Set 10.
:
This
is
done by simply
moving up one bead on the
tens'
A
rod
(Fig. 29).
Step 2
down
First
:
the
1
remove the 10 by moving
on the
tens'
rod
A
with the
move up 3 on
thumb (Fig. 30).
index finger, and then
unit rod
Note 1
B
:
with the
This example shows
one or more 1-unit beads.
applies
how
the
to
subtract 10
and add
Problems to which such procedure
33 The Japanese Abacus
•
1
10-7
10-8
10-9
10-6
In working out this example, be sure to
Note 2
:
on the
tens'
rod
A
move down
up 3 on the unit rod
before moving
B.
For the theoretical reasons for the advantages of the correct
procedure, see Note 5 (The Order of Operation) to Example 20.
6 + 4=10
Example 13.
Step 1
:
Set 6 on the unit rod B.
Move down 1 on B with the
f index finger, then move up 5 on B with
the same finger, and finally move up 1 on
Step 2
A B
on the
Fig. 32
:
A
tens' rod
index finger and the thumb with the
the
same
idea of performing the last two motions at the
Note
it
is
made by
of two digits.
the addition
time.
sum 10 when
This example shows how to form the
:
Work
with the thumb.
This procedure,
re-
quiring the subtraction of both one or more 1-unit beads and
a 5 -unit bead, applies to the problems
6+4
74-3
Example 14.
10-4 = 6
^£- ^-^T
<f
:
Set 1 on the tens' rod A.
Step 2
:
First
finger,
and move up
on B
tract
Step 1
;
10-3
9
:
1
This
the
1
on
A
move down 5
then
same
at the
time.
example shows how
to sub-
10 and add both a 5-unit bead and
one or more 1-unit beads.
Example 15.
move down
with the index
Note
10-4
9+1
8-1-2
1
S te P
=====
:
Applicable problems
10-2
+ 4=13
Set 9 on the unit rod B.
:
10-1
:
Addition and Subtraction
•
39
Move down 1 on B with the
index finger, then move up 5 on B with
the same finger and finally move up 1 on
Step 2
A
:
with the thumb.
Work
index finger
the
and the thumb with the idea of performing"
same
the last two motions at the
Note 1
time.
A B
A B
Fig. 35
Fig. 36
This example shows how to add 10 after subtract-
:
ing one or more 1-unit beads and a
Applicable
5-unit bead.
problems
9+2
8+2
9+1
9
9+4
8+4
7+4
6+4
+3
8+3
7+3
Note 2
Do
:
your operation
you
on
will find
A
not reverse motions 1 and 2 of Step
will slow
latter
procedure
example, in adding 4 to
experts, in
6
Example 16.
13
workable in
is
or
7.
This
is
some
the
first,
and moving down
fingers,
cases, for
1 next.
on AB.
:
Set 13
Step 2
:
After moving
down
the 1 on A,
1
on B
This example shows
how
more
move down 5 and move up
at the
time.
:
1
main reason
-4=9
Step 1
Note
do,
working out Example 6 (8 — 6 = 2), disfavor the
procedure of moving up 5
same
you
B and move up
same time in the manner of twisting your
at the
although this
why
If
Because after moving up 5,
down.
hard to move down 1 on
it
2.
both a 5-unit bead and one or
bead after subtracting 10.
11-2
to
add
1-unit
a B
Fig. 37
Applicable problems.
11-3
12-3
11-4
12-4
13-4
49 The Japanese Abacus
•
Example 17
6
+ 6 = 12
Step 1
Set 6
:
on the unit rod B.
Move up 1 on B (Motion 1),
move up 5 on B (Motion 2), and move up
Work the two ringers
1 on A (Motion 3).
Step 2
A B
with
Fig. 39
Note
after
the
motions
1
:
idea
at the
of performing
same
This example shows how to add
:
the
two
first
time.
rod
1 to the tens'
adding one or more 1-unit beads and subtracting a 5-unit
bead on the unit rod.
Applicable problems
+6
6+6
7+6
5+7
6+7
7+7
+8
5+9
6+8
5
5
Note 2
last
:
as
is
8+6
possible
this
it
the
at
after completing the first motion.
jrocedure should not be followed,
does
not
work in some
example, the problem, " 46 + 6
41 and 42) or
perform the
to
two motions of Step 2 above
same time
But
It
:
"96 + 6 = 102/' can
For
cases.
= 52"
(Figs.
be worked in no other way
than that indicated.
Example 18
12
—6=6
AB.
Step 1
:
Set 12 on
Step 2
:
After moving
move down 5 and
1
A B
11-6
12-6
13-6
14-6
the 1 on A,
on B in succession.
Note: This example shows
Fig. 44
down
how to add a
5-unit
bead and subtract one or more 1-unit beads
after subtracting 10. Applicable
12-7
13-7
14-7
13-8
14-8
problems
14-9
:
Addition and
Example 19.
9
:
Set 9 on the unit rod B.
Step 2
:
Move down
thumb and
Note 1
:
same time
the
chanically at
the
move
3 on B, and
Perform the two motions me-
on A.
1
as if twisting
AB
the index finger.
how
This example shows
Applicable problems
2+9
4+9
3+9
3+8
Note 2
:
6+9
7+9
4+8
8+9
8+8
8+7
7+8
tion
This operation
added.
9
:
Example 20.
Step 2
:
Move down
the
1
:
on
1,
represented by the equa-
and
;
\>4pF
V^
num-
always
AB
set
Fig. 48
first.
This example shows
one or more 1-unit beads.
15-6
A
setting a two-digit
ber on the board, as in Step
Note 2
subtracted and
B.
When
the tens' digit
may be
is
AB.
Set 16 on
:
for 10,
16-7=9
:
move up 3 on
of 7
+ 7 = 9-3+- 10 = 16
Step 1
Note 1
9+9
9+8
9+7
9+6
Since 7 cannot be added to the 9 on the unit rod
B, 3, the complementary digit
is
10,
:
4+7
10
Fig. 46
to sub-
one or more 1-unit beads from the unit rod and add
tract
41
•
+ 7 = 16
Step 1
up
Sufrtoracfcion
15-7
16-7
how
to
subtract 10
Applicable problems
15-8
16-8
17-8
and add
:
15-9
16-9
17-9
18-9
....
Note 3
:
Since 7 cannot be subtracted from the 6 on the
42 The Japanese Abacus
•
unit rod, ID
digit of 7
from rod A, and
subtracted
is
for
10,
The
added.
is
be represented by the equation
3, the
complementary
may
basis for this operation
:
16-7-16-10 + 3 = 9
Note 4
:
Mechanization of Operation.
The fundamental
and speedy
simple
is
makes abacus operation
which
principle
To
mechanization.
give
a
explanation, the mechanical operation of the abacus
to
minimize your mental labor and limit
without carrying
mentary
it
digits for
and
5,
to
is
designed
the unit rod,
by means of the comple-
to the tens' rod,
10 and
it
theoretical
the
to let
result
form
itself
mechanically and naturally on the board.
To
give an example, in adding 7 to 9, the student accustomed
to the
Western mode of calculation
the board as a
9 and 7
to
the
is
of mental
result
calculation to the effect that
But such procedure
16.
probably form 16 on
will
in
is
every
way
inferior
Not only does
above-mentioned mechanical one.
Western method require mental exertion and time but
liable to cause perplexity
When
a problem of
the board,
the
and
and subtraction
is
is
worked on
Addition and
very simple.
is
it
errors.
addition
procedure
this
sub-
which involve two rods, are simplified by means of a
traction,
complementary
that
digit,
sum 10 when added
is,
the
digit
For instance, suppose we
to a given digit.
have to add 7 on a rod where there
say, " 7
and 3
and add
is
1 to next
necessary to give the
is
9
;
then
we think
or
10," and subtract 3 from the rod in question,
rod on the
left.
When we
have to subtract
7 from 16, we think or say, " 7 from 10 leaves 3," and subtract
1
from the next rod on the
in question.
This means that 10
added or subtracted on the
ing the complementary
tens'
digit, the
is
left,
and add 3
to the
always reduced to
rod.
1,
rod
and
Therefore, after recall-
operator has simply to perform
Addition and Subtraction
two mechanical operations
either of the
subtracting the com-
:
plementary digit and adding 1 on the tens'
rod
or subtracting 1 on the tens' rod and adding
The
tary digit (in subtraction).
No
on the board.
numbers
in the
to be
complemen-
the
form
may be contained
digits
added or subtracted, the entire operation
performed by applying
is
(in addition)
result then will naturally
how many
matter
43
•
this
mechanical method to each digit
in turn.
The same mechanical method
applies to the operations which
Suppose we
require the analysis of 5 (see Examples 7 to 10).
have to add 3 to
4; then
digit of 3 for 5 (that
5 when added to
two 1-unit
digits
3),
we merely think of
is,
the
sum
digit
and
the digit necessary to give
and we move down the 5-unit
on the rod
Then
in question.
Any
appear on the board.
naturally
the complementary
the result will
attempt to calculate the
answer mentally will retard the operation.
10 has only
8 and
two
:
2,
five
groups of complementary digits
7 and 3, 6 and 4, and
4 and
1,
and 3 and
2.
5 and
9 and
1,
while 5 has only
5,
Accordingly,
mechanized method requires no more mental
:
the
effort
use
of
the
than that of
remembering one of the elements of each of these very few
complementary
pairs of
digits.
This
the fundamental reason
is
which makes calculation by means of the complementary
much
simpler and
speedier
and
less
liable to error
digit
than the
ordinary way of mental or written calculation.
The following examples
the ordinary calculation
from
we
right to
first
left.
work on.
9 from
is.
digits,
In written calculation we proceed
thinking
-f-
"9 + 8 — 17,
Next we add the 30
In the problem
7, so
show how much more laborious
For instance in the problem 99
add the unit
and 24 + 6 = 30."
will
to
the
+ 77 + 66,
17 + 7 — 24,
88
90 of 99 and
567 — 89, we cannot subtract
borrowing 10 from the 6 in the
tens' place,
the
we
44 The Japanese Abactis
Next proceeding to the
get 8.
we again
tens' place
find that
we cannot subtract the 8 from the remaining 5 of the minuend, so we borrow 1 from tiie remaining 5 in the hundreds'
place, and we get 7 in the tens' place and the answer 478.
These processes involve laborious mental exertion.
On
from
hand,
other
the
left to right, that
is,
on the abacus proceed
calculations
all
from the highest
to the lowest digit.
This accords with our natural customary practice of naming or
remembering
numbers from the highest
all
Therefore, to set numbers on the board
In
to the lowest digit.
is to
calculate numbers.
conclusion, incredibly speedy abacus operation
attributable to three reasons: mechanical operation
the complementary digit, left to right operation,
is
mainly
by means of
and the
previ-
ously explained dozen rules of rational or scientific bead mani-
These are the reasons why, no matter how rapidly
pulation.
numbers may be mentioned,
as long as they are given distinct-
the skilled abacus operator can add
ly,
error,
how
of
irrespective
many
and subtract without any
digits
the
may
numbers
contain.
Note 5
When
:
The Order of Operation.
addition involves two rods, as in the example 9 4-7
= 16,
be sure to subtract 3 from the unit rod, and next add 10 in
the
form of
The
idea
1
to the tens' rod.
of 7
"7 = 10 — 3."
the
in
rod and the tens'
7.
Now
9,
rod,
you
add
to
out,
shifts of attention
should
not
be
add 10
first
i3
and
such a procedure
between the unit
followed.
Because in
will naturally first observe the unit rod to
subtract 3
to the tens' rod,
of attention
terms of the complementary digit
But as already pointed
which involves unnecessary
add
+ 7=9-3 + 10 = 16.
So you would be tempted
subtract 3 next.
adding 7 to
Thus 9
1.
from the unit
rod,
and then, proceeding
This procedure requires only two shifts
and operation.
On
the other hand, if
you did not
Addition and Subtraction
subtract 3
you would have
first,
to subtract 3
adding
after
the
to
1
come back
to
45
to the unit rod
This inferior
rod.
tens'
•
procedure would delay your operation.
The advantage of the correct procedure becomes even clearer
in some problems containing more
3Z?E!
For instance, in
than one digit.
h#SA
adding 7 to 996, note the decided
and
advantage of forming 3
mechanically
ceeding
the
left
straight
to
hundreds' rods of their 9 and to
shown
=8, be
rod,
y y
<*s
ABCD
ABCD
Fig. 49
Fig. 50
on the thousands',
as
subtraction involves two rods, as in the example 15
—7
set
1
49 and 50.
in Figs.
When
>~S
and
the tens'
clear
to
pro-
sure to subtract 10 in
and next add 3
the
form of
1
from the
tens'
Thus
to the unit rod.
15-7-15-10 + 3-8.
In subtracting 7, naturally you will
then you will see that
and
that
vou must borrow 10 from the
form of
stant subtract 10 in the
complementary
ing this
impossible
is
it
first
digit of
natural
then add the
3.
order
If
1,
to
look at the unit rod;
subtract
7
from 6
rod.
At
this in-
tens'
and then add
3,
attention, first subtract
you were
have to shift your attention
subtract 10 after adding 3.
to
10 and
the
this
to
the
tens'
to
use
complementary
the
rod to
wrong procedure would
cause needless shifting of attention and delay operation.
that failure
the
reverse this order, you would
back again
Thus
e.,
So follow-
7 for 10, to the unit rod.
of
i.
digit
Note
would necessitate
the less efficient method of mental calculation.
The
correct
procedure
is
especially
problems containing more than one
advantageous
digit, e.g., the
in
some
problem
1,000-1-999.
Following the correct procedure, in this problem we can proceed
45 The Japanese Abacus
mechanically
from
straight
right
to
left
52), while the
(Fig.
and
incorrect procedure (Fig. 53) involves the loss of time
wvv
5HS
2S^S.
ABCD
ABCD
ABCD
Fig. 52
Fig. 53
Fig. 51
When
setting
Also,
first.
when adding
or subtracting
digits,
set
the tens'
numbers of two or more
add or subtract beginning with the highest-place
digits,
This
numbers of two or more
labor.
another fundamental rule which will produce efficiency.
is
As previously explained, when a number
beginning with the highest
is
named
board than
beginning
of
opposite to that
ward with the
with
written
calculation,
last digit after a
on the
easily
This method
lowest digit.
the
or given,
can be mentally remembered
digit, it
and calculated much more naturally and
or set
digit.
which
is
started back-
is
number has been
given.
Adding and Subtracting Two-Digit Numbers
2.
When
setting two-digit
numbers,
Also when
set the tens' first.
adding and subtracting two-digit numbers, add and subtract the
tens'
This
first.
is
On
the abacus always operate
bers, as in
is
especially true in the calculation
number
the highest
digit,
it
is
named
or given,
much more
and
beginning with the highest-place
digit
Written
calculation
is
started
easily
than
num-
in the intro-
beginning
can be mentally remembered or
naturally
calculated
efficiency
of large
As explained
Examples 26 and 27.
duction, since a
left to right.
The
a fundamental rule based on efficiency.
of this rule
place.
from
set
with
and
on
the
board,
with
the
lowest-
backward with the
last
Addlt'Oii
number has been
digit after a
number
calculated while
is
number
to set a
is
it
and Subtraction 47
given, whereas on the abacus a
being
is
to calculate
given, in other words,
it.
14 + 25=39
Example 21.
AB
~IT
Step 1
Step 2
+ .2
Step 1
rod
B
AB
AB
AB
Fig. 56
Fig. 55
Fig. 54
Result of Step 2
Step 3
Result of Step 3
3 4
5
3 9
AB, with the 4 appearing on the unit
Set 14 on
:
+
(Fig. 54).
Step 2
Add
:
A
This gives 3 on
Step 3
the 2
Add
:
of 25 to the 1 on
and 34 on
the
AB
remaining 5
This gives 9 on B.
the forefinger.
A
with the thumb.
(Fig. 55).
of 25 to the 4 on
The answer
is
39
B
with
(Fig. 56).
The same procedure can be expressed diagramatically
as
seen above.
Example 22.
45
+ 27=72
AB
~4~5
±2_
Step 1
Step 2
6 5
AB
A B
Fig. 58
Fig. 59
AB
Fig. 57
A
Step 1
:
Set 45 on
Step 2
:
Add
and 65 on
answer
the 2 of 27 to the 4 on A.
AB
72
Example 23.
Step 1
:
Step 3
This gives 6 on
(Fig. 58).
:
is
7
7 2
(Fig. 57).
Add the remaining
2 on B and 7 on A, as 1
Step 3
gives
AB
+
7 of 27 to the 5 on B.
is
carried to the 6 on A.
(Fig. 59).
79-23=--56
Set 79 on
AB
(Fig. 60).
This
The
43 The Japanese Abacus
•
Step 2
:
from the
of 23
2
Subtract the
7
on
Thi3
A.
AB
7 9
-2
A B
A B
Fig. 61
Fig. 62
A B
Fig. 60
leaves 5 on
Step 3
:
-
A
AB
and 59 on
(Fig.
61).
The answer
56
is
A B
83 on
:
Set
Step 2
:
Subtract
:
A
AB
8.
A
subtract
AB
9
from
This
64).
The answer
to
34
Subtracting
.and
more
is
in
Example 25,
1
:
Set
Step 2
:
Add
456
456
from
A
This
9 from 10 on B.
ai-d
you
65).
of Over
Two
Digits
adding or subtracting numbers containing
Two
will
1
of 9 from 10,
1
Numbers
the case of two-digit numbers.
and subtraction
subtract
(Fig.
same
digits
borrow
3,
the
Step
Step 3
from the 8 on A.
(Fig.
and enables you
The methods used
three or
4 3
9
Subtract the remaining 9 of 49 from the 3 on B.
on B.
Adding
63).
Add, to the 3 on B, the remainder
get 4
Step 1
Step 2
3 4
(Fig.
4 of 49
the
and 43 on
As you cannot
on
-
8 3
4
Fig. 65
Fig. 64
Step 1
leaves 3
•
A B
A B
Step 3
62).
AB
5Z
Fig. 63
(Fig.
83-49 = 34
Example 24
4 on
Step 3
5 6
Subtract the remaining 3 of 23 from the 9 on B,
This leaves 6 on B.
leaves
5 9
3
Step 1
Step 2
are
suffice to
make
as
problems each in addition
this clear.
+ 789-1,245
on BCD (Fig.
the 7 of
789
those just described in
to the
66).
4 on B.
This gives you
Addition and Subtraction
253^.
A B C D
A B C D
A B C D
A B C D
Fig. 66
FiG. 67
Fig. 68
Fig. 69
AB
and 1,156 on
11 on
Step 3
Step 1
Step 2
4 5 6
7
to the 5
1,1 5 6
4Step 3
8
'
,
g
gt
Step 4
is
1245.
is
1,245
Example 26.
ABCD
3,17
+
+ 5,876=9,055
ABCD
ABCD
Fig. 72
Fig. 73
Fig. 74
8,1 7 9
8
Step 3
8,9 7 9
Step 4
1_
9,0 4 9
6 Step 5
9,0 5 5
Step 1
:
Step 2
:
Set 3,179 on
Add
Step 3
:
on B.
8,979 on
ABCD
:
ABCD
Step 5
:
Add
you 55 on CD.
Example 27.
Step 1
:
Set
(Fig.
70).
A
and 8,179 on ABCD.
Add the 8 of the remaining 876
to the 1
Step 4
ABCD
the 5 of 5,876 to the 3 on
A. This gives you 8 on
This
Add
gives
(Fig.
you 9 on B and
72).
the 7 of the remaining 76 to
This gives you 904 on
the 7 on C.
9,049 on
The answer
ABCD
±_ _
+
the remaining 9 to the 6 on
Fig. 71
Step 1
Step 2
9
BC and
(Fig, 69).
3,179
ABCD
5
8 of the remaining 89
you 45 on CD.
gives
67).
(Fig. 68).
Add
:
(Fig.
ABCD
Fig. 70
+
ABCD
This
D.
1,2 4 5
the
ABCD
This gives you 23 on
on C.
1,236 on
4
Add
:
49
23S
5S5I<
ABCD
•
ABC
and
(Fig. 73).
the remaining 6 to the 9
The answer
is
9,055
623-375-248
623 on ABC (Fig.
on D.
(Fig.
75).
74).
This gives
50 The Japanese Abacus
•
ABC
Stepl
6 2 3
3
-
Step 2
3 2 3
-
ABC
ABC
Fig. 77
on
Step 3
B by
A
and 323 on
from the 6 on A.
ABC
(Fig.
answer
Subtract
:
248
is
A B C D
3,8
-
y^vy :v4vy
ABCD
ABCD
ABCD
Fig. 81
Fig. 82
Fig. 83
Step 1
Step 1
Step 2
Set 6,342 on
:
rod
thousands'
the
(Fig.
and
Step 2
Step 3
Subtract the
:
the 6 on A.
Step 4
2
Step 3
Step 5
7
8>^ 9 5
ABCD
on
j
n g 547
from the 4 on A. This
ABCD,
2
on
with 6 on
the
ones'
79).
4 2
4_
The
Fig. 80
3,8 4 2
j.
This leaves 48 on BG.
on B.
C by
6,342-2,547 = 3,795
ABCD
_5
AB
on
A B C D
Fig. 79
•-
This leaves 25
78).
^W JSgg
6,342
76).
remaining 5 from the 3 on
the
(Fig.
Example 28.
2
This
77).
borrowing 1 from the 5
4, 3
(Fig.
borrowing 1 from the 3 on A.
and 253 on
-
ABC
Step 4
^
2 4 8
Subtract the 7 of the remaining 75 from the 2 on
:
Step 4
z
Fig. 78
Subtract the 3 of 375
leaves 3
Step 3
7_
2 5 3
:
fr0
2 of 2,547 from
This leaves 4 on
A
and 4,342
(Fig. 80).
Subtract
m
leaves 38
the
the 3 on
on
AB
5 of the remain-
B by borrowing
and 3,842 on
1
ABCD
(Fig. 81).
Step 4
;
Subtract the 4 of the remaining 47 from the 4 on
Addition and Subtraction
Step 5
Subtract
:
borrowing
answer
(Fig. 82).
remaining 7 from
the
the
2
D
on
by
from the 8 on B. This leaves 795 on BCD. The
1
3,795
is
ABCD
on C on 3,802 on
This leaves
C.
51
(Fig. 83).
Exercises
4.
Probably the most convenient way of dealing with problems
containing a long
of numbers, such as the following,
file
use the top edge of the abacus as a marker.
problem
1
on page 52,
of the abacus
number,
first
and
form
move
(Fig. 84); then
under the
on
it
the abacus
the next number, 20, appears
the
one
board
down
until
right above
and quickly.
Fig. 84
the beads
Fig. 85
(Fig.
85);
end of the problem.
also greatly facilitated by having some-
is
Numbers should be read
For example, the number 123, 456, 789
numbers.
call out the successive
distinctly
the
in this fashion to the
Abacus calculation
For example, in
first
number on
beard, and add that
and continue
to
place the top edge
immediately
24,
is
should be given as " one,
two,
million
three
;
four,
five,
six
thousand; seven, eight, nine."
The
best exercise for
nine times.
If
your sum
Again, add 789
ABC
is
nine times
skill is
correct,
it
to
will
add
123,456,789
be 1,111,111,101.
on the rods GHI, with
456 on DEF nine
as the
add 123
nine times, and you will get the same sum.
Subtract
123,456,789 from 1,111,111,101 nine times, and you
with zero.
will
end
These three exercises involve every procedure used
in problems of addition
and
subtraction.
operator can work each of the
and a
I
times; finally
unit rod; next add
on
attaining
three
A
third-grade abacus
exercises in
first-grade operator in thirty seconds.
one minute,
52 The Japanese Abacus
•
Adding and Subtracting Two-Digit Numbers
I.
(1)
(2)
(3)
(4)
24
20
55
44
22
66
11
44
-33
-11
-55
-22
-88
-33
77
44
22
12
55
-66
-22
-23
-11
-55
-m
11
11
55
<
33
-44
33
:5)
33
77
88
99
88
77
99
11
(6)
(7)
(8)
55
55
66
44
55
77
-55
-22
-80
54
35
33
69
-44
-76
-55
55
88
75
-33
-66
164
99
67
Adding and Subtracting Three-Digit Numbers
II.
m
[.
(1)
(2)
(3)
(4)
(5)
222
665
778
555
335
778
222
889
443
223
345
762
473
528
981
811
176
634
367
189
561
259
846
667
445
778
289
265
778
665
621
946
158
732
-255
-345
428
564
566
444
-392
-657
734
855
216
774
-628
-476
-889
-677
5,110
5,266
5,553
2,397"
322
Adding and Subtracting Four -Digit Numbers
;
(1)
(2)
(3)
(4)
(5)
3,627
1.508
9,472
6, 345
8,160
2,079
4,384
7,819
5,623
1,950
9,105
2,746
1,809
5,321
4,684
3,263
5,162
7,038
8,574
9,970
2,456
8,193
5, 647
7,038
9,825
3,741
6,580
1,269
4,001
9,372
7,081
5,469
-2,505
6,924
8,570
1,439
3,748
4,917
-3,268
-7,015
-6,803
-6,294
9,847
5,192
2,603
50,967
57,672
58,122
1,372
9,620
8,135
-3,786
24,740
24,557
4,051
V;
MULTIPLICATION
There are several methods of multiplication on the abacus.
The one introduced
which
in the following pages
generally considered the best and
is
method taught
in
grade
schools.
standard terminology will be used.
5x2 = 10,
problem
be
will
tLe standard
describing
Thus,
called
now
the
method,
.he
for example, in the
multiplicand, 2 the
and 10 the product.
multiplier,
It is
5
In
is
method
a recent
is
customary to
multiplicand at
set the
of the abacus and the multiplier to the
three rods unused between
the
the
just
The
Abacus Committee in favor of two unused rods
part
two or
leaving
left,
two numbers,
separate them clearly but not too widely.
central
enough
to
decision of the
will
be followed
used here gives
the product
in our problems here.
The method of
multiplication
immediately to the right of the multiplicand.
method which
favored
immediately to the
The
multiplier
tiplicand.
is
gives
the
first
figure
There
of the
is
a less
product
left.
very often a smaller number than the mul-
The small number
is
easy to remember, thus saving the
time and trouble of constantly referring to
it.
The experienced
abacist often saves even the trouble of setting a small multiplier
on the abacus. This
tiplicand
is
is
believed to be the reason
why
the mul-
customarily set at the right and the multiplier at the
54 The Japanese Abacus
left,
although there
ting the
is
nothing particularly objectionable to
set-
two d umbers in the reverse order.
Although the use of a unit rod marked with a unit point
does not have as
much
bearing in problems
of multiplication
and division as in addition and subtraction,
calculation in
the
unit
many
Therefore, in the following problems
ways.
of the multiplicand
figure
the case of the multiplier,
on the third rod
figure is set
As
plier,
set
is
however,
number, the unit rod
fractional
so
on a unit rod.
long
as
is
it
In
not
a
and the unit
disregarded
is
to the left of the multiplicand.
for the order of setting the multiplicand
and the
multi-
since the unit figure of the multiplicand must be set on
a unit rod,
it
is
advisable for the
plicand ahead of the multiplier.
beginner to
It
So they very often
at a glance.
of the multiplicand, thus saving
hand back
the
set
set
and the multiplier
the multiplier
the time
ahead of
required in shifting
to the left after setting the multiplicand.
more frequently experts
set
multi-
should be noted, however,
that experts can locate both the multiplicand
the
does facilitate
it
Even
only the multiplicand on the board
not even troubling to set the multiplier.
Multiplying by One-Digit Numbers
1.
Example
•
1.
4x2=8
VVW?? YVWV4
ABCDEF
2
ABCDEF
TTcIdTTT
Fig. 86
Fig. 87
Step 1
:
numbers
J,
8
Set the multiplicand 4 on the unit rod
multip^'er 2 on rod A, thus
the
2
as in Fig,
86
leaving two
Step 1
Step 2
4
Result
D
and the
vacant rods between
Multiplication
Step 2
4 by
Multiply the multiplicand
:
8
Set the product
multiplicand,
and
on F, the second rod
rod
clear
D
of
its
the
•
55
multiplier 2.
to the right of the
Fig.
4.
87 shows the
result of Step 2.
Note
The accompanying diagram shows another way to
Here the two figures in the row
the same problem.
:
illustrate
designated Step 1 indicate that the multiplier 2 and the multiplicand 4 have been set
remaining on
A
D
and
respectively.
row designated Result show the
the
in
figures
on
A
The two
multiplier
and the product 8 which has been
set
2
on F
as the result of the multiplication.
All the following examples
will
be illustrated in the two
ways shown above.
The reasons
multiplicand after
for clearing off the
be given
tiplication will
at the
its
mul-
end the section on multiplica-
tion by two-digit numbers.
Example
8x
2.
6=48
frW+W
ABCDEF
6
A B C D E F
ABC DE
Fig. 88
Fig. 89
Step 1
Set 8 on the unit rod
:
6
F
D
8
4_8
4 8
Step 1
Step 2
and 6 on A, leaving two
vacant rods (Fig. 88).
Step 2
clear
D
Multiplying 8 by
:
of
its
8.
6,
In this step,
the multiplicand, designated E,
48
set the
the
is
product 48 on EF, and
first
rod to the right of
the tens' rod of the product
(Fig. 89).
Note
:
Some
experts
say
it
is
desirable to clear
multiplicand before setting the product.
away the
For instance, in the
above example, they say that the product 48 should be
set
on
56 The Japanese Abacus
EF
D
after clearing
of
This method
8.
its
lias
saving the time of shifting the hand back to the
off the multiplicand after setting the product.
Committee frowns upon
for beginners,
memory
plicand must be carried in the
after
it
clear
left to
But the Abacus
saying that,
this procedure,
apt to cause confusion in
is
it
the merit of
especially
the
that
multi-
has been cleared
away from the board.
Example
24x7=168
3.
ABCDEFG
2 4 (To
7
2
11
2 8
1 4
Fig. 92
Fig. 91
ABCDEFG
7
ABCDEFG
A3CDEFG
Fig. 90
Step 1
the unit rod,
Step 1
Ste P 2
Step 2
by
Step 3
7000168
and
,
7, set the
E
clear
of
set 7
on
A
Von
product 26
its
4
(Fig. 91).
2 in 24 by
product 14 on EF, thereby adding this
new product
:
on FG, and
clear
EFG, which
is
Note
1
E
of
This makes a
for setting the product 14
EF, which are one place higher than FG,
adding
14,
product
is
set
on
do
not
EF.
form
Note 2
take
the
trouble
is
:
set the
28
to the
of 168 on
to
the
previous
When
of thinking that
Instead just mechanically set the 1
14
on the rods
obvious.
140 in actual value and that therefore
and add the 4 in
result
total
7,
the answer (Fig. 92).
The reason
:
2.
its
24
4 in
j
on rt^
G, and
Multiplying the remaining
Step 3
(Fig. 90).
Multiplying the
:
„
.
24 on DE, with E as
Set
:
this
in
2 on F, and
this
must be
14 on
let
E
the
itself automatically.
In Step 2,
while in Step 3,
E
is
F
is
the tens' rod
of the product 28,
the tens' rod of the product 14.
In each
Multiplication
step of multiplication, the first rod to the right
in the multiplicand which
is
multiplied
•
57
of that figure
the tens' rod of the
is
product.
Note 3
When
:
there are two digits in the multiplicand,
multiply the last digit by the multiplier and then the
2.
first
digit.
first
Multiplying by Two-Digit Numbers
Example
8x17 = 136
4.
ABCDEFGH ABCDEFGH ABCDEFGH
Fig. 93
Step 1
A BCDEFGH
17008000
8
+5^6
Set 8 on the unit rod
:
AB
and 17 on
Step 1
Step 2
Step 3
Step 2
1 in
17000136"
Step 3
Fig. 95
Fig. 94
93).
Multiplying the 8 by the
:
the
set
17,
(Fig.
E
product 8
on
G
(Fig. 94).
Multiplying the 8 by the 7 in 17, set the product
:
56 on GH, and
on G, you
clear
E
of
on FGH, a
get,
its
Since
8.
total of 136,
you already have 8
which
is
the
answer
(Fig. 95).
Note
:
When
there
are
two
multiply the multiplicand by the
next by the
Example
last
5.
digits
first
in
the
multiplier,
digit of the multiplier
digit of the multiplier.
46x23 = 1,058
522^^^^^
ABCDEFGH
Fig. 96
ABCDEFGH
Fig. 97
first
and
53 The Japanese Abacus
A BCDEFGHI
23004600
Step 1
Step 2
Step 3
1 2
+ 18
230040138
+
+
ABCDEFGHI
23 on
:
AB
Step 2
:
:
Multiplying the 6 in
GH
unit
and
rod,
set
46 by the 2
in 23, set the
(Fig. 97).
on HI and
GH, you
member
the
Multiplying the same 6 in 46 by the 3 in 23, set
the product 18
12 on
as
(Fig. 96).
product 12 on
Step 3
46 on EF, with
Set
1 2
230001058
Fig. 98
Step 1
Step 4
Step 5
8
get
a
clear
total
that each time the
of
F
of
Since you have
6.
its
138 on GHI
same
digit
(Fig. 98).
Re-
in the multiplicand is
multiplied by one digit after another in the multiplier, the value
of the product
reduced by one rod or place.
is
wy w^^
ABCDEFGHI
ABCDEFGHI
Fig. 99
Fig. 100
Step 4
:
Multiplying the 4 in 46 by the 2 in
This makes a total of 938 on
product 8 on G.
Step 5
:
answer 1,058 on
Note
the
GHI
set
the
(Fig. 99).
GH
and
FGHI
clear
(Fig.
E
of
its
4.
This leaves the
100).
In case both the multiplier and the multiplicand have
:
digits,
first
23,
Multiplying the same 4 in 46 by the 3 in 23, set
the product 12 on
two
wyw^gM
(1) multiply
the
last
digit of the multiplicand
digit of the multiplier; (2) multiply the
same
by
digit of
the multiplicand by the last digit of the multiplier; (3) multiply
the
first
digit
of the
multiplicand
by the
first
digit
of
the
Multiplication
multiplier;
and
plicand by the
(4) multiply the
last
same
digit of the
first
This
digit of the multiplier.
is
•
59
multi-
the funda-
mental rule of multiplication.
Example
97x48=4,656
6.
i^yv^y^
y+yygggg
ABCDEFGHI
ABCDEFGHI
Fig. 101
Fig. 102
ABCDEFGHI
480097000
^yvg
4
80
9
+ 36
ABCDEFGHI
+
Fig. 103
Step
on
AB
1
:
(Fig.
Step 2
:
:
F
Step 1
Step 2
Step 3
Step 4
Step 5
7 2
4 6 5 6
4 8
Set 97 on EF, with
as the unit rod,
and
set
Multiplying the
GH
(Fig.
7
in
97 by the 4 in 48,
set the
102).
Multiplying the same 7 in 97 by the 8 in 48,
the product 56 on HI, and clear
28 on GH, you get
a total of
F
of
its
7.
336 on GHI
(Fig.
103).
55^f5SS5
ABCDEFGHI
ABCDEFGHI
Fig. 104
set the
:
set
Since you have
y^>yy^y^>y^
Step 4
48
101).
product 28 on
Step 3
2 8
+5 6
3 3 6
Fig. 105
Multiplying the remaining 9 in 97 by the 4 in 48,
product
36 on FG.
This makes a total of 3,936 on
CO The Japanese Abacus
-
FGHI
(Fig.
Step 5
104).
Multiplying the same 9 in 97 by the 8 in 48, set
:
GH, and
the product 72 on
on FGHI, a
Note
of 4,656, which
total
of clearing off
each
multiplication by all the
of
is
The preceding examples
:
sirability
its
E
clear
the answer (Fig.
will
105).
have indicated the
in the multiplicand
digit
in the
digits
This gives you,
9.
its
multiplier.
This
is
large number.
of the digits
by
the
digits
the
in
the
in
find
Second,
further to the
cedure by as
a
tell
multiplied
incorrect
this
the
beyond the product of the
right
many
of
is
hard to
it
multiplicand you had
multiplier.
you
opera-
the multiplicand
would necessitate the removal
procedure
3.
when
case
you would often
First,
which
all
the
especially
after
If
did not do so, you would be greatly inconvenienced in
tion.
de-
multiplier
correct pro-
digits as there are in the multiplicand.
Multiplying by Numbers of Over Two-Digits
No
how many
matter
principle
digits
of multiplication
by two-digit numbers.
is
the
the
You have
multiplier
same
may
have,
the
as that of multiplying
only to see that you do not
mistake the order of multiplication
and the rods on which
to
to set products.
Example
Step 1
432 on
:
7.
Set 37 on
ABC
Step 2
:
37x432 = 15,984
(Fig.
FG, with
G
as
the
unit
rod,
and
106).
Multiplying the 7 in 37 by the 4 in 432,
set the
fir
ABCDEFGHI
Fig. 108
set
J
ABCDEFGHI
Fig. 107
JK
Multiplication
•
61
<^ yyv<7^7y^
Y^y^^^Z?^^?^
ABCDEFGHIJK
ABCDEFGHI JK
Fig. 108
Fig. 109
VVYVYYi?W
ABCDEFGHI
V<?<?9W4^-
ABCDEFGHIJK
JK
Fig. 110
Fig. Ill
VVVVWYMiY
ABCDEFGHI JK
TTTIToT 7
0~0~0~0 Step 1
Step 2
2 8
2
1^
ABCDEFGHIJK
4320030302
1
Fig. 112
4
2
9
product 28 on HI
Stf
p
3
:
(Fig.
makes
set the
:
(Fig.
7
product 14 on JK, and clear
Step 5
:
HIJK
(Fig.
on
IJ.
This
108).
in
G
37 by
of
its
the
7.
2 in 432,
This makes
109).
Multiplying the 3 in 37 by the 4 in 432, set the
product 12 on GH.
This makes a total of 15,024
on
GHIjK
110).
Step 6:
product 9
(Fig.
product 21
the
set
HIJ
Step 5
Step 6
Step 7
43200015984
Multiplying the same
a total of 3,024 on
(Fig.
432,
a total of 301 on
Step 4
6
107).
Multiplying the same
7 in 61 by the 3 in
Step 3
Step 4
1
Multiplying the 3 in 37 by the 3 in 432, set the
on
I.
This
makes a
total
of 15,924 on
GHIJK
111).
Step 7
:
Multiplying the 3 in 37 by the 2 in 432, set the
product 6 on
J,
and
clear
F
of
its 3.
This makes, on GHIJF,
62 The .Japanese Abacus
a total of 15,984, which
Example
8.
the answer
is
(Fig.
112).
78x503 = 39,234
ABCDEFGHIJK
ABCDEFGHIJK
Fig. 113
Fig. 114
ABCDEFGHI JK
50300780000
S^2^^^2S2
4
5
03007 04024
+ 35
+ 21
ABCDEFGHIJK
on
ABC
Step 2
Set 78 on
:
(Fig.
:
FG, with
:
the product
total
(Fig.
and
503
set
H
(Fig.
114).
Multiplying the same 8 in 78 by the 3 in 503, set
24 on JK and
as the second
the
as the unit rod,
Multiplying the 8 in 78 by the 5 in 508, set the
of 4,024
words,
G
113).
4 of the product 40 on
Step 3
Step 4
Step 5
50300039234
Fig. 115
Step 1
Step 1
Step 2
+_2_4 Step 3
on HIJK.
figure
product
In
of the
must
G
clear
be
of the
setting
this
multiplier
set
on
8.
product skip rod
503
JK
This makes a
is
zero.
instead
I
In other
of
on
IJ
115).
^yy^^^w
4 y? ^^y^yv
<T>
<
<>
!
ABCDEFGHI JK
ABCDEFGHIJK
Fig. 116
Fig. 117
Step 4
:
Multiplying the 7 in 78 by the 5 in 503, set the
product 35 on GH.
This makes a total of 39,024 on
GHIJK
Multiplication
116).
(Fig.
Step 5
Multiplying the same
:
7
in
set the
product 21 on IJ instead of HI,
of 503
is
zero,
and
clear
a total of 39,234, which
4.
63
F
is
of
its
78 by the 3 in 503,
as
the second figure
This leaves, on GHIJK,
7.
the answer
(Fig.
117).
Ex ercises
Group
1.
2
3.
4.
5.
6
7.
8.
9.
10.
Group
.
I
34x4 = 136
23x5 = 115
12x4 = 48
32x3 = 96
21x5 = 105
33x45 = 1,485
52x56 = 2,912
23x65 = 1,495
53x75 = 3,975
25x85 = 2,125
11
12
13
14.
15
16
17
18
19.
20.
'
21x23 = 483
12x32 = 384
21x43 = 903
12x56 = 672
31x64=1,984
43x56 = 2,408
32x64 = 2,048
53x76 = 4,028
23x83 = 1,909
35x96 = 3,360
II
11.
9.
112x23 = 2,576
123x35 = 4,305
212x46 = 9,752
345x57 = 19,665
423x64=27,072
513x76 = 38,988
607x87 = 52,809
452x85 = 38,420
631x95 = 59,945
19
1,023x34 = 34,782
3,243x45=145,935
4,352x58 = 252,416
5,624x67 = 376,808
6,712x78 = 523,536
132x334 = 44,088
234x456 = 106,704
431x467 = 201,277
546x686 = 374,556
10.
603x97 = 58,976
20.
756x879 = 664,524
1
2.
3.
4.
5.
6
7.
8.
12.
13
14
15
16
17
18.
DIVISION
VI.
There are two fundamental methods of division on the abacus.
The
older method, though
out
of
general
memorization
which
is
the easier to learn because
in grade schools,
and
Strictly speaking,
requires the
one now taught
standard
will be introduced in the following pages.
not new, as
is
it
has long been used, but
it
only in a very limited use until around 1930 when
proved and publicized.
Standard terminology
is
It
the dividend, 5 the divisor,
customary
is
to
will
was im-
it
be used in
For example, in the problem 50-4-5 = 10,
describing the method.
50
fallen
uses the multiplication
it
the
is
it
has
The newer method,
of a special division table.
instead of the division table,
some,
1930 because
since about
use
by
favored
still
set
and 10 the
dividend a
the
the central part of the abacus
and the
quotient.
little
to the right of
divisor at the
two numbers are generally separated by
left.
three or four
The
unused
As the Abacus Committee favors leaving four unused
rods.
rods between the two
numbers,
the
following
examples
will
adhere to that practice.
The method of
division used here gives the
quotient between the dividend and divisor.
can
be
divisor
given
on the
for
setting
left.
One
the
is
dividend
that
since
first
digit of the
Two main
reasons
on the right and the
the
abacus
is
operated
with the right hand, the reverse order of setting the two
bers would cause the multiplier to be hidden by the
num-
hand much
65
Division
of the time,
as
in
the
of
case
by the
that in case the dividend is indivisible
verse order would cause the
The
multiplication.
quotient
other
is
divisor, the re-
extend right into the
to
divisor.
In division, as
in multiplication, the use of the unit rod
many
not too essential, but does facilitate calculation in
Therefore, the
unit
When
unit rod.
of the dividend
figure
the divisor
shall disregard the unit rod,
way
a
that
is
a
whole
and simply
ways,
on a
set
number, however, we
such
set the divisor in
on the
last digit is located
its
always
is
is
rod to the
fifth
left
of the dividend.
As
for the order of setting the
dividend and divisor, since
the last digit of the dividend must be set on a unit rod,
it
is
advisable for the beginner to set the dividend before setting the
As
divisor.
the
is
often reverse
the
with
case
procedure,
multiplication, however, experts
setting the divisor first or not at
all.
1.
Dividing by One-Digit Numbers
Example
8—2=4
1.
<^<^Z^y<^
ABCDEF
2
8
4
A B C D E F
Fig. 118
Step 1
:
ABCDEF
Step 1
Step 2
-J^
2
4
Result
Fig. 119
Set the dividend 8 on rod
F and
the divisor 2 on
rod A, with four vacant rods between the two numbers.
F
sure that
Step 2:
4 on
D
;
is
a unit rod
marked with
a unit point
Mentally divide 8 by 2 (8-^2=4);
Make
(Fig. 118).
set the quotient
the second rod to the left of the dividend;
and
clear
66 The Japanese Abacus
*
F
of
its
Fig.
8.
119 and the row of
diagram show the
in the
Example
figures designated Result
result of this step.
837-4-3=279
2.
^yvw^-^l
<^ywy^<z
ABCDEFGH
ABCDEFGH
Fig. 120
Fig. 121
ABCDEFGH
Step 1
Step 2
8 3 7
~S~0~0
Step 1
2
H
with
A
837 on the rods FGH,
as the unit rod,
(Fig.
30020237
Set
:
and
120).
Step 2
Compare
:
the
3 with the
Step 3
7
-2
8 in 837.
1
30027027
-2
2
Step 4
9
left
2 on
7
30027900
the 8 on F.
3 goes into 8 twice with
over.
D,
Set
the
quotient figure
second rod to the
the
of 8 in 837.
Next multiply the divisor
This leaves 2 on
F
(Fig.
121).
ABCDEFGH
ABCDEFGH
Fig. 122
Fig. 123
Step 3
:
Compare
the
with
3
23 on FG.
The 2 on F
figure
on E.
Step 4
2
left
over.
Set 7 as the quotient
Next multiply the divisor 3 by
tract the product
(Fig.
with
21 from
the
is
3 goes
the remainder left over as a result of the previous step.
into 23 seven times
left
and subtract the product 6 from
3 by this quotient figure 2,
G
3 on
set
23 on FG.
this 7,
and sub-
This leaves 2 on
122).
:
Compare the 3 with 27 on GH. The 2 on
G
is
the
Division
remainder
left
over
27 nine times.
into
a
result
Set
the
as
second
of the
quotient
3 goes
step.
9 on F.
figure
67
•
Next
multiply the 3 by this 9, and subtract the product 27 from the
27 on GH.
DEF
This
Answers
:
and
the
leaves
answer 279 on
123).
(Fig.
Note
GH
clears
to
problems in division can be easily checked
by multiplication. Thus, to check the foregoing answer, simply
the
number you
originally divided
FGH,
product 837 on
DEF
279 on
multiply the quotient
i.
837 as the dividend. By
e.,
the
this
by the divisor
and you
by,
that
3,
get
will
is,
the
same rods on which you had
checking the student will see that
the position of the quotient in division
that of the multipli-
is
cand in multiplication, and that the position of the dividend in
division
that
is
we may say
of the product in multiplication.
that the
Therefore,
and
methods of multiplication
division
introduced in this book form the counterpart of each other.
Example 3
6,013^7 = 859
IWWiW?
>At< ><^7
<T>< ><r
T
ABCDEPGHl
ABCDEFGHI
Fig. 124
Fig. 125
ABCDEFGHI
TOl)
13
6
8
Step 1
Step 1
Step 2
-_5 6
7
0008 0413
A
the
as
(Fig.
:
6,013
unit
on
rod,
and
FGHI,
set
7
124).
Compare
with the 6 in 6,013.
3 5
6 3
5
the
divisor 7
-
into 6.
7 will not go
So compare the 7 with the
Step 4
9
60 in 6,013.
6 3
700085900
figure 8
on
I
Step 2
-
008
with
Set
:
Step 3
5
7
T
on E» the
times.
first
rod to the
7 goes into 60 eight
In this case set the quotient
left
of the
first digit
of the
68 The Japanese Abacus
•
Next multiply the divisor 7 by
dividend.
56 from the 60 on FG.
the product
(Fig.
This
4
leaves
on
G
125).
^yyv^i^^Y
^<r>99^$^7
ABCDEFGHI
CDEFGHI
Fig. 127
Fig. 126
Step 3
Compare
:
7 by this 5,
:
H
(Fig.
Compare
63 nine times.
multiply the 7
by
the 63 remaining
859 on
Note
:
GH.
on
the 7 with 63 remaining on HI.
this
9,
on HI.
dividend, compare
it
quotient
cedure
is
vision.
is
divisor
is
with the
larger than the
two
first
The
figure
digits of the dividend.
rod to the
left
first
rod to the
left
chief merit of this pro-
multiplied by the di-
the
same
as
the
principle
write
of graphic
di-
the quotient figure 1
But in dividing 36 by 4, you write the
On
the abacus board the
cannot be put above the dividend.
viding 36 by 2, the
digit of the
to its division.
quotient figure 9 above the 6 in 36.
quotient
first
on the very rods on which the dividend
In dividing 36 by 2, you
above the 3 in 36.
Next
127).
visor gives the product
This procedure
G.
This clears HI, and leaves the answer
checking, the quotient
was located previous
on
7 goes
and subtract the product 63 from
digit of the dividend.
that, in
9
figure
In this case set the quotient figure on the
first
Next multiply the
126).
Set the
EFG (Fig.
When the
of the
7 goes into 41
and subtract the product 35 from the 41 on GH.
This leaves 6 on
Step 4
the 7 with 41
Set the quotient figure 5 on F.
five times.
into
and subtract
this 8,
first
quotient figure 1
is
se":
So in
di-
on the second
of 36, while in dividing 36 by 4, the quotient
Division
figure
9
is
set
on the
rod to the
first
left
69
of 36.
Dividing by Two-Digit Numbers
2.
Example
4.
552-f-23=:24
ABCDEFGHI
ABCDEFGHI
Fig. 128
Fig. 129
www?y?
ABCDEFGHI
230000552
2
- 4
_~
ABCDEFGHI
2 3
01
Step 1
with
I
23 on
:
on
as the unit rod,
AB
Step 2
:
(Fig.
Compare
left
and
Step 3
set
of the 5 in 552.
Now
2 goes
Set the quotient figure 2 on E, the second
Next multiply the 2 in 23
and subtract
This leaves 1 on
:
Step 5
230024000
the 2 in 23 with the 5 in 552.
this quotient figure 2,
5 on G.
Step 4
8
-12
GHI,
Step 3
128).
into 5 two times.
rod to the
6_
Step 1
Step 2
9 2
-
552
Set
2
4
Fig. 130
by
•
G
(Fig.
the product
4 from the
129).
multiply the 3 in 23
by
the
same quotient
and subtract the product 6 from 15 on GH.
9 on H
(Fig. 130).
figure 2,
leaves
??WVVV¥V
W'QV^VVW
ABCDEFGH
ABCDEFGHI
Fig. 131
Fig. 132
This
J
70 The Japanese Abacus
•
Step 4
:
-
Compare the 2
2 in 23
by
this
quotient
product 8 from the 9 on H.
Step 5
Next mul-
and subtract the
H
This leaves 1 on
(Fig. 131).
from the 12 remaining on HI. This
24 on EF
leaves the answer
Example
figure 4,
2 goes
H.
Multiply the 3 in 23 by the same 4, and subtract
:
the product 12
and
23 with the 9 on
Set the quotient figure 4 on F.
into 9 four times.
tiply the
in
(Fig.
clears
HI
132).
6,303-83 = 76
5.
gWggEg
ABCDEFGHIJ
ABCDEFGHIJ
Fig. 133
Fig. 134
ABCDEFGHI
8300006308
7
-
ABCDEFGHI
J
6
Set 6,308 on GHIJ,
:
with J as the unit rod, and set
83 on
AB
Step 2
(Fig.
not go into
in
this
left
of the
7,
:
Set the quotient figure 7
6 in
8
on F, the
Next multiply the 8
6,308.
This leaves 7 on
H
Multiply the 3 in 83 by the
(Fig.
(Fig.
135).
134).
same quotient
and subtract the product 21 from 70 on HI.
49 on HI
8 will
7 in the quotient, and subtract the product 56
from the 63 on GH.
Step 3
Step 5
8300076000
So compare the 8 with the 63 in 6,308.
6.
rod to the
83 by
Step 4
the 8 in 83 with the 6 in 6,308.
goes into 63 seven times.
first
-48
-18
133).
Compare
:
Step 3
1
4 9 8
7
Fig. 135
Step 1
5 6
-2
8 3
Step 1
Step 2
figure
This leaves
71
Division
^vyv^^^
4lWV<i>44W^
ABCDEFGHIJ
ABCDEFGHIJ
Fig. 136
Fig. 137
Step 4
into
49
Compare
:
8 in 83 by
Step 5
6,
leaves the answer
Note
:
In case the
76 on
FG
divisor
is
take the trouble of comparing
136).
two
with that of the dividend.
larger than that
of the
two digits of the dividend.
incorrect, correct
the
dividend,
In
137).
number, do not
digits with the first
work out the
When
figure
This clears IJ
IJ.
two-digit
a
Simply compare the
figure mentally.
7,
(Fig.
its
or three digits of the dividend to
is
(Fig.
I
and subtract the product 18 from 18 on
and
Next mul-
Multiply the 3 in 83 by the same quotient
:
8 goes
and subtract the product. 48 from
this 6,
This leaves 1 on
the 49 on HI.
on G.
Set the quotient figure 6
six times.
tiply the
49 on HI.
the 8 in 83 with the
first
first
two
correct quotient
digit of
the divisor
digit of the divisor
compare
case quotient
them by the methods shown
it
with the
first
figures tried are
in
Examples
and 8 instead of perplexing yourself with mental
6,
arithmetic.
Thus make the most of the chief advantage of the abacus, the
complete mechanical process which minimizes mental labor, and
experience will enable you to find correct quotient figures at a
glance.
Example
tf.
4,698-54=87
S5255W
^W94 VHA
ABCDEFGHIJ
ABCDEFGHIJ
Fig. 138
Fig. 139
J
The Japanese Abacus
72
how
This example shows
when
revised
9
Step 1
rod,
and
(Fig.
138).
Step 2
AB
54 on
The 5
:
in
54
OS
6 9 8
5 with the 46 in 4,698.
3 7 8
Step 5
7
-
So compare the
4,698.
Step 4 (revision)
3 2
-3
go into the 4 in
will not
+_5_
0~0~F0
5
5
2 8 Step 6
goes
46
into
Now
times.
5400087000
nine
suppose you
have tried 9 as the quo-
tient figure instead of the correct
you
set
Step 3 (revision)
1
5 4
J as the unit
4 5
5400090198
540
Set 4,698 on
:
GHIJ, with
Step 1
Step 2
-(36)
-
must be
of division
process
too large a quotient figure has been used.
ABCDEFGHI
5400004698
-
the
54 by
will multiply the 5 in
45 from the 46 on GH.
8 and have
set
it
on F.
Then
and subtract the product
9,
This leaves
on H.
1
Next multiply-
ing the 4 in 54 by the same 9, you will find that the product
36
to
is
larger than the 19 remaining
have tried a quotient figure one
^
WW^4M
ABCDEFGHI
on HI and that you ought
than 9
less
:
subtract 1
8 on
To
revise
ABCDEFGHI
J
the
from the 9 on
Step 4
8,
quotient
incorrect
F,
and you
between the quotient figures 9 and
to the 1
on H.
:
J
Fig. 141
get the
Next multiply the 5 in 54 by
F.
139).
<k»<?<S<r><S^<?^&'&
Fig. 140
Step 3
(Fig.
Now you
8,
have 6 on
1,
figure
9
to
8,
new
quotient figure
e.,
the difference
i.
and add the product 5
H
Multiply the 4 in 54 by the
(Fig.
new
140).
quotient figure
and subtract the product 32 from 69 on HI.
This leavea
J
Division
37 on HI
(Fig.
•
73
141).
WWSB2
ABCDEFGHIJ
ABCDEFGHIJ
Fig. 142
Fig. 143
Step 5
into
Compare
:
37
seven
the 5 in 54 with the 37 on HI.
So
times.
Next multiply the 5 in 54 by
35 from the 37 on HI.
Step 6
7,
the
set
quotient
this 7,
5 goes
on
7
figure
and subtract the product
This leaves 2 on
(Fig.
I
142).
Multiply the 4 in 54 by the same quotient figure
:
and subtract the product 28 from 28 remaining on
clears IJ
and
Example
7.
leaves the answer
the
first
87 on
FG
(Fig.
IJ.
This
143).
1,666-5-17=98
This example shows
when
G.
how
a
problem of division
digit of both divisor
is
worked
and dividend are the same.
<?4><r><?<?<^ <?AS^>
ABCDEFGHIJ
ABCDEFGHIJ
Fig. 144
Fig. 145
ABCDEFGHI
1700001666
9
-
ABCDEFGHIJ
:
AB
Set 1,666 on GHIJ,
(Fig.
Step 4
9
with J as the unit rod, and
17 on
Step 3
1700090136
Fig. 146
Step 1
9
6 3
_
17
-
set
1 7
9
-(6
3)
9
9046
-
1
Step 5
+
___
144).
Step 1
Step 2
1
j-_56 Step
9 8
6
74 The Japanese Abacus
•
Step 2
When
t
the
are the same, as in
of the two numbers.
of the dividend
In such a situation,
smaller
is
example.
one
figure
If
than
If 9 is
the quotient figure.
this
compare the second
example,
this
8
is
and the dividend
digit of the divisor
first
still
that
too
large,
try
too large, go on
less till the correct
one
is
the second digit
if
of the
found.
digits
try 9 as
divisor,
8 as in Step 5 of
trying a quotient
In such a case 9
the figure likeliest to be correet.
is
Now
9 as the quotient figure and
try
rod to the
left
of the
set
it
on F, the
of the dividend.
first digit
Next multiply
the 1 in 17 by this 9 and subtract the product 9
GH.
H
This leaves 7* on
Step 3
(Fig.
first
from 16 on
145).
Multiply the 7 in 17 by this same 9, and subtract
:
This leaves 13 on
the product 63 from 76 on HI.
HI
(Fig.
146).
Step 4
The
:
and the
1
H
the
are
1 in
V^VW44Wi
17
remaining on
So
same.
compare the 7 in 17 and
3
the
3
is
try
on
remaining
smaller than 7.
ABCDEFGHI
I.
So
9 as the quotient figure and
set it
the 1 in 17 by this 9 and
subtract
13 on HI.
on
This leaves 4
17 by this same
9,
you
than 46 remaining on
J
Fig. 147
I.
the
So you
have tried 8 as the quotient figure
product
Now
9
multiply
from the
Next, multiplying the 7 in
will see that the
IJ.
on G.
product
will find that
(Fig.
63
is
larger
you ought to
147).
V4VW44 V#^
^(SVV^^ff
ABCDEFGHI
ABCDEFGHI
Fig. 148
J
Fig. 149
J
J
Division
Step 5
tract
To
:
So multiply
in Step 4.
the
17
in
1
by
between the 9 and 8, and add the product
ing on
Then you
I.
on
get 5
(Fig.
I
|
Step 6
Multiply the 7 in 17 by the
:
98 on
leaves the answer
Note
In cases where the
:
FG
(Fig.
8, sub-
difference
the
to the
4 remain-
148)..
new
quotient figure 8
This clears IJ
IJ.
149).
both the divisor and
digits of
first
1,
1
and subtract the product 56 from 56 on
and
75
Next you must revise the division
from the 9 on G.
1
9 to
revise the incorrect quotient figure
•
the dividend are the same, if the second digit of the dividend
larger than that of the divisor, set 1
is
on the second rod
An
instance
Example
is
as the quotient figure
to the left of the first digit of the dividend.
given in Example 9.
7,644-84=91
8.
This example
is
to
show how division
the quotient figure tried
is
is
to be revised
when
too small.
^< >< >< >< >y^^<^7
T
T
r
T
ABCDEFGHI
ABCDEFGHI
J
Fig. 150
J
Fig. 151
ABCDEFGHI
.
•
.
•
8 4 (TO
7 6
4T
8
ABCDEFGHI
-6
+
Set 7,644 on GHIJ,
:
with J as the unit rod, and
84 on
AB
Step 2
:
(Fig.
The 8
set
84
Step 4
1
-
8 4
8"4
will
(Revision)
8T00T9T0'
Step 5
1
-
150).
in
Step 3
8400080924
Fig. 152
Step 1
4
-32
J
Step 1
Step 2
8
-I
"8400091000
Step 6
76 The Japanese Abacus
*
So compare the 8 with the 76 in
not go into the 7 in 7,644.
8 goes into 76 nine times.
7,644.
So you ought
But suppose by mistake you have tried 8
the quotient figure.
as the quotient figure instead of the correct
Then you
on F.
GH
(Fig.
Step 3
than 8
1,
and
9,
the
e.,
i.
This
4
have tried
9,
84 by the same quotient
in
product
i.e.,
is
on
:
between
I
(Fig.
add
the two quotient figures, 8
The 8
This
153).
4V9V94 V9^
ABCDEFGHI
J
J
Fig. 154
in
84 and the 8 remaining on
see that they
are
the quotient figure 1 on G.
subtract the product 8
the 4 in
154).
more
I
are
the
So compare the 4 in 84 with the 4 remaining on
and you can
J.
84 and
a quotient figure one
Fig. 153
on
larger than
Next multiply the divisor 84
on F.
difference
ABCDEFGHI
1,
GHL
32 from 124 on
revise the incorrect quotient figure 8 to 9,
^^y^^Y^vy
Step 5
on
12
leave
and subtract the product 84 from the 92 on HI.
leaves 8
same.
will
152).
To
:
to
to the quotient figure 8
by
it
8 in 84 by 8 and subtract
remainder 92
will find that the
(Fig.
Step 4
the
will subtract the
you ought
that
1
Multiplying
you
Then you
9 and have set
151).
:
figure 8,
will multiply the
64 from the 76 on GH.
the product
9 as
to try
84 by the same
also the
Now, multiplying
from the 8 on
1,
same.
I.
J,
Therefore, set
the 8 in
84 by
Next multiplying
subtract the product 4
from the 4
This clears IJ and leaves the quotient 91 on
FG
(Fig.
Division
77
Dividing by Numbers of Over Two-Digits
3.
Example
3,978 + 234=17
9.
^2?W?g
ABCDEFGHI
ABCDEFGHI JK
JK
Fig. 156
Fig. 155
ABCDEFGHI
ABCDEFGHIJK
JK
Fig. 158
Fig. 157
Step 1
with
K
:
Set 3,978 on HIJK,
as the unit rod,
and
set
ABCDEFGHIJK
23400003978
1
ABC
234 on
Step 2
(Fig.
Compare
:
234 with the 3
in
the
figure
1
2
on
2340010
in 3,978.
left
Set the
F,
-
this
quotient figure
-
234 by
1,
and subtract the product 2
on
H
Step 3
1,
I
and
:
This leaves
Step 6
±2_
2 3 8
7
- 2
-
1
2 8
Step 7
Step 8
23400170000
Multiply the 3 in 234 by the same quotient figure
and 167 on HIJ
:
3 8
156).
subtract the product 3
Step 4
1,
(Fig.
Step 5
1 6
1
10
2 3 4
in
1
16 38
18
of the 3
multiply the 2
Step 3
Step 4
- (2 4)
the
Now
from the 3 on H.
3
- 4
8
2 3 4
second rod to the
2
-
2 in
3,978.
goes into 3 one time.
quotient
-
155).
Step 1
Step 2
(Fig.
from 9 on
I.
This leaves 6 on
157).
Multiply the 4 in 234 by the same quotient figure
and subtract the product 4 from 7 on
J.
This leaves 3 on
78 The Japanese Abacus
and 1,638 on HIJK
J
(Fig.
153).
W2W53^W*
ABCDEFGHIJK
ABCDEFGHI JK
Fig. 159
Fig. 160
Step 5
Compare
:
the 2 in
234 with the 16 remaining on
Suppose you have tried 8 as
2 goes into 16 eight times.
HI.
and have
the quotient figure instead of the correct 7
Then you
G.
the product 16
will multiply the
from the 16 on
234 by
2 in
This
HI.
set it
on
and subtract
8,
HI.
clears
Next,
multiplying the 3 in 234 by the same 8, you will find that the
product 24
is
larger than 3 remaining on
have tried a quotient figure one
to
Step 6
:
To
and
that
than 8
less
you ought
(Fig.
159).
revise the incorrect quotient figure 8 to 7, sub-
from the 8 on G, and you get the new quotient
tract 1
7 on G.
Next multiply the 2 in 234 by
between the quotient figures 8 and
Now you
I.
J,
have 2 on
I
7,
1,
and
and 234 on
i. e.,
the difference
product 2 on
set the
UK
~ -
figure
(Fig.
160).
*"
"IF
ABCDEFGHI JK
ABCDEFGHIJK
Fig. 161
Fig. 162
Step 7
7,
on
:
Multiply the 3 in 234 by the new quotient figure
and subtract the product 21 from 23 on
J
and 28 on JK
Step 8
:
This leaves 2
161).
Next multiply the 4 in 234 by the same new quo-
tient figure 7,
This clears
(Fig.
IJ.
and subtract the product 28
JK and
leaves the answer 17
on
om 28 on JK.
FG (Fig. 162).
i
Division
Example 10.
7,061 -=-307
79
= 23
ABCDEFGHI JK
ABCDEFGHI JK
Fig. 163
Fig. 164
A BCDEFGHI JK
wiwwy^f
•
•
•
•_
3070000706
1
2
-
ABCDEFGHI
Step 1
JK
with
set
K
as the
unit
numbers
Step 2
(Fig.
as always
Step 3
and
HIJ.
:
163).
in
307 and the 7 in 7,061 you
Set the quotient figure
from the 7 on H. This
and subtract
this 2,
leaves 1
H
on
(Fig. 164).
Multiply the 7 in 307 by the same quotient figure
setting the product 14
This leaves 92 on
zero, see that
(Fig.
vacant rods between
four
Next multiply the 3 in 307 by
the product 6
Step 5
3070023000
can see that 3 goes into 7 two times.
2 on F.
9
-21
and
rod,
Comparing the 3
:
Step 3
Step 4
-
307 on ABC, leaving
the two
6
-14
3
Set 7,061 on HIJK,
:
Step 1
Step 2
30700200921
Fig. 165
2,
•
you
set
IJ,
subtract
it
from 106 on
Since the second digit in 307
IJ.
the
on
14
product
on
U
instead
of
is
HI
165).
yy*yyyyy?yy~
*£
^ vyyyyyyy
ABCDEFGHI JK
ABCDEFGHI JK
Fig. 166
Fig. 167
Step 4:
The 3
in 307 goes into
the
9
on
I three
times.
80 The Japanese Abacus
•
So
set the quotient figure
307 by
this quotient figure 3
Step 5
Next multiply the
on G.
3
in
and subtract the product 9 from
This leaves 21 on
L
the 9 on
3
3
JK
(Fig.
166).
Multiply the 7 in 307 by the same quotient figure
:
and subtract the product 21 from the 21 on JK. This dears
JK and
leaves the answer
Key
4.
digit in the
(Fig.
167).
the
first
digit in the divisor is 5,
twice the
first
dividend often proves a correct quotient figure.
When
2.
FG
to Finding Correct Quotient Figures
When
1.
23 on
the
first digit
in the divisor is 9, the first digit in
the dividend often proves a correct quotient figure.
When
3.
ond,
it is
ond,
5.
it is
first digit
in the divisor is larger than the sec-
safe to try a small quotient figure.
When
4.
the
the
first digit
in the divisor is smaller than the sec-
safe to try a large quotient figure.
Exercises
Group
I
1.
24-4-2 = 12
2.
36-^3 = 12
3.
115-5-5
= 23
8.
1,495-j-65-23
4.
204-4-6-34
9.
2,451-4-57
5.
357-4-7 = 51
10.
4,293-4-81
Group
= ll
6.
132-?-12
7.
441-^-21-21
= 43
= 53
II
1.
8,298-4-68
= 122
6.
6,342-5-453
2.
4,270-4-14 = 305
7.
9,728-4-304 = 32
= 148
3.
11,100-j-75
4.
7,560-4-28 = 270
5.
24,957-4-47 = 531
= 14
8.
38,920-4-895 = 56
9.
46,113-5-809 = 57
10.
26,460-5-147 = 180
VIL
DECIMALS
mark
In addition and subtraction the unit point serves as the
of a decimal point, and the calculation of decimal problems
quite the
is
as that of whole numbers.
same
However, in multiplication and division, you cannot
and that of the quotient unless
find the unit rod of the product
you know two
easily
rules covering the position of the decimal point
of the product and
two others
covering
the
position
These four rules may be best
decimal point of the quotient.
explained and illustrated in paired counterparts.
of rules applies
to
of the
The
first
pair
whole or mixed-decimal numbers, and the
second to decimal fractions.
Rule A.
When
the multiplier
the unit rod of the product
multiplicand by as
many
is
a whole or a
moves
rods
mixed decimal,
to the right of that of the
one as there are whole
plus
digits in the multiplier.
Rule B.
When
the divisor
is
a whole
or
a
mixed decimal
number, the unit rod of the quotient moves to the
unit rod of the dividend by as
many
left
of the
rods plus one as there are
whole digits in the divisor.
Rule C.
first
significant
the product
digit
When
is
the multiplier
figure
is
in
formed on the
of the multiplicand.
is
a
tens
the
first
decimal
fraction
place, the last digit of
rod to the right of the
Call this the basic rod.
time the value of this multiplier
is
whose
last
Then, each
reduced by one place, the
82 The Japanese Abacus
•
by one rod to the
of the product shifts
last digit
of this
left
basic rod.
Rule D.
When
the divisor
significant figure is in
quotient
is
the
formed on the
of the dividend.
is
a decimal fraction whose
tens
first
the last
place,
rod to the
left
is
of the last digit
Then, each time the
Call this the basic rod.
value of this divisor
first
of the
digit
reduced oy one place, the
last digit
of
the quotient shifts by one rod to the right of this basic rod.
Example
(A)
1.
4x2=8
Unit rod of the multiplicand
Unit rod of the product
.
(A)
8-2=4
(B)
ABCDEF
(B)
ABCDEF
8
8
Unit rod of the quotient
Unit rod of the ^dividend
As seen
the
in the
multiplier
first
—J
diagram above, showing the position of
(Rod A),
multiplicand
the
product (Rod F), when the multiplier
a
is
and
(Rod D),
one-digit
number,
the unit rod of the product moves by two rods to the right
that of the multiplicand.
product
is
the
of
In other words, the last digit of the
formed on the second rod
to
the right
of that
of
the multiplicand.
As seen
in the second
diagram above, showing the position
of the divisor (Rod A), the dividend (Rod F), and the quotient
(Rod D), when the
divisor
is
a one-digit number, the unit rod
of the quotient moves by two rods to the
dividend.
In
other
words,
formed on the second rod
Example
The
first
2.
(A)
the
last
digit
left
of that of the
of the quotient
is
to the left of that of the dividend.
25 x 15 = 375
(B)
375- 15 = 25
diagram below shows that when the
& two-digit number, the last digit of the product
multiplier
is
is
formed on
.
Decimals
•
83
the third rod to the right of that of the multiplicand.
The second diagram below shows
two-digit
number, the
of the
digit
last
when
that
the divisor
quotient
a
is
formed on
is
the third rod to the left of that of the dividend.
!_«
Unit rod of the multiplicand
Unit rod of the product
:
ABCDEFGHI
(A)
o
~nr
ABCDEFGHI
(B)
tit
~
si~k
2 5
3 7 5
Unit rod of the quotient
Unit rod of the dividend
Example
3.
(A) 405 x
123=49,815
(B) 49,815
+ 123-405
Unit rod of the multiplicand
Unit rod of the product
ABCDEFGHIJKL
(A)
12
3
ABCDEFGHIJKL
(A)
405
49815
123
49815
405
Unit rod of the quotient
Unit rod of the dividend
The
first
diagram above shows that when the multiplier
number, the
three-digit
last digit
of the product
is
a
formed on
is
the fourth rod to the right of that of the multiplicand.
The second diagram shows
digit
number, the
that
when
of the
last digit
the divisor
quotient
is
a three-
is
formed on the
fourth rod to the left of that of the dividend.
Note on Example 3 (B)
from the
quotient
In case the dividend
:
is
divisor with four vacant rods, as in this example, the
product
is
distinguishable
clearly
i
r
,
i
from the
i
since two vacant rods are left between them, thus
:
if the
divisor,
ABCDEFGH
—
12
But
separated
3
4
5*
dividend were separated from the divisor with only
three vacant rods, the quotient produced
would be hardly
dis-
tinguishable from the quotient, since only one vacant rod would
i_i/>i
lett between them
be
,
ABCDEFG
i
thus
:
«
VTW0
„
.
4
5
From
this
example
r
—
—
8
84 The Japanese Abacas
•
the reader will see that in case
second figure of the quo-
the
tient is a cipher, the quotient is hardly distinguishable
divisor.
Thus
is
(A)
4.
from the
always preferable to set the divisor on the
of the fourth rod to the
fifth instead
Example
it
of the dividend.
left
34x1.2=40.8
40.8-1.2=34
(B)
Unit rod of the multiplicand
Unit rod of the product
.
(A)
ABCDEFGHIJ
1.2
ABCDEFGHIJ
(B)
1.2
3 4
4
4 0.8
3 4
0.
Unit rod of the quotient
Unit rod of the dividend
Observe the
the multiplier
first
mixed number, the
a
is
diagram above, and you
will find that
whole
last
when
digit of the
product moves to the right of that of the multiplicand by as
many
rods plus one as there are whole digits in the multiplier.
Observe the second diagram,
the divisor
tient
mixed number, the
when
that
find
will
whole
digit of the quo-
to the left of that of the dividend
by as many rods
is
moves
and you
a
last
plus one as there are whole digits in the divisor.
Example 5
32x0.4 = 12.8
(D) 12.8-0.4=32
(C) 98x0.32=31.36
(C)
(D')
31.36-0.32 = 98
Unit rod of the multiplicand
--.
i
Unit rod of the product
;
(C)
ABCDEFGHI
0.4
1
(DO
2.8
ABCDEFGHI JKL
3 1.3 6
9 8
0.3 2
3 2
Unit rod of the dividend
J
S
i
Diagrams C and
J
3 1.3 6
2.8
ABCDEFGHI
0.4
\
ABCDEF6HI
0.3298
32
1
(D)
(CO
I
C
decimal fraction whose
— Unit rod of the quotient
above show that when the
first
significant
figure
is
divisor is a
in the tenth
—
Decimals 85
•
place, the last whole digit of the product is
formed on the
first
rod to the right of that of the multiplicand.
D
Diagrams
and
decimal fraction whose
place, the last
first
rod to the
Example 6
whole
left
show
D', above
first
digit
that
significant
the divisor
figure
of the quotient
the
in
is
a
is
tenth
formed on the
is
of that of the dividend.
(C) 98x0.032 = 3.136
32x0.04=1.28
(C)
when
(D) 1.28-5-0.04
= 32
(D')
3.136-f-0,032=98
Unit rod of the multiplicand
(
Unit rod of the product
•
(C)
ABCDEFCHI
(CO
32
0.04
j
ABCDEFGHI JKLM
98
0.032
3.13
1.2 8
(D)
ABCDEFGHI
0.0 4
(DO
ABCDEFGHI JKLM
3.136
0.0 3 2
1.2 8
6
3 2
9 8
—
Unit rod of the dividend
Unit rod of the quotient
Diagrams C and Of above show
a decimal fraction whose
dredth place, the
first
that
when
the multiplier
significant figure is in the
whole digit of the product
last
the very rod on which that of the multiplicand
Diagrams
D
and D' above show
decimal fraction whose
dredth place, the last
the very rod
Example?.
significant
when
figure
located.
is
the divisor
is
hun-
formed on
is
whole digit of the quotient
on which
(C)
first
that
in
is
the
is
hun-
formed on
32x0.004=0.128 (C) 98x0.0032=0.3136
-T-
0.004
= 32
(D') 0.3136-^-0.0032
= 98
.Unit rod of the multiplicand
;
:-- Unit
rod of the product
!
ABCDEFGHI' J KL
0.004
a
that of the dividend is located.
(D) 0.128
(C)
is
3 2
0.128
(CO
•.
ABCDEFGHI iKLM
0.0032
98
0.3136
86 The Japanese Abacus
-
(D)
ABODEFGHI JKLM
0.0
0.12
4
(DO
ABCDEFGHI JKLMN
0.0
8
0.3
9 8
3 2
3 2
1—4— Unit rod of the dividend
C
j
the
first
last
rod to the
-'
above show that when the multiplier
a decimal fraction whose
sandth place, the
6
'
•—-Unit rod of the quotient
Diagrams C and
13
first
whole
left
significant figure is in the thou-
digit of the product is
of the
is
last
whole
digit
formed on
of the multi-
plicand.
Diagrams
D
fraction whose
and D' show
first
that
when
significant figure
the last whole digit of the quotient
is
is
the divisor
is
a decimal
in the thousandth place,
formed on the
to the right of the last whole digit of the dividend.
first
rod
MENTAL CALCULATION
VIII.
abacus
All
experts
On an
rapidity.
calculation
can
they
average
are
on the abacus.
as
mentally with miraculous
calculate
It
twice
is
quick in mental
as
for
possible
anyone
by proper
attain astonishing rapidity in such mental calculation
The
practice.
secret
in
lies
applying
abacus
to
calculation
to
mental arithmetic by visualizing abacus manipulation.
Here are the
vital points
:
For example, in adding 24 to 76, close your eyes and
1.
visualize the beads of
an abacus
24 onto the beads, aiding your
flicking the index finger
really
digits
adding a
series of
At
first,
days
as if
numbers, say, 24 + 76 + 62
+ 50,
which come up
Remember,
is
of your
practise the addition of
to a
222 + 555 + 223, and the
4.
of the abacus by
hand
memory by folding one
sum has come up to 100.
3.
visualization
and thumb of your
aid your
the
Then mentally add
right
calculating on an abacus.
When
2.
set to 76.
left fingers
each time
numbers of two or more
round sum, for example, 76 + 24, or
like.
practising a few minutes at a time for
many
worth more than practising hours on a single day.
EXERCISES
Constant daily practice
ficient
in
use
the
of
is
the
one
essential if
to
is
become pro-
The following
abacus.
exercises,
prepared and arranged in accordance with the most up-to-date
methods, have been kindly furnished by Professor Miyokichi
They
Ban, an outstanding abacus authority.
good beginning
the
for
more problems
in
who can then
student,
serious
provide
will
Also
any ordinary arithmetic book.
and
that problems in multiplication
find
note
may be used
division
a
as
problems in addition and subtraction respectively.
The
exercises are arranged so that a student can
measure
his
progress against the yardstick of the Japanese licensing system,
the required standard of proficiency
being given at the beginning
of a
first,
each
of
second or third grade
Abacus Committee,
is
the
for
possessor
awarded by the
as
for
qualified
officially
The
group.
license,
grade
particular
employment
in
Licenses for the lower
a public corporation or business house.
grades are given on the basis of unofficial
examinations
con-
ducted by numerous private abacus schools.
The
variety
exercises
to
receiving
the
equal
are
also
problems,
attention
chosen
with
—an
provement in abacus operation.
just initiated
to
each
give
digit
essential
the
maximum
from zero
to
of
nine
requirement for im-
The system followed
is
that
by the Central Abacus Committee after long and
careful research.
Exercises
/
•
Sixth Grade Operator
Group
A
(1 set per minute, or entire
group with 70 o/Q
accuracy in 10 minutes.)
(2)
(3)
(4)
(5)
528
967
482
106
815
160
239
251
543
302
427
650
147
928
491
(1)
951
108
598
710
719
243
120
954
452
758
329
852
106
491
-643
-839
267
-169
-401
-958
843
536
304
695
576
385
702
987
514
740
690
871
439
870
183
308
-235
-673
.
724
460
381
629
203
984
-671
-305
-526
632
340
579
315
760
807
927
634
870
215
481
264
7,782
8,823
1,319
8,940
3,054
(8)
796
(6)
(7)
(9)
(10)
360
769
241
654
185
829
420
952
516
730
213
195
309
740
698
308
-513
-854
756
495
246
487
536
809
953
497
932
508
360
589
274
617
690
421
508
-785
-320
-197
147
963
873
201
617
674
-307
-631
-175
420
873
164
196
124
902
689
596
765
286
204
-482
-968
480
148
840
795
319
342
756
392
138
203
875
7,959
2,588
7,545
1,909
8,688
850
201
721
370
89
90 The Japanese Abacus
Group
(1
set per
B
m
minnte, or entire group with
70^
accuracy in 10 minutes.)
(4)
(5)
(1)
(2)
(3)
728
36
627
52
271
631
87
75
713
6,104
4,089
52
238
60
42
50
705
-94
42
35
487
92
68
96
-426
-81
8,096
175
621
93
47
349
7,513
97
312
904
2,138
59
481
-78
-5,083
72
510
60
305
593
74
-3,041
574
20
61
421
-52
85
961
86
907
139
19
8,260
53
807
73
354
619
40
9,038
-854
-69
-705
13
8,042
968
264
96
16,155
14,157
6,832
20,520
1,632
(6)
(?)
(8)
(9)
(10)
936
3,094
806
4,105
624
54
78
21
65
51
-716
-81
29
6,084
413
712
817
13
785
439
67
-18
-745
41
5,021
834
-50
71
2,095
48
203
62
938
36
27
40
329
26
984
-78
526
-3,605
81
-1,203
-459
7,041
63
542
-953
9,036
48
259
807
130
17
76
95
1,068
69
395
597
508
70
92
58
30
672
92
247
649
817
40
7,650
1,500
16,830
6,525
14,148
379
23
85
Exercises
Group
C
Group
(70o/o accuracy, 5 minutes.)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
187x53=
245x21 =
309x19 =
408x38 =
561x60 =
620x42 =
716x90 =
832x57 =
954x74 =
973x86 =
Group
5,145
(2)
5,871
(3)
(4)
(5)
2,698x24 =
3,980x30 =
4,509x65 =
5,062x73 =
6,874x68 =
7,431x80 =
8,146x12 =
9,357x49 =
8,230x97 =
33,660
26,040
70,596
(6)
(7)
(8)
(9)
83,678
(10)
64,440
47,424
E
Group
(10o/o accuracy, 5 minutes.)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(7096 accuracy; 5 minutes.)
1,725x51 =
15,504
960-4-24 = 40
810-4-45
= 18
= 95
6,640-4-80 = 83
7,505-5-79
5,920-5-16-37
4,080-^-68 = 60
3,127-5-53
= 59
2,160^-30 = 72
91
D
(1)
9,911
•
87,975
64,752
119,400
293,085
369,526
467,432
594,480
97,752
458,493
798,310
F
(10o/o accuracy, 5 minute
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
1,152-4-72 = 16
(9)
2,1844-91=24
(10)
¥986-4-34= ¥29
¥855-4-19 =
¥45
¥7,280-5-80= ¥91
¥6,240-4-78= ¥80
¥76
¥4,128-^-96 = ¥43
¥390-5-13= ¥30
¥2,320-5-40= ¥58
¥1,550-5-25= ¥62
¥884-5-52= ¥17
¥5,092-5-67=
92 The Japanese Abacus
•
//.
Fifth Grade Operator
Group
(10%
A
accuracy, 10 minutes.)
(1)
(2)
(3)
(4)
425
619
502
7,245
167
839
153
698
461
9,035
5,302
762
2,013
956
716
791
8,523
147
179
3,540
514
-478
-694
9,684
317
295
1,283
5,726
420
138
960
-7,081
409
8,096
869
2,048
377
971
5,609
(5)
683
1,049
3,056
-543
-835
4,067
812
843
2,684
952
794
9,235
329
708
4,786
327
3,176
-504
760
1,032
128
952
-6,380
4,215
8,073
805
426
837
671
905
158
-319
-903
-6,527
23,310
7,724
30,348
12,966
35,285
(6)
(7)
(8)
895
3,471
912
7,043
584
463
(9)
409
241
(10)
237
498
4,093
6,3.09
509
628
9,324
308
146
2,918
309
647
7,410
285
451
5,672
1,509
5,037
734
491
854
1,086
6,082
427
-236
-7,018
-984
-2,536
-841
165
965
219
308
4,763
978
8,074
867
872
517
240
129
421
164
902
2,086
715
1,653
6,217
8,059
731
-5,268
820
9,620
3,108
-607
9,046
597
259
714
35
-126
-340
-895
375
136
935
31,842
16,781
28,071
12,090
31,230
563
708
3,572
Exercises
Group
(70%
(1)
(2)
B
accuracy, 10 minutes.)
(3)
(4)
.
(5)
3,128
504
196
837
6,059
940
6,142
2,501
7,590
3,240
8,437
879
385
2,601
918
1,056
-5,263
6,210
9,082
136
9,582
-198
1,479
•2,847
365
7,920
8,937
5,297
691
7,068
-649
-3,078
-429
7,809
4,702
927
254
9,087
4,253
2,375
6,815
718
3,960
6,042
4,301
-1,536
-8,473
-348
843
5,914
3,159
1,649
2,057
3,521
4,726
5,341
52,116
26,939
60,117
34,909
53,970
(8)
(9)
(10)
163
(6)
463
0)
9,325
1,706
695
8,563
930
8,014
5,406
-358
4,587
8,045
273
712
-1,876
-6,145
7,208
9,634
7,201
462
592
958
7,342
8,594
149
3,180
1,048
403
3,027
4,807
453
9,731
1,780
-7,380
-2,759
-902
758
9,508
5,109
6,314
3,146
4,261
8,925
509
6,291
937
-2,346
7,921
6,178
530
8,679
-697
2,078
4,813
2,085
518
7,163
8,637
632
5,239
6,024
3,508
981
9,264
7,621
2,187
4,386
3,146
850
1,375
9,840
-6,029
4,065
8,254
217
1,096
8,407
5,706
-275
-5,709
-641
4,362
695
-5,314
672
9,024
1,039
430
5,890
613
55^365
726
852
9,763
26 703
61,005
8,175
25,701
61,095
93
94 The Japanese Abacus
-
Group
(10%
(1)
$
359
accuracy, 10 minutes.)
(2)
(3)
9,535
$
C
$
604
(4)
$
(5)
413
$ 2,190
647
7,569
174
2,895
390
408
812
731
706
574
163
3,720
1,048
6,054
1,697
914
269
318
481
792
-647
-1,093
817
8,249
156
5,021
356
4,580
3,078
8,630
2,680
932
325
906
126
-634
-267
-9835
423
310
206
185
658
142
962
127
-263
-8,472
-598
9,053
5,061
835
316
975
5,289
470
-7,529
206
6,057
704
3,749
-807
4,058
874
419
527
182
739
36,774
8,418
26,775
3,418
21,645
4,381
948
(6)
$ 4,157
823
(7)
$
516
(8)
$
8,594
(9)
$
740
(10)
$ 1,507
7,082
250
6,294
960
496
395
913
103
8,023
273
279
649
153
-341
-7,082
857
5,082
5,019
752
836
761
6,042
361
431
-845
-3,978
-109
821
2,473
520
6,395
3,150
749
648
548
264
605
3,976
125
634
4,987
128
837
409
2,014
730
9,086
2,719
287
361
674
301
9,168
4,658
314
285
7,431
-580
968
-6,792
401
-597
-1,632
-805
905
5,893
467
961
270
3,202
32,355
13,169
28,349
37,674
Exercises
Group
(70o/o
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
accuracy, 5 minutes.)
942x495 ==466,290
839x457 ==383,423
723x980 ==708,540
680x134 == 91,120
508x268 ==136,144
417x873 ==364,041
396x629 ==249,084
204x316 = 64,464
165x501 = = 82,665
751x702 ==527,202
Group
(70%
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
D
Group
(70^
9,724^-26 = 374
8,151-^13 = 627
= 109
62,560-^80 = 782
5,556^-12 = 463
49,572-f-54 = 918
30,150-f-67 = 450
23,128^-98 = 236
17,535-^-35 = 501
43,855-^-49 = 895
7,739^-71
E
accuracy, 5 minutes.)
348x276= 96,048
(2) 854x965 = 824,110
(3) 902x804 = 725,208
(4) 627x108= 67,716
(5) 105x519= 54,495
(6) 570x843 = 480,510
(1)
(7)
(8)
(9)
(10)
F
accuracy, 5 minutes.)
93
489x751 = 367,239
613x397 = 243,361
236x632 = 149,152
791x420 = 332,220
Group
G
(10o/o accuracy, 5 minutes.)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
630,207-f-90 = 7,023
533,484^-87 = 6,132
420,616^-74 = 5,684
113, 148-f-63
= 1,796
278,772-^52 = 5,361
85,075-^41 = 2,075
366,873^-39 = 9,407
98,504^-28 = 3,518
72,885-^15 = 4,859
214,240-i-26 = 8,240
93 The Japanese Abacus
•
///.
Fourth Grade Operator
Group
A
(70o/o accuracy, 10 minutes.)
(1)
(2)
(3)
(4)
(5)
3,458
9,526
1,459
9,753
2,983
4,198
3,146
3,670
4,120
8,973
4,723
9,265
-1,256
6,309
7,269
2,368
4,537
-5,904
5,092
9,085
6,912
5,084
8,329
2,148
5,279
8,762
2,048
5,871
7,190
6,827
1,397
-6,450
-8,317
-3,802
3,856
7,539
1,236
3,047
2,710
7,084
8,102
8,152
-7,895
-3,047
-2,739
6,087
6,735
5,680
2,409
6,180
8,264
-4,391
-5,140
5,091
6,470
6,374
4,561
5,021
7,913
4,280
9,834
6,371
5,912
1,948
4,106
9,561
2,604
9,603
1,438
3,406
5,172
7,538
90,765
31,983
73,620
33,782
80,370
(6)
(7)
4,675
2,905
7,816
(8)
(9)
(10)
6,753
7,201
2,951
5,482
8,954
6,759
9,160
8,035
4,710
1,832
1,093
3,294
4,973
3,986
8,094
2,346
8,643
5,106
5,603
2,869
8,712
3,683
9,215
7,150
1,290
5,087
3,908
3,874
-4,712
-1,035
-5,368
9,624
1,693
9,201
4,158
7,214
2,067
6,854
4,127
6,507
8,527
6,870
6,541
2,401
9,382
4,096
5,907
3,712
5,240
3,648
-7,582
-4,326
6,154
-6,049
-8,731
-3,278
6,395
7,564
5,930
7,298
8,476
2,489
8,059
4,870
7,142
1,360
9,781
7,951
9,320
5,013
9,125
6,403
8,309
7,562
4,671
75,825
45,259
85, 095
54,469
90,675
Exercises
Group
(10%
*
B
accuracy, 10 minutes.)
(5)
(1)
(2)
(3)
(4)
2,453
5,906
3,629
8,431
5,192
98,710
41,568
7,846
1,029
20,753
85,934
-
9,504
67,941
4,825
9,205
36,029
5,174
80,453
4,501
2,318
1,683
9,631
59,327
4,308
3,712
-2,687
-31,405
8,574
20,893
8,716
3,086
-7,840
-43,126
-2,735
3,129
8,074
4,932
64,279
3,052
48,531
2,689
9,607
60,243
6,912
17,269
73,215
87,069
-17,043
-5,890
-9,134
1,058
4,067
2,769
8,573
7,698
2,750
1,574
4,716
90,635
10,896
6,458
76,352
5,804
-54,618
-6,481
7,208
53,961
2,879
8,140
70,396
9,645
2,307
5,021
31,972
4,125
285,621
871,226
171,312
323,211
97,007
(6)
CO
(8)
(9)
(10)
6,712
72,934
1,846
4,165
8,679
3,950
1,653
30,782
1,594
15,032
4,167
3,817
4,935
3,872
6,985
60,843
9,051
8,071
51,943
2,803
5,679
1,329
6,329
92,381
41,697
1,703
95,874
-6,705
-4,372
-87,156
9,306
8,634
-53,682
-8,306
-2,148
4,068
39,581
80,597
78,491
9,034
37,459
7,824
5,431
40,982
13,507
2,516
2,307
14,970
2,417
5,420
70,128
40,592
-7,026
-25,469
7,650
7,263
8,749
91,486
63,208
3,290
60,175
286,055
29,078
3,105
5,763
6,251
4,205
9,016
-20,489
-8,016
8,095
6,748
2,569
2,698
816,903
104,104
361,038
65,139
5,821
9,347
9%
93 The Japanese Abacus
•
Group
(10
(1)
%
C
accuracy, 10 minutes.)
(2)
(4)
(3)
(5)
% 2,701
342
$ 58,976
4,109
917
50,746
601
60,179
249
4,586
7,340
45,981
$
378
$
683
$ 8,472
9,630
80,651
1,469
-1,597
-265
71,524
1,590
70,354
619
790
8,495
3,042
7,301
13,267
50,723
-468
-5,107
793
25,478
6,154
268
-67,328
2,138
9,032
24,896
956
782
96,205
-359
974
814
3,245
7,621
32,058
5,839
-40,168
-8,274
50,789
9,523
613
5,209
92,361
43,167
873
19,057
63,921
1,083
70,934
415
34,580
548
80,246
904
862
86,937
-3,012
-654
68,120
2,817
1,035
342,576
237,201
361,180
145,418
203,355
(6)
$ 4,583
(7)
(8)
$ 839
(9)
(10)
$ 13,597
$ 9,218
13,659
$ 75,219
327
523
763
40,137
732
139
19,768
27,458
572
403
-6,875
-906
5,017
89,162
1,065
29,581
240
430
56,387
690
50,864
6,754
2,049
941
32,071
8,542
30,482
-98,625
259
947
41,038
1,937
8,765
684
-506
-4,317
32,596
93,521
5,231
483
95,820
-751
-60,218
-2,496
8,075
70,318
60,725
7,415
9,547
42,816
178
37,081
7,109
-2,679
-985
24,139
80,264
320
24,365
6,140
8,074
6,304
282,708
4,693
156,524
690
242,820
804
108,840
1,690
244,458
1,296
7,308
864
Exercises
Group
D
(70 <?£ accuracy, 5 minutes.)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
92,854x84 = 7,799,736
86,213x59 = 5,086,567
73,041x90 = 6,573,690
60,378x16 = 966,048
51,762x27 = 1,397,574
47,609x70 = 3,332,630
30,427x32 = 973,664
29,185x63 = 1,838,655
18,596x45 = 836,820
54,930x81 = 4,449,330
Group
(70o/,
Group
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
F
(70^
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
9,108x379 = 3,451,932
8,240x568 = 4,680,320
7,894x740 = 5,841,560
6,372x953 = 6,072,516
5,423x182 = 986,986
4,617x194 = 895,698
3,581x807 = 2,889,867
2,056x625 = 1,285,000
1,905x401 = 763,905
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
3,769x236 = 889,484
(10)
E
1,375x562 = 772,750
2,610x148 = 386,280
3,784x625 = 2,365,000
4,208x201 = 845,808
5,429x874 = 4,744,946
6,057x903 = 5,469,471
7,906x417 = 3,296,802
8,591x730 = 6,271,430
9,832x986 = 9,694,352
4,163x359 = 1,494,517
Group
accuracy, 5 minutes.)
99
accuracy, 5 minutes.)
(10o/o
(10)
•
G
accuracy, 5 minutest
2,647x3,740 = 9,899,780
3,068x2,698 = 8,277,464
9,854x7,219 = 71,136,026
1,370x4,805 = 6,582,850
8,401x6,457 = 54,245,257
4,936x9,523 = 47,005,528
6,125x5,184 = 31,752,000
2,519x8,306 = 20,922,814
7,093x1,962 = 13,916,466
5,782x3,071 = 17,756,522
100 The Japanese Abacus
•
Group
(10%
H
accuracy, 5 minutes.)
379,428+42: =9,034
706,860-^85: =8,316
Group 1
(10o/o accuracy, 5 minutes)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
406,164^-68: =5,973
94,240-^-76 =1,240
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
325,134-i-54: =6,021
(10)
235,662^-31: =7,602
658,145^-97: =6,785
87,362^-19: =4,598
115,710-^-30: =3,857
55,637^-23: =2,419
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
accuracy, 5 minutes.)
636,768^-792 = 804
35,984-*-208 = 173
194,394+537 = 362
403,425-*-815 = 495
86,163-*-373 = 231
531,34O+-620 = 857
932,832-*-984 = 948
286,090-*-469 = 610
363,726-*-501 = 726
74,314-5-146 = 509
(10%
(1)
(2)
(3)
(4)
= 235
87,040-*- 256 = 340
752,128-*- 832 = 904
64,220-*- 380 = 169
548,784-*- 927 = 592
431,748
603 = 716
385,746
478 = 807
219,537-*- 519 = 423
107,912-* 164 = 658
620,895-* 795 = 781
Group
Group J
(10%
94,235-*- 401
K
accuracy, 5 minutes.)
553,149-*-6,829 = 81
273,375-*-3,645 = 75
97,351-4-1,453 = 67
761,664^-7,934 = 96
81,018^-4,501 = 18
(5)
(6)
(7)
(9)
(9)
424,901-*-8,017 = 53
(10)
394,044-5-9,382=42
84,942-*-2,178 = 39
120,048-*-5,002 = 24
611,640-^6,796 = 90
Exercises
IV.
101
•
Third Grade Operators
Group
A
(70 0/0 accuracy, 5 minutes.)
(1)
(2)
(4)
(3)
(5)
$ 41,306
$ 28,640
835
7,962
135
86,029
$ 8,127
$ 526
$ 105,942
659
4,192
17,492
60,271
94,516
95,641
961,037
358,604
62,481
529
401,286
5,208
-963
83,672
890,375
514
€38,125
1,450
238,107
-6,813
-380,276
37,269
80,734
-71,850
-5,397
9,270
409,715
396
784
903,851
25,816
842
740,138
978,250
18,394
401,369
-17,438
-732,609
253
3,795
549,076
57,048
12,047
3,702
90,386
6,729
604,518
153
756
36,594
2,478
520,943
6,127
78,915
-421
695,718
481
28,459
406,329
4,367
74,308
813,045
9,067
-53,608
-29,134
70,925
2,780,919
943,910
1,885,878
2,164,955
2,802,537
*
(6)
$ 350,624
(7)
79,328
(9)
(8)
952
(10)
$ 14,538
93,041
8,653
$ 60,382
847
38,207
751
-68,729
184,705
2,415
706,394
26,374
-134
31,894
531
2,460
652,096
791,560
936
891,270
875
8,725
45,287
507,269
63,092
410,936
479,163
2,759
9,283
680
$
$
421
-726
69,641
60,584
65,148
13,849
-6,358
-804,293
-15,807
245,139
-308,619
-15,974
931,054
804,975
372
704,368
729
5,812
4,715
54,071
30,427
983
693,807
-3,051
-57,249
13,567
209
7,410
16,042
425,698
6,182
90,412
362
2,610
9,703
207,648
687,936
201,876
80,597
56,184
89,513
3,160
658,222
1,917,072
1,828,871
2,537,019
2,719,107
103 The Japanese Abacus
•
Group
(J0%
accuracy, 10 minutes.
to the nearest thousandth
;
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
14,902x52 = 774,904
7,105x0.098 = 696.29
9,674x603 = 5,833,422
63.25x7.64 = 483.23
853x4.017 = 3,426,501
0.3081x0.926 = 0.285
2,984x351 = 1,047,384
(10)
0.2176x87.5 = 19.04
(11)
(12)
(13)
(14)
(15)
(16)
to the nearest thousandth;
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
594x376 = $975,344
$4,608x0. 189 =£871
$7, 832 x 897 = $7, 025, 304
$94, 120x6. 4 = $602, 368
$8 029 x 738 = $5 925 402
$975x45. 12 = $43, 992
$2,
,
,
,
(19)
(20)
$6, 549x643 = $4,211, 007
(18)
accuracy, 10 minutes.
to the nearest dollar.)
|5,176x0.625 =$3,236
$3, 061x903 = $2, 764, 083
$6, 843x201 = $1,375, 443
(17)
Group
1—10
Calculate problems
11-20
4,097x238 = 975,086
5,638x149 = 840,062
(J0%
B
C
Calculate problems
11-20
1-10
to the nearest dollar.)
937,015-5-965 = 971
(11)
$9 9-*- 0.368 = $269
0.08988-5-6. 42 = 0.014
(12)
$83, 619-5-27
0.070654-M). 136 = 0.520
(13)
$71,967-5-149
= 89.5
=
55.426-5-214 0.259
63,366-5-708
= $3, 097
= $483
(14)
$649,612-5-7,061
=$92
= $640
(15)
$560-5-0 875
415,473-^-591
(16)
$415, 693-5-593 =$701
(17)
$33, 154-5-6 0.5 = $548
280,932-5-82 = 3,426
(18)
$2, 485-5-2. 84
= 468
(19)
$122, 536-4-901
(20)
$54-*-0.432 = $125
= 703
315.333-5-45.9 = 6.87
17.316-5-0.037
(10) 241,893-5-7,803 = 31
.
= $875
=$136
SBN
8048 0278-5
ABA/NACS: MATHEMATICS
THE JAPANESE ABACUS
Its Use & Theory
SIMPLE and primitive though
it
seem, the Japanese
abacus is capable of amazing speed and accuracy.
In numerous
tests it has outclassed the best electric
computers of the Western world. Here for the first
time in English
is
both an explanation of the mystery
and a complete book of instruction for the use of
this amazing pocket calculator.
clarity of Mr. Kojima's text
you
The simplicity and
now make it
possible for
to learn the basic principles in a single reading -
much more
easily than
you could
learn, say, the slide
rule- and with practice of the graded exercises
come an abacus expert in your own right.
be-