Call me old fashioned but today I was wondering how an abacus works. Yes I mean those old slates with rods and balls that used to be calculators...Anyway lets get down to business.
In an abacus today, each rod to the left is to the power of ten, as in our decimal mathematics system, because there are ten beads on each rod. . This simple decimal abacus operates like this, using the example of starting with 458 and intending to add 247:
First of all the 458 has four beads at the bottom of the 100 rod, five beads at the bottom of the tens rod and eight at the bottom of the units rod. Of the 247 to add, the calculation starts with the units at the right. Seven is added to the eight at the bottom, but because there were only two left at the top, only two can go downwards:
There are five still to add. The solution is that five, instead of coming down, must go up (including the two that came down).
However, because five go up, there is a price to pay. One bead must come down in the tens rod before the calculation continues. This completes the units calculation, as below:
Having added seven of the 247, now is the time to add 40. This means four beads of the tens rod (that is 40) must come down. However, something odd happens. All the beads are down and although it might be thought that this completes the movement, it does not...
This is crucial. The above cannot be allowed. All the beads cannot stay down, because this is the next power up (10 X 10 is 100) and it shows that the all important Indian to Arab to the West and subsequently to the world zero is those ten beads when together. This is the design fault in the abacus, or rather in our interpretive needs, discussed at the end. We need zero, and so all ten beads must go up, leaving a gap - nothing at all - or the zero, as shown:
The price to pay, before the calculation can move on, is a bead then must drop in the hundred column. So this happens:
Finally, with the forty seven added, the remaining two hundred can be added by dropping down two beads on the hundred power rod:
The result is seven beads down on the first rod, or seven hundred, no beads down on the second rod, and therefore zero tens, and five beads down on the last rod, representing five. This is 705 and the calculation is successful.
The abacus therefore shows a need for rules of movement:
Beads go down.
Any remaining beads then go up.
The zero rule demands that if all beads go down they must also then all go back up (this additional rule is needed because, just as the Romans thought with numbers, the abacus deals with solid objects; so when after all have gone down and none it seems need to go up, they in fact all need to go up thus giving zero a visual property for us: to appear as completely nothing).
If any beads go up, one must drop down in the next rod to the left.
The abacus therefore changes arithmetic into counting and movement:
After reading through the explanation I hope you realize that using an abacus is quite similar to doing arithmatic the traditional way we learned it, by writing it down or visualizing it in your hear. Or said correctly, doing arithmatic our traditional way resembles the abacus because after all this age old calculator is the original. I hope you enjoyed and will follow me