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Clearing

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To clear a soroban, the right index finger and thumb is placed between the five-beads and the one-beads of the leftmost rod. The right hand is then moved from left to right, such that all beads are pushed away from the reckoning bar. This sets all rods to zero.

Addition

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The addition of two positive numbers is carried out from left to right.[1] If there are sufficient unused beads on the current rod, the current digit of the addend can be added directly. Otherwise, one may employ one of the following techniques:[2]

Addition techniques[3]
+ 5 series + 1 = + 5 − 4 + 10 series + 1 = − 9 + 10
+ 2 = + 5 − 3 + 2 = − 8 + 10
+ 3 = + 5 − 2 + 3 = − 7 + 10
+ 4 = + 5 − 1 + 4 = − 6 + 10
+ 5 = − 5 + 10
− 5 + 10 series + 6 = + 1 − 5 + 10 + 6 = − 4 + 10
+ 7 = + 2 − 5 + 10 + 7 = − 3 + 10
+ 8 = + 3 − 5 + 10 + 8 = − 2 + 10
+ 9 = + 4 − 5 + 10 + 9 = − 1 + 10
Example: 2 + 6 = 8 (direct addition)
{{Soroban diagram|dots = 2|0|2}} {{Soroban diagram|dots = 2|0|8*}}
2 ... ... + 6 = 8
Example: 4 + 3 = 7 (uses "+ 3 = + 5 − 2")
{{Soroban diagram|dots = 2|0|4}} {{Soroban diagram|dots = 2|0|9*}} {{Soroban diagram|dots = 2|0|7*}}
4 ... ... + 5 ... ... − 2 = 7
Example: 24 + 8 = 32 (uses "+ 8 = − 2 + 10")
{{Soroban diagram|dots = 2|2|4}} {{Soroban diagram|dots = 2|2|2*}} {{Soroban diagram|dots = 2|3|2*}}
24 ... ... − 2 ... ... + 10 = 32
Example: 27 + 6 = 33 (uses "+ 6 = + 1 − 5 + 10")
{{Soroban diagram|dots = 2|2|7}} {{Soroban diagram|dots = 2|2|8*}} {{Soroban diagram|dots = 2|2|3*}} {{Soroban diagram|dots = 2|3|3*}}
27 ... ... + 1 ... ... − 5 ... ... + 10 = 33
Example: 162.07 + 393.94 = 556.01
{{Soroban diagram|dots = 3|1|6|2|0|7}} {{Soroban diagram|dots = 3|4*|6|2|0|7}}
162.07 ... (3 is added directly to rod A)
... + 300 ...
{{Soroban diagram|dots = 3|4|5*|2|0|7}} {{Soroban diagram|dots = 3|9|5*|2|0|7}} {{Soroban diagram|dots = 3|5|5*|2|0|7}}
("+ 9 = − 1 + 10" on rod B)
... − 10 ...		 (The second step of "+ 9 = − 1 + 10" on rod B involves adding 1 to rod A. Doing so directly is not possible. Instead, "+ 1 = + 5 − 4" on rod A has to be used.)
... + 500 ...		 ... − 400 ...
{{Soroban diagram|dots = 3|5|5|7*|0|7}} {{Soroban diagram|dots = 3|5|5|5*|0|7}}
("+ 3 = + 5 − 2" on rod C)
... + 5 ...		 ... − 2 ...
{{Soroban diagram|dots = 3|5|5|5|9*|7}}
(9 is added directly rod D)
... + 0.9 ...
{{Soroban diagram|dots = 3|5|5|5|9|1*}} {{Soroban diagram|dots = 3|5|5|5|0|1*}} {{Soroban diagram|dots = 3|5|5|6|0|1*}}
("+ 4 = − 6 + 10" on rod E)
... – 0.06 ...		 (The second step of "+ 4 = − 6 + 10" on rod E involves adding 1 to rod D. Doing so directly is not possible. Instead, "+ 1 = − 9 + 10" on rod D has to be used.)
... − 0.9 ...		 ... + 1 = 556.01

Subtraction

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The subtraction of a positive subtrahend from a larger minuend is carried out from left to right.[4] If there are sufficient used beads, the digit of the subtrahend can be subtracted directly. Otherwise, one may employ one of the following techniques:[5]

Subtraction techniques[3]
− 5 series − 1 = − 5 + 4 − 10 series − 1 = − 10 + 9
− 2 = − 5 + 3 − 2 = − 10 + 8
− 3 = − 5 + 2 − 3 = − 10 + 7
− 4 = − 5 + 1 − 4 = − 10 + 6
− 5 = − 10 = 5
− 10 + 5 series − 6 = − 10 + 5 − 1 − 6 = − 10 + 4
− 7 = − 10 + 5 − 2 − 7 = − 10 + 3
− 8 = − 10 + 5 − 3 − 8 = − 10 + 2
− 9 = − 10 + 5 − 4 − 9 = − 10 + 1
Example: 8 − 6 = 2 (direct subtraction)
{{Soroban diagram|dots = 2|0|8}} {{Soroban diagram|dots = 2|0|2*}}
8 ... ... − 6 = 2
Example: 7 − 3 = 4 (uses "− 3 = − 5 + 2")
{{Soroban diagram|dots = 2|0|7}} {{Soroban diagram|dots = 2|0|2*}} {{Soroban diagram|dots = 2|0|4*}}
7 ... ... − 5 ... ... + 2 = 4
Example: 31 − 8 = 23 (uses "− 8 = − 10 + 2")
{{Soroban diagram|dots = 2|3|1}} {{Soroban diagram|dots = 2|2|1*}} {{Soroban diagram|dots = 2|2|3*}}
31 ... ... − 10 ... ... + 2 = 23
Example: 23 − 6 = 17 (uses "− 6 = − 10 + 5 − 1")
{{Soroban diagram|dots = 2|2|3}} {{Soroban diagram|dots = 2|1|3*}} {{Soroban diagram|dots = 2|1|8*}} {{Soroban diagram|dots = 2|1|7*}}
23 ... ... − 10 ... ... + 5 ... ... − 1 = 17
Example: 63.02 − 33.561 = 29.459
{{Soroban diagram|dots = 2|6|3|0|2|0}} {{Soroban diagram|dots = 2|1*|3|0|2|0}} {{Soroban diagram|dots = 2|3*|3|0|2|0}}
63.02 ("− 3 = − 5 + 2" on rod A)
... − 50 ...		 ... + 20 ...
{{Soroban diagram|dots = 2|3|0*|0|2|0}}
(3 is subtracted directly from rod B)
... − 3 ...
{{Soroban diagram|dots = 2|2|0|0*|2|0}} {{Soroban diagram|dots = 2|2|9|0*|2|0}} {{Soroban diagram|dots = 2|2|9|5*|2|0}}
("− 5 = − 10 + 5" is to be used on rod C. However, the first step of that involves subtracting 1 from rod B. Doing so directly is not possible. Instead, "− 1 = − 10 + 9" on rod B must first be carried out.)
... − 10 ...		 ... + 9 ... (The second step of "− 5 = − 10 + 5" is to be carried out on rod C.)
... + 0.5 ...
{{Soroban diagram|dots = 2|2|9|0|2*|0}} {{Soroban diagram|dots = 2|2|9|4|2*|0}} {{Soroban diagram|dots = 2|2|9|4|7*|0}} {{Soroban diagram|dots = 2|2|9|4|6*|0}}
("− 6 = − 10 + 5 − 1" is to be used on rod D. However, the first step of that involves subtracting 1 from rod C. Doing so directly is not possible. Instead, "− 1 = − 5 + 4" on rod C must first be carried out.)
... − 0.5 ...		 ... + 0.4 ... (The second and third steps of "− 6 = − 10 + 5 − 1" is to be carried out on rod D.)
... + 0.05 ...		 ... – 0.01 ...
{{Soroban diagram|dots = 2|2|9|4|5|0*}} {{Soroban diagram|dots = 2|2|9|4|5|9*}}
("− 1 = − 10 + 9" on rod E)
... − 0.01 ... 		... + 0.009 = 29.459

Multiplication

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Multiplication table
× 0 1 2 3 4 5 6 7 8 9
0 00 00 00 00 00 00 00 00 00 00
1 00 01 02 03 04 05 06 07 08 09
2 00 02 04 06 08 10 12 14 16 18
3 00 03 06 09 12 15 18 21 24 27
4 00 04 08 12 16 20 24 28 32 36
5 00 05 10 15 20 25 30 35 40 45
6 00 06 12 18 24 30 36 42 48 54
7 00 07 14 21 28 35 42 49 56 63
8 00 08 16 24 32 40 48 56 64 72
9 00 09 18 27 36 45 54 63 72 81

Let the multiplicand and multiplier be a × 10p and b × 10q respectively in scientific notation.

For all m and n, the mth significant digit of the multiplicand is multiplied with the the nth significant digit of the multiplier to give a two-digit product (if this product is less than 10, keep a leading zero). These two digits are added to the (m + n – 1)th and (m + n)th rod from the left.[6]

The product is given by

(a × 10p) × (b × 10q) = ab × 10p + q

⌊ab⌋, which is represented by first two rods of the product, is either a two-digit number or a one-digit number with a leading zero. The decimal point is inserted just after the (p + q + 2)th rod to obtain the product.

Example: 123 × 0.45 = 55.35
{{Soroban diagram|dots = 1|4*|5*|0|0|0|0|1*|2*|3*|0|0|0|0|0|0*|0*|0*|0*|0*}}
(For clarity, the significant digits of the multiplicand 123 are set on rods G, H and J while those of the multiplier 0.45 are set on A and B. Multiplication will take place on rods Q, R, S, T, and U.)
{{Soroban diagram|dots = 1|4*|5|0|0|0|0|1*|2|3|0|0|0|0|0|0*|4*|0|0|0}}
(m = 1 and n = 1. Add the product of rods G and A to rods Q and R. Rod Q is the first rod of the product.)
1 × 4 = 04
{{Soroban diagram|dots = 1|4*|5|0|0|0|0|1|2*|3|0|0|0|0|0|0|4*|8*|0|0}}
(m = 2 and n = 1. Add the product of rods H and A to rods R and S.)
2 × 4 = 08
{{Soroban diagram|dots = 1|4*|5|0|0|0|0|1|2|3*|0|0|0|0|0|0|4|9*|2*|0}}
(m = 3 and n = 1. Add the product of rods J and A to rods S and T.)
3 × 4 = 12
{{Soroban diagram|dots = 1|4|5*|0|0|0|0|1*|2|3|0|0|0|0|0|0|5*|4*|2|0}}
(m = 1 and n = 2. Add the product of rods G and B to rods R and S.)
1 × 5 = 05
{{Soroban diagram|dots = 1|4|5*|0|0|0|0|1|2*|3|0|0|0|0|0|0|5|5*|2*|0}}
(m = 2 and n = 2. Add the product of rods H and B to rods S and T.)
2 × 5 = 10
{{Soroban diagram|dots = 1|4|5*|0|0|0|0|1|2|3*|0|0|0|0|0|0|5|5|3*|5*}}
(m = 3 and n = 2. Add the product of rods J and B to rods T and U.)
3 × 5 = 15
{{Soroban diagram|dots = 1|4|5|0|0|0|0|1|2|3|0|0|0|0|0|0*|5*|5*|3*|5*}}
123 × 0.45 = (1.23 × 102) × (4.5 × 10–1) = ...
(Counting rod Q as the first rod of the product, the decimal point is located just after rod number (2 + (−1) + 2), i.e. the 3rd rod, rod S.)
... = 55.35

Division

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Several methods have been devised to divide one positive number by another. The traditional Chinese technique[7] would require all the beads of the suanpan and hence cannot be conducted on the soroban. The following method has emerged to be the most popular since the 1930s.[8]

For clarity, the significant digits of the divisor can be set near the left of the abacus. The dividend is initially set near the right. Division is then executed from left to right using a method similar to long division. In this process, the quotient gradually occupies the rods representing the dividend.

Let the dividend and divisor be a × 10p and b × 10q respectively in scientific notation. The first rod of the quotient refers to the rod two to the left of the dividend. Let n be the number of significant digits of the divisor. (n + 1) digits of the divisor will be handled at each step.

First, find the largest integer x1 such that b x1 is smaller than or equal to the number represented on rods 2 through (n + 2). Set the 1st rod as x1, and subtract b x1 from rod 2 through (n + 2). Next, find the largest integer x2 that such that b x2 is smaller than or equal to the number represented on rods 3 through (n + 3). Set the 2nd rod as x2, and subtract b x2 from rods 3 through (n + 3). Repeat this process until the dividend vanishes, or until desired level of precision has been achieved.

The quotent is given by

(a × 10p) ÷ (b × 10q) = ⁠a/b⁠ × 10p - q

⌊⁠a/b⁠⌋, which occupies the first rod, is either the first digit of the quotient or its leading zero. The decimal point is inserted just after the (p - q + 1)th rod to obtain the quotient.

Example: 1580 ÷ 25 = 63.2
{{Soroban diagram|dots=2|2*|5*|0|0|0*|0|1*|5*|8*|0|0}} Equivalent step in long division: 
          
25)1580
For clarity, the significant digits of the divisor, 25, are set on rods A and B. The significant digits of the divisor, 1580, are set on rods G, H, and J. Rod E is considered as the first rod of the quotient. Since the divisor has 2 significant digits, 3 digits of the divisor will be handled at each step.
{{Soroban diagram|dots=2|2*|5*|0|0|0|0*|1*|5*|8|0|0}} {{Soroban diagram|dots=2|2*|5*|0|0|0|0*|1*|5*|8|0|0}} Equivalent step in long division: 
    0     
25)1580
25 × 0 = 000
(000 is the largest multiple of 25 that is smaller than or equal to the 015 on rods F, G, and H. Rod E is set as 0. Subtraction takes place on rods F, G, and H.)
015 − 000 = 015
{{Soroban diagram|dots=2|2*|5*|0|0|0|0|1*|5*|8*|0|0}} {{Soroban diagram|dots=2|2*|5*|0|0|0|6|0*|0*|8*|0|0}} Equivalent step in long division: 
    06    
25)1580
  −150
     8
25 × 6 = 150
(150 is the largest multiple of 25 that is smaller than or equal to the 158 on rods G, H, and J. Rod F is set as 6. Subtraction takes place on rods G, H, and J.)
158 − 150 = 008
{{Soroban diagram|dots=2|2*|5*|0|0|0|6|0|0*|8*|0*|0}} {{Soroban diagram|dots=2|2*|5*|0|0|0|6|3|0*|0*|5*|0}} Equivalent step in long division: 
    063   
25)1580
  −150
     80
    −75
      5
25 × 3 = 075
(075 is the largest multiple of 25 that is smaller than or equal to the 080 on rods H, J, and K. Rod G is set as 3. Subtraction takes place on rods H, J, and K.)
080 − 075 = 005
{{Soroban diagram|dots=2|2*|5*|0|0|0|6|3|0|0*|5*|0*}} {{Soroban diagram|dots=2|2*|5*|0|0|0|6|3|2|0*|0*|0*}} Equivalent step in long division: 
    063.2
25)1580.0
  −150
     80
    −75
      5.0
     −5.0
        0
25 × 2 = 050
(050 is the largest multiple of 25 that is smaller than or equal to the 050 on rods J, K, and L. Rod H is set as 2. Subtraction takes place on rods J, K, and L.)
050 − 000 = 000
{{Soroban diagram|dots=2|2|5|0|0|0|6*|3*|2*|0|0|0}}
1580 ÷ 25 = (1.58 × 103) ÷ (2.5 × 101) = ...
(The dividend has vanished. Since rod E is the first rod of the quotient, the decimal point is located just after rod number (3 − 1 + 1), i.e. the 3rd rod, rod G.)
... = 63.2

Other operations

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Methods have been devised to handle negative numbers,[9] as well as compute the square root[10][11] and cube root[12] of a number.[13]

References

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  1. ^ Abacus Tutorial – How to use an Abacus. Japanese, Chinese abacus techniques. Addition. Totton Heffelfinger and Gary Flom.
  2. ^
    Single-digit addition techniques
    Current rod
    0 1 2 3 4 5 6 7 8 9
    Addend digit + 1 + 1 + 1 + 1 + 1 + 5 - 4 + 1 + 1 + 1 + 1 - 9 + 10
    + 2 + 2 + 2 + 2 + 5 - 3 + 5 - 3 + 2 + 2 + 2 - 8 + 10 - 8 + 10
    + 3 + 3 + 3 + 5 - 2 + 5 - 2 + 5 - 2 + 3 + 3 - 7 + 10 - 7 + 10 - 7 + 10
    + 4 + 4 + 5 - 1 + 5 - 1 + 5 - 1 + 5 - 1 + 4 - 6 + 10 - 6 + 10 - 6 + 10 - 6 + 10
    + 5 + 5 + 5 + 5 + 5 + 5 - 5 + 10 - 5 + 10 - 5 + 10 - 5 + 10 - 5 + 10
    + 6 + 6 + 6 + 6 + 6 - 4 + 10 + 1 - 5 + 10 + 1 - 5 + 10 + 1 - 5 + 10 + 1 - 5 + 10 - 4 + 10
    + 7 + 7 + 7 + 7 - 3 + 10 - 3 + 10 + 2 - 5 + 10 + 2 - 5 + 10 + 2 - 5 + 10 - 3 + 10 - 3 + 10
    + 8 + 8 + 8 - 2 + 10 - 2 + 10 - 2 + 10 + 3 - 5 + 10 + 3 - 5 + 10 - 2 + 10 - 2 + 10 - 2 + 10
    + 9 + 9 - 1 + 10 - 1 + 10 - 1 + 10 - 1 + 10 + 4 - 5 + 10 - 1 + 10 - 1 + 10 - 1 + 10 - 1 + 10
  3. ^ a b IN HOUSE TRAINING FOR "ABACUS" Published by user orlyaiman on Scribd. Slides 44 and 45.
  4. ^ 算盤 Abacus Tutorial – How to use an Abacus. Japanese, Chinese abacus techniques. Subtraction. Totton Heffelfinger and Gary Flom.
  5. ^
    Single-digit subtraction techniques
    Current rod
    0 1 2 3 4 5 6 7 8 9
    Subtrahend digit – 9 - 10 + 1 - 10 + 1 - 10 + 1 - 10 + 1 - 10 + 5 - 4 - 10 + 1 - 10 + 1 - 10 + 1 - 10 + 1 - 9
    - 8 - 10 + 2 - 10 + 2 - 10 + 2 - 10 + 5 - 3 - 10 + 5 - 3 - 10 + 2 - 10 + 2 - 10 + 2 - 8 - 8
    - 7 - 10 + 3 - 10 + 3 - 10 + 5 - 2 - 10 + 5 - 2 - 10 + 5 - 2 - 10 + 3 - 10 + 3 - 7 - 7 - 7
    - 6 - 10 + 4 - 10 + 5 - 1 - 10 + 5 - 1 - 10 + 5 - 1 - 10 + 5 - 1 - 10 + 4 - 6 - 6 - 6 - 6
    - 5 - 10 + 5 - 10 + 5 - 10 + 5 - 10 + 5 - 10 + 5 - 5 - 5 - 5 - 5 - 5
    - 4 - 10 + 6 - 10 + 6 - 10 + 6 - 10 + 6 - 4 - 5 + 1 - 5 + 1 - 5 + 1 - 5 + 1 - 4
    - 3 - 10 + 7 - 10 + 7 - 10 + 7 - 3 - 3 - 5 + 2 - 5 + 2 - 5 + 2 - 3 - 3
    - 2 - 10 + 8 - 10 + 8 - 2 - 2 - 2 - 5 + 3 - 5 + 3 - 2 - 2 - 2
    - 1 - 10 + 9 - 1 - 1 - 1 - 1 - 5 + 4 - 1 - 1 - 1 - 1
  6. ^ 算盤 Abacus Tutorial – How to use an Abacus. Japanese, Chinese abacus techniques. Multiplication. Totton Heffelfinger and Gary Flom.
  7. ^ 算盤 Short Division on a Chinese Abacus - Chinese Suan Pan Totton Heffelfinger and Gary Flom.
  8. ^ 算盤 Abacus Tutorial - How to use an Abacus. Japanese, Chinese abacus techniques. Division. Totton Heffelfinger and Gary Flom.
  9. ^ 算盤 Abacus Tutorial – How to use an Abacus. Japanese, Chinese abacus techniques. Negative Numbers from Subtraction. Totton Heffelfinger and Gary Flom.
  10. ^ 算盤 Square Roots on an Abacus – Kato Fukutaro's Method by Edvaldo Siqueir. Totton Heffelfinger and Gary Flom.
  11. ^ Square roots as solved by Takashi Kojima. Totton Heffelfinger and Gary Flom.
  12. ^ 算盤 Abacus Cube Roots. Totton Heffelfinger and Gary Flom.
  13. ^ 算盤 Advanced Abacus Techniques. Tutorial Instruction. Advanced abacus,soroban,suan pan. Totton Heffelfinger and Gary Flom.
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