How Things Work: The Abacus

This week, How Things Work is going to set the record straight on how to operate an abacus in the unfortunate event that your graphing calculator dies during your mathematics exam.
An abacus is a calculation tool that was commonly used by the Chinese as early as the year 3000 BCE and the Babylonians in 2400 BCE. The concept of a mechanistic counting device gained widespread appeal, giving rise to the Japanese abacus, the Roman abacus, the Greek abacus, the Russian abacus, the Danish abacus, the Native American abacus, and the Abacus for the Visually Impaired.
Unfortunately there is no Pittsburgh abacus, so the following directions will assume that the reader is acquainted with the popular Chinese model.
An abacus consists of a series of parallel rods held together by a rectangular wooden frame. A horizontal beam divides the frame into an upper deck and a lower deck. Each rod holds five beads ? one on the top deck and four on the bottom. The number of rods can vary from one abacus to the next, but most are equipped with at least 10. An abacus with 45 beads can thus be used to represent
numbers as high as 999,999,999 ? which is quite impressive.
Although abaci were primarily used to count merchandise and animals, this shouldn?t discourage the reader from using them to solve differential equations.
To set the abacus to zero, all of the beads are pushed towards the outer frame, exposing the central bar. Beads are ?counted? the moment they touch the central bar. The rightmost column of beads is designated the ones column. The adjacent column represents the tens column, the next is the hundreds column, and so on. Beads on the bottom deck have single digit values, while those on the top deck are given a value of five. The number ?1? is represented by sliding the top bead on the rightmost column ?up? to meet the horizontal bar. The numbers two, three, and four can be formed by joining one, two, or three beads, respectively, to the first bead. For the number five, the lone bead in the top deck is pushed down against the bar. Six, seven, eight, and nine can be formed by again raising beads on the bottom deck to meet the central bar.
In this manner, numbers get larger and larger as you progress further and further to the left. The number 999,999,999 can be formed by pushing all 45 beads against the central bar so that none of them are in contact with the outer frame.
Still, your abacus will be of no help on your astrophysics final unless you can perform higher calculations. To do this, it is necessary to learn a host of finger techniques.
On the Chinese abacus, addition is performed by moving beads ?up? with the thumb and ?down? with the index finger. Some complex operations require the index finger to move the beads up, however. Failure to switch fingers will ruin your calculation and offend the abacus-using community.
Theoretically, it is possible to perform almost any mathematical functon on the abacus. One can do multiplication, division, square root calculations, or cube root calculations. Those of you willing to master these techniques may want to read Lee Kai-Chen?s ?How to Learn Lee?s Abacus.? I can take triple integrals on abaci in my sleep, but it looks like I?ve run out of space to go into it here. I hope you remembered extra batteries.