The Chinese Abacus

1
The Chinese Abacus
Number Sense
Cheryl Ooten, Math Professor Emeritus Santa Ana
College cheryl.ooten_at_yahoo.com
2
Overview

• ?A little bit about China
• ? Abacus development worldwide
• ? Back to China how to use the Suan Pan

3
China U.S.
? 4,000 year history ? 4 times as many
people ?Same size
Photos Linda Koluvek
4
Middle Kingdomindependent interdependent in
intellectual growth
World map
5
Mathematical Development

• ? 5000 BCEEvidence in Mesopotamia Similar
  process in China India
• ? Contact between China India
• ? Contact between China West

6
Why dont we know more about early Chinese
mathematics?

• ? Manuscripts on bark bamboo decayed.
• ? Books were rarely preserved from generation to
  generation.
• ? All books from earlier periods burned by order
  of emperor Shi Huang in Qin Dynasty, 220 BCE
• 
  Ref Berlinghoff

7
Early Chinese Recording Devices

• Bones/Tortoise Shells Knotted
• 16th-11th century BCE Strings

Picture of Columbian quipu Bergamini, p. 13
8
What do we know about Chinese mathematics?

• Scraps of writing survived orally or by copying.
• Ten manuscripts were collected
• ? Called Ten Computational
• Canons
• ? Used as study material for
  
• imperial exams
• Ref Hemenway

9
Confucian Order

• Upward mobility
• possible
• by passing exams to serve the emperor
• Ref Hemenway

Picture of ConfuciusHemenway, p. 117
10
Six Arts of Confucius

• Ritual
• Music
• History
• Archery
• Drivers Training (chariots)
• Math
• Ref Chu Hemenway

Picture of ConfuciusHemenway, p. 117
11
What math was needed to serve the emperor
empire?

• Practical Applications for
• construction
• calendar making
• accounting
• farming issues
• division of resources

12

• Contents of Jiuzhang suanshu
• Nine Chapters on Mathematical Art
• Field Mensuration
• Percentages Proportion (Exchange)
• Decreasing Shares (Interest)
• Decrease (length) to Benefit of Width
• Estimation of works (Civil Engineering)
• Fair Distribution of Goods (Taxes)
• Too Much Not Enough (Estimating)
• Rectangular Arrays (Linear Systems)
• Base-Leg Ref Martzloff Mikami

13

• Jiuzhang suanshu
• Nine Chapters on Mathematical Art
• ?Predated Qin dynasty burning manuscripts. Scraps
  remained.
• ?Earliest written work in the world
  systematically discussing manipulation of
  fractions
• (Ref Calinger)
• ?Describes extracting square cube roots using
  calculating pieces

14
Development of Abacus Worldwide

• What is an abacus?
• Generic for a computational aid to record
  mathematical work done in a persons head

Ref Tom
15
Derivation of word abacus

• Greekabax or abakon
• means table or tablet
• Hebrewavaq means dust
• Early abaci were tables or tablets with dust or
  sand for keeping track of computations.
• 
  

Ref Ryerson Hemenway
16
Early Roman Counter Abaci
Counters shifted column to column
Picture of original simplified Roman counter
abacus Ifrah, p. 107
Ref Ifrah
17
Oldest Counting Board4th century BCE
Picture of Salamis tablet from 300 B.C. Boyer,
p. 200

• 4th century BCE, White marble
• Greek numerical symbols
• Found 1846 Greek Island Salamis
• Now National Museum in Athens
• Ref Dilson Boyer Ifrah

18
Early Roman Seal
R.H. abacus on table L.H. writing tablet Ref
Hemenway
Picture Hemenway, p. 75
19
1st Century BCE Sculpture
Picture of marble sculpture Hemenway, p. 75
Slave calculates bequests of dying man.
Ref Hemenway
20
Aztec Vase from Guatamala
Picture of Aztec vase Hemenway, p. 75
Calculating tax collected as cacao beans
Ref Hemenway
21
Inca Abacus12th-16th centuries

• In Inca Empire of Peru,
• Grand Treasurer used an abacus with maize kernels
  to calculate accounts, transposing results to
  quipu (wool cords on rope w knots recording
  data).
• (

Picture Bergamini, p. 13
Ref Bergamini
22
Russian Schoty(Ref Martzloff)
Picture of schoty Martzloff, p. 214
23
Chinese Abacus (Suan Pan)

• Modern Form2/5

Suan Pan means counting tray
Ref Tom
24
Japanese Abacus (Soroban)

• ? Is 1/4
• ? 1 bead on top, 4 on bottom
• ? Also used in Korea

Picture of soroban Gullberg, p. 170
Ref Gullberg
25
Lee Kai-chen abacus, 1958

• Soroban on top
• Suan Pan on bottom
• Ref Ryerson

26
Abacus Derivative20th century
Picture from Bergamini, p. 20

• School children in Ohio w Abacounters
• 
  Ref
  Bergamini

27
Common U.S. School Abacus
28
How did the modern abacus develop?

• Begin w counting boards sprinkled w sand
• Add lines pebbles.
• Add x to mark thousands .
• Add pebble halfway for 50.

Picture Dilson, p. 36
Ref Hemenway Dilson
29

• Turn sideways
• it begins to look like
• the modern Chinese abacus

Same picture as last slide Dilson, p. 36
Ref Dilson
30
Add a frame, wires, beads

• The Chinese improved it
• Added crossbar.
• 2 beads in heaven. (Each 5 x lower bead!)
• 5 beads in earth.

Ref Dilson
31
BACK TO CHINA
Photos Linda Koluvek
32
Chinese used Calculating Pieces or Counting Rods
(6th century BCE)
? Bamboo, ivory, or iron rods ? Red for and
Black for - ? No zero (just empty space) ? The
4 in 40 and 400 would look different.
Ref Mikami Boyer
33
Calculating Pieces orCounting Rods or Sangis
(Nelson, p 49)
Ref Nelson
34
16th Century
Earliest diagram in China of using a counting
board
Picture of frontispiece Hemenway, p. 65
Ref Hemenway
35
Solved Linear Systems
System of linear Equations Ifrah, p. 120
36
Explanation of Suan Pan in 1593 Manuscript Suan
Fa Thung Tsung
Picture of early printed picture of
abacus Boyer, p. 201
Ref Ifrah
37
Chinese Merchant Using Suan Pan
Picture Ifrah, p. 122
Ref Ifrah
38
Importance of 5

• ? We have 5 fingers on a hand often think of 6,
  7, 8, 9 as 51, 52, 53, 54
• ? Roman Numerals V, VI, VII, VIII
• ? Chinese Calculating Pieces
• ? Chinese ( Japanese) Abacus

39
Compare Chinese Rod Numerals Bead Columns of
Suan Pan
Ref Martzloff
40

• Europeans didnt keep abacus after 1500 but
  Chinese did.
• Why did Chinese keep using it?
• Calculation was difficult w traditional Chinese
  numerals.

Picture of traditional national
numeral Gullberg, p. 44
Ref Dilson Angel
41

• ? Chinese children learn early.
• ? Special classes help mental math preserve
  culture.
• ? Contests help motivate practice.
• ? Licenses issued to experts in China Japan
• (Dilson, 1968)

Picture of Chinese girl with Suan pan Wu, p. 2
Ref Wu
42
Speedy?

• Soroban (Matsusaki) vs Electrical Calculator
  (Wood) late 1940s
• Chinese banking student at Columbia on abacus
  beat computer late 1940s
• Chinese prof Lee Kai-chen won in 1959 in Seattle
  against computer.
• Kai-chen won again in New York.
• Later, abacus won in contests in Taipei.
• 
  Ref Dilson

43
Advantages of Abacus?

• ?Cheap.
• ?Easy to learn fun.
• ?Portable.
• ?Doesnt need electricity or battery.
• ?Fast if you practice.
• ?Can calculate in the air

44
Advantages for Number Sense?

• ? Provides concrete model
• ? Helps students understand carrying
  borrowing.
• ? Improves mental math
• ? Primes working with negatives
• ? Helps students understand base 10 number system
  better.

45
Disadvantages?

• ? Beads are easily disturbed erase work.
• ? No record of previous work.
• ? Must repeat calculation to check.
• ? Cannot be used for higher math.

46
What will a Suan Pan do?

• ? Add, subtract, multiply, divide.
• ? Take square roots cube roots.
• ? Calculate in other bases.

47
Chinese Abacus

• Most 9 or 13 columns
• Some 11, 17, 21, 23, or 31 columns
• Need more columns? Put second abacus left of one
  that you are using. (Green)

Ref Green
48
How does the Suan Pan work?

• Columns from right are 1s, 10s, 100s,
• Top beads are worth 5 x bottom beads.

49

• Lay abacus flat steady it with left hand.
• Begin with all heaven beads up and earth beads
  downagainst the outside frame.
• Use thumb forefinger to move beads.

What number?
Picture of abacus hand Dilson, p. 44
Ref Dilson
50
Lets practice.

• (Be curious about how the Suan Pan could
  increase number sense for U.S. grade school
  children if it was available from grade one.)

51
Practice reading these numbers

• 1

52
Practice reading these numbers

• 2

53
Practice reading these numbers

• 3

54
Practice reading these numbers

• 4

55
Register these numbers

• 1 4
• 2 8
• 3 19
• 4 25
• 5 24
• 6 542,705
• 7 306,503

56
How do you add?

• Level I-
• 1 213 235
• 2 111 231

57
How do you add when not enough beads below the
bar?

• Level II-
• 1 425 132
• 3 3823 6145

58
How do you add when not enough beads in column?

• Level III-
• 1 2346 870
• 3 26,478 92,054

59
How do you add when column next column dont
have enough beads?

• Level IV-
• 1 999 5
• 4 99,997 4

60
How do you subtract?

• Level I-
• 1 984 522
• 4 64,752-13,250

61
How do you subtract when not enough beads in
column?

• Level II-
• 1 73 - 4
• 5 263 76
• 9 64,532 - 28,714

62
How do you subtract when not enough beads in next
column either?

• Level III-
• 1 100 - 47
• 4 100,001 - 284

63
Bibliography-- are my favorites

• Angel, A., Porter, S. (2001). A survey of
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• Bergamini, D. (1980). Mathematics. Alexandria,
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• Berlinghoff, W. P. Gouvea, F. Q. (2004). Math
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• Boyer, C. B. Merzbach, U. C. (1991). A history
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• Calinger, R., editor. (1995). Classics of
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• Chu, D. (1973). China. New York Scholastic Book
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64
Bibliography

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65
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