How to Use a Chinese Abacus

Date: 06/24/2004 at 21:16:05
From: Carlos
Subject: how to use Chinese abacus

Can you tell me how to use a Chinese abacus?  Thanks!

Date: 06/26/2004 at 16:43:10
From: Doctor Katy
Subject: Re: how to use Chinese abacus

Hi Carlos!

Thanks for writing to Dr. Math.

The first step in knowing how to use an abacus is knowing how to 
"read" an abacus.  I hope that you have an abacus yourself, because 
trying examples and playing around with it a little will make learning 
how to use it much, much easier.

The abacus consists of 13 columns, each one divided into an upper deck 
and a lower deck; the lower deck consists of 5 beads per column, the 
upper deck has 2 beads per column.  I included a little diagram, I
hope you can make sense of it:  each "o" represents a bead.  Note 
there is always space for you to move some of the beads away from the
others.

Diagram of a typical abacus

---------------   -
|             |    |
|ooooooooooooo|    |  upper deck
|ooooooooooooo|   -
|-------------|
|             |   -
|ooooooooooooo|    |
|ooooooooooooo|    |
|ooooooooooooo|    | lower deck
|ooooooooooooo|    |
|ooooooooooooo|   -
---------------
 ^^^^^^^^^^^^^
 the 13 columns

Each column represents a digit, so with 13 columns you can do 
calculations with numbers up to 10 trillion!

The way to express a number is to move up the beads of each column to 
represent a digit, starting from the right.  The value of a bead in
the lower deck is 1, the value of a bead in the upper deck is 5.  For 
example, to express the digit 7, you would move up two beads in the 
lower deck and one bead in the upper deck, so that (1+1) + (5) = 7. 
Make sense?

Knowing that, this is what the number 237 would look like:

---------------
|            o|
|oooooooooooo |
|ooooooooooooo|
|--------------
|          ooo|
|ooooooooooooo|
|oooooooooo o |
|ooooooooooo o|
|ooooooooooooo|
|ooooooooooooo|
---------------
           237


You might notice that each column can take on a value from 0-15 and 
think it's silly because in our number system, each digit only takes 
on values from 0-9.  However, this becomes handy when you need to
carry digits that exceed 10, just like you do when doing addition with
pen and paper.  Also, it allows you to do calculations with other
number systems that are not 10-based, if you want to.

How about another example?

---------------
|            o|
|oooooooooooo |
|ooooooooooooo|
|--------------
|          ooo|
|ooooooooooooo|
|ooooooooooo o|
|ooooooooooooo|
|oooooooooooo |
|oooooooooo oo|
---------------
           529


Notice that the digit 5 in the third column can alternatively be 
expressed by moving up one bead in the upper deck instead of the five 
beads in the lower deck.

If all this starts making sense, I think you can already see how using 
an abacus can be useful for simple arithemtic.  However, the abacus is 
only useful if one can "read" it quickly enough and do the simple 
additions in your head (namely 1 upper bead and 4 lower beads = 1*5 + 
4*1 = 9), but that really isn't hard to learn.

Now, to do something like 625 + 536, first put down the number 625. 

---------------
|          o o|
|oooooooooo o |
|ooooooooooooo|
|--------------
|          oo |
|oooooooooo oo|
|ooooooooooo o|
|ooooooooooooo|
|ooooooooooooo|
|ooooooooooooo|
---------------
           625

Then you proceed to add each digit, starting from the right.  To add 
the 6 (from the number 536), simply add 1 bead to the upper deck and 1 
bead to the lower deck (since 1*5 + 1*1 = 6).  Next, to add the 3 to 
the second column, just move up 3 beads from the lower deck (1*3 = 3). 
To add the 5, add one bead from the upper deck. Your final result will 
look like this:

---------------
|          o o|  <-- Notice that whenever you have 2 beads up in the
|ooooooooooooo|      upper deck, you can move them down and add one 
|oooooooooo o |      from the lower deck in the next column.  If you
|--------------      don't, you will have numbers greater than 10 in 
|          ooo|      a column, which is confusing.
|oooooooooo o |
|ooooooooooooo|
|ooooooooooooo|
|ooooooooooooo|
|ooooooooooo o|
---------------

This simplifies to:

---------------
|           o |
|ooooooooooo o|
|ooooooooooooo|
|--------------
|         oooo|
|ooooooooo    |
|ooooooooooooo|
|ooooooooooooo|
|ooooooooooooo|
|ooooooooooooo|
---------------
          1161  

It looks tedious, but with practice it becomes much easier (1.3
billion Chinese can't be wrong!).

So what the abacus does for you is that it "holds" the first number 
for you while you're adding the second one digit by digit.  When you 
want to do addition or subtraction problems, all you're doing is 
shifting beads up and down each column (i.e. digit) at a time, so that 
even 23509725 - 9438558 isn't too difficult to do. 

There are lots more "tricks" that one should know to be able to do 
more addition and subtraction problems, but they are hard to explain 
without showing.  But just for your information, calculations such as 
multiplication, division, and even taking the square/cube root are all 
possible, but much more difficult.

For further information on this and many other things, you can visit a 
website dedicated entirely to the abacus:

  Abacus - The Art of Calculating With Beads
    http://www.ee.ryerson.ca:8080/~elf/abacus/ 

Also, you can look at another response in our archive given by a 
different math doctor.

  Using an Abacus
    http://mathforum.org/library/drmath/view/59147.html  

Feel free to write back if you have any further questions or need 
clarifications on anything.

- Doctor Katy, The Math Forum
  http://mathforum.org/dr.math/