Chinese Rod-Numeral System

Date: 11/09/98 at 22:35:23
From: Carmel Ruby
Subject: Chinese rod-numeral system

How exactly does the Chinese rod-numeral system work?

Date: 11/13/98 at 23:38:44
From: Doctor Ameis
Subject: Re: Chinese rod-numeral system

Hello Carmel,

The Chinese rod-numeral system was a base 100 system. It had a ones 
place, a hundreds place, a ten-thousands place, and so on. However, the 
rod-numeral system was not a complete place value system in that it did 
not make use of a symbol for zero. Zero was shown by the absence of 
rods in a counting board compartment. Also, the rod-numerals were used 
for doing arithmetic, not for writing purposes. The rods were used to 
represent positive and negative numbers. Black rods were used to form 
negative numbers and red rods were used to form positive numbers (today 
we associate negative with the color red, not with the color black).

The rods were carried in a pouch and placed on a counting board which 
had compartments corresponding to the ones place, the hundreds place, 
and so on. Each compartment was split into two parts. The right part 
for the heng rods (see below) and the left part for the tsang rods 
(see below). 

There were two types of rod numbers: the hengs (from 1 to 9) and the 
tsangs (from 10 to 90). Whole numbers were represented by combining 
these two types of rods.

Here are the heng rods:

   1     |
   2     ||
   3     |||
   4     ||||
   5     |||||
         ___
   6      |
         ____
   7      ||
         ___
   8     |||
         ____
   9     ||||
            

Here are the tsang rods:

   10     ___
   
          ___
   20     ___

          ___
   30     ___
          ___

          ___
          ___
   40     ___
          ___

          ___
          ___
   50     ___
          ___
          ___

   60      |
          ___
 
           |
   70     ___
          ___

           |
   80     ___
          ___
          ___

           |
   90     ___
          ___
          ___
          ___

Let's see how 3614 would be represented. Because the rod system is 
based on hundreds, we need to look at 3614 in terms of ones, hundreds, 
etc. 3614 is made up of 36 hundreds and 14 ones. This means that one 
group of rods is used to represent 14 and another group is used to 
represent 36. It would look like this (the ones place is on the right 
and the hundreds place is to the left of the ones):

   ___            
   ___ ___   ___ ||||
   ___  |


Note how the heng rods are to the right of the tsang rods. That is how 
it would look on the counting board within each compartment (split into 
heng and tsang parts).

Let's see how 735 would be represented. It has 7 hundreds and 35 ones.  
735 would look like this:

   ____   ___ 
    ||    ___ |||||
          ___
  
Note that there are no tsang rods needed in the hundreds place (because 
7 is less than 10).

After the rod-numerals were placed on the counting board in the 
appropriate compartments (ones, hundreds, etc.), they were then 
manipulated by repositioning and reforming them as required for doing 
the arithmetic. A simple example follows. You might want to use 
cuisenaire rods to form the rod-numerals and then manipulate them to 
do the arithmetic.

Suppose you are adding 17 and 64. First, each number is represented by 
placing the appropriate rod-numerals on the counting board. This is 
what it would look like on the counting board in the ones compartment 
(in this case, only the ones compartment is involved because we are 
adding 17 ones and 64 ones).

   ___  ____   (17 ones)
         ||

    |    
   ___  ||||   (64 ones)
     
Now the arithmetic thinking begins. All work is done mentally with the 
rods serving as a memory aid for what is currently the situation. 
(Isn't this also the case when doing pencil and paper arithmetic? The 
actual arithmetic is done mentally. The stuff written on the paper 
serves as a memory aid.)

Combining the vertical rods in the heng part we obtain six of them. 
Five of these six vertical heng rods trade for one horizontal heng rod. 
This leaves one vertical rod remaining. But now there are two 
horizontal heng rods. These two rods trade for one horizontal tsang rod 
(5 + 5 = 10). This leaves us with only one heng rod - the leftover 
vertical one - in the heng part of the compartment.

Now there are three horizontal tsang rods and one vertical tsang rod in 
the tsang part of the ones compartment. No trading is required for this 
situation. The final result would look like this.

    |    
   ___   |  (81 ones)
   ___
   ___

Hope this helps. Feel free to ask for further assistance with this.

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