User:Jccsvq/sandbox

Data used
ABCDEFGHIJKLM| 4567890123|Entering radicand starting in CD (first group) 2          |First root digit in B   -4         |Subtract square of B from first group 2 567890123|Null remainder 4 567890123|Doubling B. Appending next group to remainder 41 567890123|5/4≈1, try 1 as next root digit -4       |Continue division by 41, subtract 1✕41 from EF     -1| 41 157890123|15 as remainder 42 157890123|Double second root digit 42 157890123|Append next group 423157890123|157/42≈3, try 3 as next root digit -12      |Continue division by 423, subtract 3✕423 from E-H -06|     -09|  423 30990123|309 as remainder 426 30990123|Double third root digit 426 30990123|Append next group etc.|

Generated table
After changing caption:

Abacus diagram
9 18 8  0  0  1              9  9  9 ╔═════════════════════════════════════════╗ ║  │  │  │  ●  ●  ●  ●  ●  ●  ●  │  │  │  ║ ║  │  │  │  │  │  │  │  │  │  │  │  │  │  ║ ║  ●  ●  ●  │  │  │  │  │  │  │  ●  ●  ●  ║ ╠═════════════════════════════════════════╣ ║  ●  ●  ●  │  │  ●  │  │  │  │  ●  ●  ●  ║ ║  ●  ●  ●  │  │  │  │  │  │  │  ●  ●  ●  ║ ║  ●  ●  ●  ●  ●  │  ●  ●  ●  ●  ●  ●  ●  ║ ║  ●  │  │  ●  ●  ●  ●  ●  ●  ●  ●  ●  ●  ║ ║  │  │  │  ●  ●  ●  ●  ●  ●  ●  │  │  │  ║ ║  ●  ●  ●  ●  ●  ●  ●  ●  ●  ●  │  │  │  ║ ║  │  ●  ●  ●  ●  ●  ●  ●  ●  ●  ●  ●  ●  ║ ╚═════════════════════════════════════════╝    A  B  C  D  E  F  G  H  I  J  K  L  M

Math
$$x_i-(y_i+1)^2=x_i-y_i^2-2y_i-1<0$$

Table of content

 * Foreword
 * Traditional versus modern methods
 * The principle of least effort
 * Addition and subtraction
 * Learning addition and subtraction
 * Use of the 5th lower bead
 * Division
 * Modern and traditional division; close relatives
 * Guide to traditional division (帰除法)
 * How to learn the division table
 * Dealing with overflow
 * Specialized division tables
 * Division by powers of two
 * Traditional division examples
 * Multiplication
 * Roots
 * Square root
 * Cube root
 * A non-traditional method
 * Practicing without exercise sheets
 * The 123456789 exercise
 * Abbreviated operations

Suanpan article (contrib)
The most mysterious and seemingly superfluous fifth lower bead, likely inherited from counting rods as suggested by the image above, was used to simplify and speed up addition and subtraction somewhat, as well as to decrease the chances of error. Its use was demonstrated, for example, in the first book devoted entirely to suanpan: Computational Methods with the Beads in a Tray (Pánzhū Suànfǎ 盤珠算法) by Xú Xīnlǔ 徐心魯 (1573, Late Ming Dynasty). The following two animations show the details of this particular usage:

Subpages
Testing a reference to a subpage.