Calculation with abacus (soroban)

number display

o o

---

o o
o o
o o
o o
this is 0 (default state)

o o

---
  o <- one lower bead means 1 when pulled up
o  
o o
o o
o o
this is 1

o
  o -> upper bead means 5 when pulled down
---
  o
o o
o
o o
o o
this is 7

o
  o
---
o o
  o
o
o o
o o
this means 17 (1 at 10's place, 7 at 1's place)

add

to do 1+2, first put 1
o o

---
  o
o  
o o
o o
o o
then try to move 2 lower beads up, it becomes
o o

---
  o
o o
o o
o 
o o
3 is the answer

to do 4+2, first put 4
o o

---
  o
o o
o o
o o
o 
then try to move 2 lower beads up,
but we can't so we add 5 (upper bead down)
and subtract 3 (3 lower beads down)
o 
  o
---
  o
o 
o o
o o
o o
6 is the answer

+1 could be upper bead & 4 lower beads down
+2 could be upper bead & 3 lower beads down
+3 could be upper bead & 2 lower beads down
+4 could be upper bead & 1 lower beads down
In this case,
use the thumb to move the lower bead,
and the index finger the upper bead.
abacus is designed cleverly so they move in the same direction.

to do 9+2, first put 9
o 
  o
---
  o
o o
o o
o o
o 
we can't move 2 lower beads up,
we can't do +5 -3 either because 5 is already there
so we do +10 -8
remove 8 from 1's place and add 1 to 10's place
o o
  
---
o o

o o
o o
o o
11 is the answer

+1 could be subtract 9 and add 1 in the higher digit
+2 could be subtract 8 and add 1 in the higher digit
...

subtract

• subtract directly, e.g.
• 4 - 2: just move down 2 lower beads
• 9 - 6: just move up the upper bead and 1 lower bead (so it’s 3)

• subtract 5 and add (5 - the number), e.g.
• 7 - 4: move upper bead up(-5), and 1 lower bead up (+1)
• Like the second mode of addition, the thumb and index finger do the moving simultaneous in the same direction (up)

• subtract 1 in the higher digit and add (10-the number) in the current digit, e.g.
• 24 - 5:
• Add 5 to the 1’s place (now it becomes 9) and remove 1 from 10’s place
• 57 - 9:
• Add 1 in 1’s place to make it 8
• Subtract 1 from 10’s place. This means move up the upper bead (-5) and move up 4 lower beads (+4)

multiply

• 3*5=15 so add 1 to the 1000’s place and 5 to the 100’s place

• Then 3*6=18, +1 to the 100’s place and +8 to the 10’s place

• Then 4*5=20, +2 to the 100’s place and +0 to the 10’s place

• Finally +2 to the 10’s place and +4 to the 1’s place

• eg 3000 * 20000, 3*2 = 06, 3 is at the 4th digit, 2 is at the 5th digit, so 06 goes to the 9th and 10th digit from the right.

divide

• single digit divisor
• Eg 72 / 3
• First put 72 (dividend) on the abacus
• 7 can be divided by 3 (to get 2 and some remainder), but we always want to use 2 digits to be divided by 1 digit, so we use 07 (3rd and 2nd digit from the right), and get a 2, we place the result 2 on the 4th digit.
• Then use 2 (the result) to multiply by 3 (the divider) and get 6, subtract this 6 from 07(dividend). Now we get 01 on the 3rd and 2nd digit, the abacus looks like: 2012 → the first 2 is the result (for now) and 12 at the end is the remaining number to be divided.
• Use 12 to be divided by 3, we get 4, so we put 4 to the left of 12, now the whole board becomes 2412
• Take 4(the new result) to multiply with 3(the divider), we get 12. Subtract that from the 12(dividend) at the rightmost digits, so the rightmost two digits becomes 00, and the board becomes 2400
• 24 is the final answer
• When the divider has n digits, the final result will end at the n+2 digit from the right. The rightmost n+1 digits will be the final remainder of the division. E.g. 100 / 3 should end as 3301, meaning that 33 is the quotient and 01 is the remainder.

• multi-digit divisor (without return)
• E.g. 231 / 21
• First put 231
• Try 02 / 2 (divisor’s first digit), which gets 1, put the 1 (the quotient) to the left of the 0
• Use the 1 (the quotient) to multiple 2 (the divisor’s first digit), which is 02, subtract that that from 02 (the dividend), so 02 is changed to 00.
• Use the 1 (the quotient) to multiple 1 (the next digit of divisor), which is 01, subtract that from 03 in the dividend area (hundred place changed from 2 to 0 from last step, tenth place is still 3), and get 02.
• Now keep iterating:
• Use 02 (first two digits from the current dividend, which is 21 now) to be divided by 2 (divisor’s first digit), get 1, put it to the left of 02 (current dividend)
• Use 1 (quotient) to multiply 2 (divisor’s first digit) and subtract that from the 02 in dividend
• Use 1 (quotient) to multiply 1 (divisor’s next digit) and subtract that from the 01 fin dividend
• Now the dividend is cleared to be all 0s. The abacus shows 11000
• Since the divisor has 2 digits, the right most 3 digits (divisor digits + 1) are for remainders (0 for this case), and on its left, the 11 is the answer.

• multi-digit divisor with return
• E.g. 45 / 15
• Put 45
• 04 / 1 is 4, put 4 to the left of 04
• 4 (quotient) times 1 (divisor’s first digit) is 4, remove 4 from 04 in the dividend and get 00 there
• 4 (quotient) times 5 (divisor’s next digit) is 20, try to remove 20 from 05 in the dividend and there isn’t enough. So we reduce the quotient to 3, and return 1 (divisor’s first digit) to the tenth place of the dividend (where we subtracted 4 last step), so the dividend becomes 15
• Now try 3 (new quotient) times 5 (divisor’s next digit) and it’s 15, subtract it from 15 in the dividend. Now it has enough, so subtract and get 0. If it’s still not enough, e.g. in the case of 60 / 15, then repeat the return step (reduce quotient by 1) again until the dividend has enough to subtract.
• Now the dividend is 0, and the answer is 3, the right most two digits (remainder) is 0.

• multi-digit divisor with multi-digit return
• E.g. 65771 / 739
• First put 9 as the first digit of the quotient (65 / 7)
• Remove 63 from the top 2 digits of dividend (9 * 7), dividend becomes 02771
• Remove 27 from the 27 in dividend, dividend becomes 00071
• Try to remove 81 from the 07 in dividend, not enough
• So reduce quotient to 8, and return 73 (the first two digits of the divisor) back to the digits where they were subtracted. Dividend becomes 07371
• Now try to remove 72 (new quotient 8 times 9) from the 737 in the dividend, dividend becomes 06651.
• 66 / 7 is 9, remove 63 and dividend becomes 00351
• Remove 27 and dividend becomes 00081
• Remove 81 and dividend becomes 00000
• The answer is 89.

subtract to negative number

• Begin by borrowing 1 from the hundreds column.

• Subtract 95 from 100 → 5

x 
  o
---
  
x x
x x
x x
x x
(unmoved beads are marked as x)

• The unmoved beads (x) add up to 94.  Add 1 to it to get 95. -95 is the answer.

• Add 118 to the soroban (5) → 123

• Since we borrowed 1 from the hundreds column earlier, remove 1 from the column of 100.

• This leaves 23, the correct answer.