Some people collect stamps or coins. I collect abaci (or, if you prefer, abacuses). My current collection includes ten abaci, most made of wood, but a few formed from ivory and jade. But enough about my own personal love of the abacus. I am more interested in explaining how to use one of these “ancient Chinese calculators.” So read on if you’re ready to throw out those four-function calculators that always seem to need fresh Energizer batteries and instead adopt this older (but oh so much cooler) bead arithmetic tool.
General Abacus Info
The standard Chinese abacus consists of a wooden frame which is divided into 2 sections by a horizontal bar. The abacus has columns containing 7 movable beads, with 2 beads in each column located above the horizontal bar (altobeads) and the remaining 5 beads below the divide (hypobeads). One altobead is the equivalent of 5 hypobeads.
Different models of the Chinese abacus can vary in how many columns of beads are contained in the abacus frame. For this post I will be demonstrating with an abacus with 13 columns. However, you can use an abacus with fewer columns and still do the same arithmetic problems that I do. This is because the value of any bead on the abacus depends on which column is designated as the unit column. Here I will assume that the rightmost column of my abacus is the unit column. This means that hypobeads in this column are each have value 1 and each altobead in this column will have value 5. As we move from right to left, the value of beads in subsequent columns increase by a factor of 10. For instance, assuming (as I just said above) that the value of each hypo- and altobead in the rightmost column are 1 and 5, respectively, this implies that each hypobead in the next column (to the left) has value 10, and each altobead has value 50. Think back to 2nd grade when you learned about the “one’s place”, “tens place”, “hundreds place,” etc. Columns of the abacus are thought of in this same way.
Represent an Integer Using the Abacus
To set your abacus to “neutral” (i.e. the Clear screen on your calculator), place all altobeads against the upper part of the frame and adjust all hypobeads so they are stacked against the bottom of the frame (see above picture). The middle horizontal beam acts as the axis along which the active beads are placed. For instance, if you want to represent the number 3 on your abacus, start from the neutral mode and simply push 3 hypobeads from the units column upward until they touch the middle beam.
Suppose you wish to represent the number 15 on your abacus. Then you need to have 5 represented in the units column and 10 represented in the tens column. You can achieve this in one of two ways. The first option is to move all 5 hypobeads in the units column and 1 hypobead in the tens column from neutral to the middle bar. This gives 15.
As an alternative, recall that 1 altobead in the units column represents 5. On your abacus simply adjust the position of 1 altobead in the unit column and 1 hypobead in the tens column and you will have the number 15.
Addition With the Abacus
To add integers using the abacus we start by placing the first integer on the frame by using the instructions provided above. Then we manipulate beads to add the second integer. Before I show an example let me just mention that one important distinction between addition using bead arithmetic and using pencil and paper to solve an addition problem is that in pencil arithmetic we begin to add at the rightmost column and move left. An addition problem with the abacus starts with the left-most column and moves right.
Now let’s do an example. Suppose I want to compute 24 + 12 with my abacus. I start with a representation of the number 24 where 2 hypobeads are raised to the middle bar in the second column (represents 20) and 4 hypobeads from the units column are arranged similarly (represents 4).
To add 12 we start with the tens column and add 10 by shifting a single hypobead up to the middle bar. Now we have represented the number 34 by adding 24 + 10.
What is left is to add 2 to the unit column. Looking to the unit column, however, we notice that there is only 1 hypobead in neutral position. We remember that 1 altobead is equivalent to 5 hypobeads, so we can add 2 to the 4 that are already in the units column (to get 6) by shifting 1 alto bead down to the middle bar to get 5 and leave only 1 hypobead away from neutral. Now, looking at the new number that results from our number shifting we can read off an answer of 36!
Subtraction With the Abacus
Subtraction using the abacus is similar to the process of addition, where you start with your first number represented by the beads and work the columns left to right to subtract the other number and determine a final answer. If you can follow the addition example from above, then you should be capable of reversing the procedure (36 - 12) to get 24!
Coming Soon: Multiplication and Division
Multiplication and division using the abacus are far more complicated than the simple processes of addition and subtraction. Check back soon for my detailed description of these processes.
For more information on using your Chinese abacus look here
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