The

Chinese abacus has 13

wires on a

wooden frame, with a wooden divider not quite down the
middle of all the wires. Each wire has seven

beads, five on one side of the division
(the "earth" beads, worth one each) and two on the other (the "heaven" beads, each of
which is worth five.)

The modern Japanese abacus has only one bead above the divider and four below, and usually
has 21 wires.

To represent a number on the abacus, beads are moved toward the divider. The rightmost
wire represents the ones, the next one over the tens; the third from the right stands
for hundreds, and so on. For example, here's a diagram of a Chinese abacus showing
54,927:

heaven beads
-----------------------------
| O O O O O O O O O O O O O |
| O O O O O O O O | O | O | |
| | | | | | | | | O | O | O |
-----------------------------
| | | | | | | | | | O O O O |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O | | |
| O O O O O O O O O O O O O |
| O O O O O O O O O | | O O |
| O O O O O O O O O O O O O |
-----------------------------
earth beads

Note that the beads furthest away from the divider don't need to be used to represent

base 10
numbers on the Chinese abacus; that's why they were able to be removed in the Japanese
abacus. The extra beads help, though, when you are doing

math. Also, you can use the
Chinese abacus to represent (and do

calculations with) numbers other than base 10.
E.g., you can do

binary calculations by working just with the heaven beads. You can
also slide the

decimal point around, and do

arithmetic with

fractions if you so desire.

To

add numbers with an abacus, enter the first number. Suppose we wanted to calculate
54,927 + 18. Start with the bigger number:

-----------------------------
| O O O O O O O O O O O O O |
| O O O O O O O O | O | O | |
| | | | | | | | | O | O | O |
-----------------------------
| | | | | | | | | | O O O O |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O | | |
| O O O O O O O O O O O O O |
| O O O O O O O O O | | O O |
| O O O O O O O O O O O O O |
-----------------------------

Then go column by column, adding the second number to the first. It's best to work
from the right to the left. There aren't enough beads in the ones column to add
8 to 54,927, so instead add one to the tens ...

-----------------------------
| O O O O O O O O O O O O O |
| O O O O O O O O | O | O | |
| | | | | | | | | O | O | O |
-----------------------------
| | | | | | | | | | O O O O |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O **O** | |
| O O O O O O O O O O O | O |
| O O O O O O O O O | | O O |
| O O O O O O O O O O O O O |
-----------------------------

... and take away two from the ones:

-----------------------------
| O O O O O O O O O O O O O |
| O O O O O O O O | O | O | |
| | | | | | | | | O | O | O |
-----------------------------
| | | | | | | | | | O O O | |
| O O O O O O O O O O O O **O** |
| O O O O O O O O O O O O **O** |
| O O O O O O O O O O O | O |
| O O O O O O O O O | | O O |
| O O O O O O O O O O O O O |
-----------------------------

Now, add one to the tens column (the "1" in "18"):

-----------------------------
| O O O O O O O O O O O O O |
| O O O O O O O O | O | O | |
| | | | | | | | | O | O | O |
-----------------------------
| | | | | | | | | | O O O | |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O **O** O |
| O O O O O O O O O | | | O |
| O O O O O O O O O O O O O |
-----------------------------

... and we are left with 54,945.

Subtraction is just like addition, but in reverse. To calculate 54,945 - 290, start
with 54,945:

-----------------------------
| O O O O O O O O O O O O O |
| O O O O O O O O | O | O | |
| | | | | | | | | O | O | O |
-----------------------------
| | | | | | | | | | O O O | |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O O |
| O O O O O O O O O | | | O |
| O O O O O O O O O O O O O |
-----------------------------

Subtracting 0 is a

no-op; to subtract 90, first subtract 100 ...

-----------------------------
| O O O O O O O O O O O O O |
| O O O O O O O O | O | O | |
| | | | | | | | | O | O | O |
-----------------------------
| | | | | | | | | | O O O | |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O O |
| O O O O O O O O O O | O O |
| O O O O O O O O O | **O** | O |
| O O O O O O O O O O O O O |
-----------------------------

Then add 10:

-----------------------------
| O O O O O O O O O O O O O |
| O O O O O O O O | O | | | |
| | | | | | | | | O | O **O** O |
-----------------------------
| | | | | | | | | | O O | | |
| O O O O O O O O O O O **O** O |
| O O O O O O O O O O O **O** O |
| O O O O O O O O O O | **O** O |
| O O O O O O O O O | O **O** O |
| O O O O O O O O O O O O O |
-----------------------------

Now, subtract 200 to get the answer (54,655)

-----------------------------
| O O O O O O O O O O O O O |
| O O O O O O O O | O | | | |
| | | | | | | | | O | O O O |
-----------------------------
| | | | | | | | | | O O | | |
| O O O O O O O O O O | O O |
| O O O O O O O O O O **O** O O |
| O O O O O O O O O O **O** O O |
| O O O O O O O O O | O O O |
| O O O O O O O O O O O O O |
-----------------------------

To perform multiplication with an abacus, basically you perform addition repeatedly.
In order to keep track of how many times you have added the number, you can use one of the
unused leftmost columns of beads. This works well with smaller numbers. If you want to

multiply
numbers that have multiple

digits, then memorize your

multiplication table and do it
like you would on paper. E.g., 37 times 23 would be 21 + 90 + 140 + 600.

Division can be performed as repeated

subtraction, using one of the leftmost columns
to keep track of how many times the subtraction has been performed. You have to decide
before you start how many

significant digits will be allowed in your answer. For example,
to compute 19/17 to the nearest hundredth, declare the ones column to be third from the
left, with tenths and hundredths in the right two columns. Enter 19:

-----------------------------
| O O O O O O O O O O O O O |
| O O O O O O O O O O | O O |
| | | | | | | | | | | O | | |
-----------------------------
| | | | | | | | | | O O | | |
| O O O O O O O O O | O O O |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O O |
| O O O O O O O O O O | O O |
| O O O O O O O O O O O O O |
-----------------------------

Subtract 17 and

increment the far left column as a counter:

-----------------------------
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O O |
| | | | | | | | | | | | | | |
-----------------------------
| O | | | | | | | | | O | | |
| | O O O O O O O O O O O O |
| O O O O O O O O O O | O O |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O O |
-----------------------------

We are left with 2; you can't subtract 17 from 2 again. But, we could imagine
2 is 20 by using the

tenths column. Do this, and instead of using the leftmost
column as your counter, use the next one over:

-----------------------------
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O O |
| | | | | | | | | | | | | | |
-----------------------------
| O O | | | | | | | | | O | |
| | | O O O O O O O O O O O |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O | O |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O O |
-----------------------------

We are left with 3 in the tenths column; you can't subtract 17 from 3, so scoot over
again, and imagine the

hundredths are the ones and the tenths are the tens. Subtract
17, and use the THIRD column at the left as your

counter:

-----------------------------
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O O |
| | | | | | | | | | | | | | |
-----------------------------
| O O O | | | | | | | | O O |
| | | | O O O O O O O O | O |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O | |
| O O O O O O O O O O O O O |
| O O O O O O O O O O O O O |
-----------------------------

We are left with 13, and you can't subtract 17 from 13, but 13 is more than half of 17,
so we can probably round up the last digit of our answer. Reading the counters on the
left, we come up with an answer of 1.11, rounded up to 1.12.

The analogy to normal long division with pencil and paper should be evident.