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About the author
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Advanced Techniques
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I Abacuses in General
I: Counting Rods
I: Counting Rods vs. Medieval European Abacus
I: Fixed beads abacus
I: Free counters abacus or table abacus
I: Origin of the Abacus
I: Subitizing
I: The Eastern Abacus
I: What is an Abacus?
II Close-up of the Eastern Abacus
II: Active and inactive beads
II: Description of the eastern abacus
II: Using your fingers to activate/deactivate beads
III Addition and Subtraction
III: Addition and subtraction with counting rods
III: Addition and Subtraction with Other Abacuses
III: Addition and subtraction with the Russian abacus
III: Addition and subtraction with the school abacus
III: And now... the Practice
III: Examples
III: External Resources
III: Introduction to Addition and Subtraction
III: Monetary simile
III: Multi-Digit Addition and Subtraction
III: One-Digit Addition and Subtraction
III: Order of operations
III: Preliminary examples
III: Probable origen de las reglas de adición y sustracción
III: Rules to use
III: Sakidama, atodama and multiple carries
III: Two pieces of advice
III: Types of one-digit addition and subtraction
III: Ways to practice addition and subtraction
III: What you need to know
III: Working from left to right
Introduction
IV Modern Multiplication
IV: Embedded Zeros
IV: External Resources
IV: Further reading
IV: Introduction to multiplication on the abacus
IV: Multi-digit multiplication
IV: Multiplicand and multiplicator
IV: Multiplication layout
IV: Multiplication of 1 × 1 digits
IV: Multiplication of 1 × 2 digits
IV: Multiplication of 2 × 2 digits
IV: The modern method of multiplication
IV: The multiplication table
IV: The unit rod and decimals
IV: Traditional Chinese layout
IV: Traditional Japanese layout
IX Variations of Exercise 123456789
IX: Conclusion
IX: Exercise 987654321
IX: Introduction
IX: Starting with subtraction
IX: Using a background
IX: Using alternate operating direction
IX: Using the lower fifth bead
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Pánzhū Suànfǎ rules of addition and subtraction
Part I Basic Concepts
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Table of Contents
Tamaoki1913
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Traditional Methods
Translating
Translation state
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V Modern Division
V: Chunk division
V: Divisor multiplication table
V: Euclidean division
V: Example: 1÷327
V: Example: 1225 ÷ 35 = 35
V: Example: 123456789÷9 = 13717421
V: Example: 634263 ÷ 79283 = 7.999987..., a difficult case
V: External Resources
V: Further reading
V: Introduction and first methods
V: Key point of division with the abacus
V: Layout on the abacus of modern division
V: Modern division examples
V: Modern method
V: Modern vs. Traditional Division
V: Multiplication and division as inverse operations
V: Placing the quotient figure
V: Traditional Chinese division layout
V: Traditional Japanese division layout
V: Unit rod and decimals
VI Introduction to Traditional Methods
VI: External Resources
VI: Learning the abacus in the past
VI: Main differences between traditional and modern methods
VI: Modern abacus versus traditional abacus
VI: Modern and traditional methods
VI: Tables of procedures and some terms and notations
VI: The principle of least effort
VII Particularities of Traditional Addition and Subtraction
VII: Fifth lower bead
VII: Introduction to traditional addition and subtraction
VII: Reverse operation (from right to left)
VIII Use of the 5th Lower Bead
VIII: About the advantage
VIII: Additional Rules
VIII: Detail of subtraction with 5th bead; exercise 123456789
VIII: Detail of the sum with 5th bead; exercise 123456789
VIII: Examples of use of the 5th bead
VIII: Exercise 123456789 and the 5th bead
VIII: Extension of the example
VIII: External Resources
VIII: Further reading
VIII: Introduction to the use of the 5th lower bead
VIII: Rules for addition
VIII: Rules for subtraction
Volkov2018
Volkov2018a
WikipediaJAKuku
Williams1856
Williams1990
Wilson2005
Woods2017
X Synopsis of the Traditional Division
X: Chapters on traditional division
X: Introduction
X: Sunzi's Division: explanation
XI Modern and Traditional Divisions; Close Relatives
XI: Further reading
XI: Indexing the multiplication table; the division table
XI: Modern Division (商除法)
XI: The division table
XI: The hidden beauty of traditional division
XI: The source of mental effort
XI: Traditional division (帰除法)
XII Guide to the Traditional Division
XII: Further reading
XII: Introduction
XII: Modern Division Layout (MDA)
XII: Multi-digit divisors
XII: On the efficiency of traditional division
XII: One-digit divisors
XII: The division table
XII: Traditional division layout (TDA)
XII: Why do division rules include remainders?
XIII Learning the Division Table
XIII: "Hard" rules
XIII: Division by 8
XIII: Easy Rules
XIII: Memorizing the division table
XIII: Statistical Rules
XIII: Subdiagonal rules
XIII: The combined multiplication and division table
XIV How to Deal with Overflow
XIV: Conclusion
XIV: First Way: Brute Force
XIV: Introduction
XIV: Second way: Suspended lower beads
XIV: Third way: Memorization
XIX Roots
XIX: Chapters
XIX: Checking your exercises
XIX: Introduction
XTRAS: €-Abacus
XTRAS: 5+3 Abacus
XTRAS: Lee's Abacus
XTRAS: Medieval European Abacus
XTRAS: Mesopotamian abacus
XTRAS: Roman Abacus
XTRAS: Russian Abacus
XTRAS: School Abacus
XTRAS: Simulators
XTRAS: Soroban Trainer
XTRAS: Soroban Trainer Help
XTRAS: Square Root Tutor with Kijohou
XTRAS: Square Root Tutor with Kijohou Help
XTRAS: Worksheets
XTRAS: Worksheets Help
XuXinlu1573
XuYues.II
XV Examples of Traditional Division
XV: 123456789 dividded by 7
XV: 123456789 divided by 2
XV: 123456789 divided by 3
XV: 123456789 divided by 4
XV: 123456789 divided by 5
XV: 123456789 divided by 6
XV: 123456789 divided by 8
XV: 123456789 divided by 9
XV: Demonstration of 123456789 divided by 2 in a 5+2
XV: Demonstration of 123456789 divided by 3 in a 5+2
XV: Demonstration of 123456789 divided by 4 in a 5+2
XV: Demonstration of 123456789 divided by 5 in a 5+2
XV: Demonstration of 123456789 divided by 6 in a 5+2
XV: Demonstration of 123456789 divided by 7 in a 5+2
XV: Demonstration of 123456789 divided by 8 in a 5+2
XV: Demonstration of 123456789 divided by 9 in a 5+2
XV: Demonstration of 412/896 on a 5+2
XV: Demonstration of 888122/898 on a 5+2
XV: Demonstration of 888122/989 on a 5+2
XV: Demonstration of 998001/999 on a 4+1
XV: Demonstration of 998001/999 on a 5+1
XV: Demonstration of 998001/999 on a 5+2
XV: Demonstration of 998001/999 on a 5+3
XV: Demonstration of 998001/999 with counting rods
XV: Division of 412 by 896
XV: Division of 888122 by 898
XV: Division of 888122 by 989
XV: Division of 998001 by 999
XV: External Resources
XV: Introduction
XV: Multi-digit divisors (long division)
XV: One-digit divisors
XVI Specific Division Tables
XVI: "Short" and "long" division
XVI: Diagonal rules
XVI: Foundation
XVI: Further reading
XVI: Some examples
XVI: Two-digit division tables
XVII Division by Powers of 2
XVII: Division by 2 in situ
XVII: Introduction
XVII: Powers of five and multiplication by 2 in situ
XVII: Powers of two
XVIII Traditional Multiplication
XVIII: Colophon: How many multiplication methods are there?
XVIII: Further reading
XVIII: Introduction
XVIII: Multiplication with counting rods
XVIII: Proposed exercises
XVIII: Traditional multiplication and the unit rod
XVIII: Traditional multiplication method
XX Square Roots
XX: Conclusion
XX: Examples
XX: External Resources
XX: First digit
XX: Further reading
XX: Next digits
XX: Preparing the dividend
XX: Preparing the divisor
XX: Procedure
XX: Square root of 456.7890123
XX: Square root of 961
XX: Square root of 998001
XX: Theory
XX: Using the modern method
XXI Cube Roots
XXI: Appendix: Cubes of two-digit numbers
XXI: Cube root of 110591 (eight digits)
XXI: Cube root of 157464
XXI: Cube root of 237176659 (three digits)
XXI: Cube root of 666
XXI: Cube root of 830584
XXI: Examples of cube roots
XXI: First digit
XXI: From elementary arithmetic to numerical analysis
XXI: Modern Method
XXI: Next digits
XXI: Preparing the dividend
XXI: Preparing the divisor
XXI: Procedure
XXI: Theory
XXII Abbreviated Operations
XXII: Cube root
XXII: Division
XXII: Introduction
XXII: Multiplication
XXII: Other useful abbreviations
XXII: Square root
XXIII Negative Numbers
XXIII: Abacuses and negative numbers; the other side of the abacus
XXIII: Addition and Subtraction
XXIII: Addition of a complement
XXIII: Arithmetic with a fixed number of digits
XXIII: Complement of the n-digit complement
XXIII: Complements and the modern abacus
XXIII: Complements and the traditional abacus
XXIII: Definition of n-digit complement
XXIII: Division
XXIII: Extension and reduction of a complement
XXIII: Further reading
XXIII: Introduction
XXIII: Meaning
XXIII: Method of complements
XXIII: Multiplication
XXIII: Obtaining the complement
XXIV Other Multiplication Methods
XXIV: Approximate values
XXIV: Case of numbers with three or more digits
XXIV: Case of two-digit number
XXIV: Further reading
XXIV: Introduction
XXIV: Multifactorial multiplication
XXIV: Multiplication by rounding the multiplier to a multiple of a power of 10
XXIV: Multiplier ending in 1
XXIV: Multiplier slightly greater than unity
XXIV: Multiplier slightly less than unity
XXIV: Multiplier starting with 1
XXIV: Squaring
XXV Other Division Methods
XXV: Division by rounding the divisor to a multiple of a power of 10
XXV: Division with excessive quotient
XXV: Divisor slightly greater than one
XXV: Divisor slightly less than one
XXV: Downward revision from the other side
XXV: Examples of division with excessive quotient
XXV: Foundation
XXV: Further reading
XXV: Introduction
XXV: When to use the method?
XXVI Newton's Method for Square, Cube and Fifth Roots
XXVI: Before starting
XXVI: Conclusions
XXVI: Cube roots
XXVI: Example of a cube root on a 13-rod abacus
XXVI: Example using the bc calculator
XXVI: Examples on the abacus
XXVI: Extension to other prime order roots
XXVI: Fifth roots
XXVI: Introduction
XXVI: Manual calculation example
XXVI: Seventh roots
XXVII RADIX Method for Decimal Logarithms and Antilogarithms
XXVII: Appendix A
XXVII: Appendix B
XXVII: Basis
XXVII: Before starting
XXVII: Further reading
XXVII: Introduction
XXVII: Obtaining antilogarithms
XXVII: Obtaining antilogarithms on the abacus
XXVII: Obtaining logarithms
XXVII: Obtaining logarithms on the abacus
XXVII: The Radix Method
XXVII: The Radix method with the abacus
XXVII: Use of logarithms
XXVII: Use of logarithms on the abacus
XXVIII Lunar Phases and Ocean Tides
XXVIII: Age of the Moon
XXVIII: Introduction
XXVIII: Oni Oni Nishi Algorithm
XXVIII: Tide abacus
yang1989
Yoshida1634
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