Introduction: Pocket Abacus

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The abacus is a calculation tool that works by sliding beads along columns to represent numbers and to compute arithmetic. Almost any sliding tool used to record calculations can be called an abacus, and over the years there has been many iterations and adaptations of this classic calculator.

For centuries the abacus ruled as the calculator for traders and merchants the world over. Today, much of the world now embraces new technology and the once mighty abacus has been replaced with solar-powered calculators and excel spreadsheets. Yet in some places, the abacus is still used as a learning tool for elementary school students and as a method of calculation for traders.

Though many cultures have used the abacus throughout the years, the two most common types that exist today are the Japanese abacus (called soroban) and the Chinese abacus (called suanpan). The main difference between the two is the Japanese abacus has one row of beads on the top deck where the Chinese has two rows, allowing the suanpan to compute to hexadecimal.
This abacus is modeled after the suanpan.
If you're in the mood to nerd it up, check out some of the other types of abaci used over the years.

The simplicity of this 'computer' belies the complexity of computations achievable. Word on the street is there are techniques to solve for square and even cube root using the abacus!
(this instructable also covers elementary arithmetic, jump to step 7 to see.)

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enough talk, let's abacus!

Step 1: A Brief Introduction

The abacus is placed flat on a table with the columns of 5 beads towards you. The abacus is 'reset' when all beads are pushed away from the center bar.
The upper deck of 2 rows of beads are called heavenly beads
The lower deck of 5 rows of beads are called earthly beads

The beads are counted by moving them up or down towards the centre bar, the abacus is read from left to right and each column corresponds as a zero placeholder (each column can represent a factor of 10). The decimal location is defined by use, or can be marked on the frame (a low-tech option would be an elastic around the frame to indicate your decimal).
The earthly beads (lower deck) are counted up to reach 5, to continue counting one heavenly bead (upper deck) is pushed down to represent 5 and the remaining earthly beads are pushed back down and counted up once again to reach 10. The process is continued on to the next column.
(using the second row of beads in the upper deck is covered in step10).

Step 2: Tools + Materials

this pocket abacus was made from materials found at the dollar store:

  • needle-nosed pliers
  • ruler
  • pencil
  • drill (with thin bit)
  • marker
  • small picture frame
  • coloured glass beads
  • stiff wire
  • white card (or paper)
  • wood paint stick (length used dependent on size of frame)
  • bamboo skewer

Step 3: Drill

Remove the back of the frame along with backing and glass, then measure the inside opening dimension. We'll use this dimension to space our columns and cut the bar that divide the top and bottom decks of beads.

dividing bar:
Cut the paint stick to the the same dimension as the opening, trim stick to suit frame.

column spacing:
Select a drill bit that is about the same size as the wire used, then drill at even increments for the columns.
(here's an instructable on how to drill small precise holes)

for reference:
Your frame may differ, but from the materials I found here's the dimensions.
frame measurement:      10cm x 7cm (4" x 2.8")
opening measurement:   7.5cm (3")
accounting for gap required between rows ~5mm o.c = 14 rows (14 / 7.5 = 5.35)

Step 4: String Beads

The frame is made from a soft wood, so i was able to bend over the wire and use a friction fit to hold the end of the wire in place.

Thread on your beads, feed wire through centre bar, then thread on remaining beads to complete the column. Wrap the wire straight through the frame and loop it back down the next drilled hole and repeat the process.

Step 5: Backing

Since this frame has been modified, and there is no picture to be inserted, the inside backing of the frame is visible and has an unfinished look. A white backing was installed in place of a picture to give this abacus a clear backdrop to read our numbers.

Using the back of the frame as a guide, cut a piece of white card stock (or piece of paper) to the same size and install in place of picture. Then, seal up the back of the frame with the backing and stand.

Step 6: Stylus

Most abaci are large enough to operate with your fingers, my pocket abacus is not. This abacus requires a stylus to move the beads accurately.

A bamboo barbeque skewer was cut to the same length as the frame used and covered in black ink to match. The sharp end that this bamboo skewer came with was designed to pierce food and was too sharp for this purpose, the end was re-cambered then re-inked to black.

Step 7: How to Add + Subtract

Start by resetting the abacus (all beads away from centre bar).

Without markings on my abacus, the decimal place can be moved according to application requirements.
In this example, add the number 218.25 to 30.12.

To start, we tally 218.25 on the abacus, then simply add 30.12 counting up from right to left just like normal addition. Only beads closest to the middle are counted.
Adding the two we get 248.37

Let's move on to something more complex..

Step 8: How to Multiply

here's where the abacus comes into it's own.

multiplication vocabulary:
multiplicand : the number to be multiplied 
multiplier : the number to be multiplied by
(In multiplication these terms are generally interchangeable)
product: result of the multiplication

Let's take 2 larger numbers:
                                                    5286 x 654
The trick here is to count the digits in the equation, in this example there is 7 digits. This means we'll need 7 columns of beads to compute our answer. Reading right to left that puts us at the red column, this is where we'll start.

Now we take our top row and multiply it by our first multiplicand (5286 x 6). Our first calculation is 5 x 6 which we know is 30, this makes our 7th column (red) at +3 and the 6th column (yellow) left at +0. Continue over and take 2 x 6 which is 12, this makes our 6th row +1 bead and our 5th row +2 beads. Continue this process until the multiplier is exhausted.

Dropping the first digit of the multiplier there are now only 6 digits, this means we will start on 6th column over for the next multiplicand. Repeat the multiplication, adding beads in each column and carrying over the remainders.
When you exhaust the multiplicand the calculation is done.

We can verify this on a calculator and we see that it is correct.

Step 9: How to Divide

Dividing on the abacus is a little more difficult, however is really just multiplication in reverse.

division vocabulary:
dividend: number to be divided
divisor: number to be divided by
quotient: result of the division

As with multiplication, we can count the numbers in the equation to figure out which column to start on the abacus. With multiplication we added the multiplicand digits with the multiplier, with division we take away dividend from the divisor.

In this example:       
                                      8965 / 5
The dividend has 4 digits and the divisor has 1.
The trick with division on the abacus is to add 1 to the total to know how many columns we will use.
4 (dividend) - 1 (divisor) +1 (abacus rule of division) = 4, this is how many digits will be in our quotient.

With multiplying, I used the abacus to keep track of the multiplicand. In division, I only keep track of the divisor (on an empty left column) and remainders from division (on an empty right column). For this reason I use the middle of the abacus to compute.
Start from left to right dividing 5 into 8 which is 1, keep track of the remainders on an empty column on the right (in this case 3. Next take that remainder as the first digit and use 9 (the second number in our dividend) and you have 39, now divide 5 into 39 to get 7 remainder 1. Continue this process over until you exhaust your dividend and you will have your answer.

Step 10: How to Use for Weights

The Chinese abacus has an additional row on the top deck. This second row of beads are used to count in 16ths (hexadecimal). This allows the abacus to be used to tally weight in pounds and ounces.

There's 16 ounces to each 1 pound. We use the rows as before but instead of going to the next row over at a 10 count you count 15 in the same row, the 16th bead is counted on the next row over, indicating 1 pound.


Step 11: Final Thoughts and Further Readings

With practice, the abacus can be used as fast as a calculator.
Calling all nerds: there's even advanced techniques for more complex numbers and even square and cube root functions.

There are many tricks to the abacus to help with speed and simplify calculations, feel free to add your tips in the comments below.

have fun!
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