算盤 Multiplying and Dividing Negative Numbers on an Abacus

ABACUS: MYSTERY OF THE BEAD
The Bead Unbaffled - An Abacus Manual

Multiplying & Dividing Negative Numbers - by Shane Baggs

Shane Baggs shares his method in a message to the Yahoo soroban/abacus newsgroup.

#2 is a good point. You solved it by adding 9's, I solved it by working with the "owed" rod. There's more than one way to do things on an abacus!

After multiplying a positive and a negative, subtract the positive from the "owed" (usually 1000's) column of the product.

Before dividing a negative by a positive, add one less than the positive to the "owed" (usually 1000's) column. If there's a remainder in the final answer, convert the positive remainder to negative by adding 1 to the quotient and subtracting the divisor from the remainder.

Until I read your post, I didn't think you could multiply a negative by a negative and get a positive! Here's how I'd handle that:

After multiplying a negative and a negative, subtract each number from the others' "owed" column. Read what's left as a positive number.

**Here's Totton's example 2: -58 * 37 = -2146**

370042000 The "42" represents -58; we owe 100.
x 37
370001554
- 37 Now we must subtract 37 from the 100's column.
370007854 We now owe 10,000; so 7854 represents -2146.

Now, let's work it backward -2146 / 37 = 58.

370000007854 I owe 10,000; so 7854 represents -2146.
+36 I must add 37-1=36 to the 10,000's column.
370000367854
÷ 37 Now you can divide by 37.
370009942000 That's 9942. As I owe 10,000; it represents -58.

Notice that I had to leave extra space when setting up the problem -- one space for each column in the divisor.

**What about remainders? The Bead Unbaffled contains 356/25 = 14.24. If it were a remainder problem, it would be 14 remainder 6. I'll work its negative.**

2500000064400 The "644" represents -356; Negative 365 divided by 25.
               Why did I leave 6 empty rods instead of 4?
    + 24     Add 1 less than 25 to the "owed" rod.
2500002464400 Now there are 4 empty rods.
         ÷ 25
2500098501900 Dividing, we get -15 remainder +19.
               Convert the remainder to negative.
      +1-25 Add 1 to the quotient and subtract 25 from the remainder.
2500098609400 The answer is -14 remainder -6.

Reference:
Soroban Abacus Group message 4429

▪ Abacus: Mystery of the Bead
▪ Advanced Abacus Techniques

September, 2007

Multiplying & Dividing Negative Numbers - by Shane Baggs

Here's Totton's example 2: -58 * 37 = -2146

370042000 The "42" represents -58; we owe 100. x 37 370001554 - 37 Now we must subtract 37 from the 100's column. 370007854 We now owe 10,000; so 7854 represents -2146.

Now, let's work it backward -2146 / 37 = 58. 370000007854 I owe 10,000; so 7854 represents -2146. +36 I must add 37-1=36 to the 10,000's column. 370000367854 ÷ 37 Now you can divide by 37. 370009942000 That's 9942. As I owe 10,000; it represents -58.