beads. Each bead on the upper deck has a value of five, while each bead on the lower
deck has a value of one (thus, it is called a 1/5 abacus). To use the abacus, users place
the abacus flat on a table or their laps; they then push all the beads on the upper and
lower decks away from the horizontal beam. From there, the beads are manipulated,
usually with the index finger or thumb of one hand, to calculate a problem. For exam-
ple, if you wanted to express the number 7, you would move two beads in the lower
deck and one bead in the upper deck: (1 1) 5 7.This modern abacus is still used by shopkeepers in Asia and many so-called “Chi-
natowns” in North America. Students continue to be taught how to use the abacus in
Asian schools, especially to teach children simple mathematics and multiplication. In
fact, it is an excellent way to remember multiplication tables and is useful for teaching
other base numbering systems, because it can adapt itself to any base. (For more
about base numbers, see “History of Mathematics” and “Math Basics.”)What is a khipu?
Khipus (or quipu, in Spanish) were used by the Incas of South America. A khipu is a
collection of knotted strings that record certain information. The approximately 600
surviving khipus use an arrangement of knotted strings hanging from horizontal
cords. But these knots are nothing like those made by other cultures: They include
long knots with four turns, single knots, figure-eight knots, and a whole host of other
knot types. Historians believe these strings and knots represent numbers once used
for accounting, inventory, and population census purposes.There are also researchers who believe the khipus may contain certain messages in
some sort of code—a kind of language used by the Incas—based on the strings, knots,
350 and even a khipu string’s type (usually alpaca wool or cotton) and color. But it may turn
What is the world’s smallest abacus?
I
n 1996 scientists in Zurich, Switzerland, built an abacus with beads being
replaced by individual molecules that all had diameters of less than one
nanometer, or one millionth of a millimeter. The beads of the world’s smallest
abacus were not moved by a mere finger, but by the ultrafine, conical-shaped
needle in a scanning tunneling microscope (STM). The scientists succeeded in
forming stable rows of ten molecules along steps just one atom high on a copper
surface. These steps acted like the earliest form of the abacus (with grooves
instead of rods keeping the beads in line). Individual molecules were then
pushed back and forth in a controlled way by the STM tip, allowing the scientist
to manipulate the molecules and “count” from 0 to 10.