**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.

▪
*Abacus: Mystery of the Bead*

▪
*Advanced Abacus Techniques*

September, 2007