April 3, 2008

Amazing Aztecs Were Math Whizzes Too

Long known for their cool circular calendars and practice of
human sacrifice, Aztecs were also math whizzes.

Aztecs used hand, heart and arrow symbols to represent
fractional distances when calculating areas of land, scientists have
discovered. The researchers pored over Aztec agricultural manuscripts trying to
understand how the indigenous people arrived at area calculations. Only when
they factored in the pictorial glyphs did the figures make sense.

The term Aztec refers to certain ethnic groups that were
socially and politically dominant in Central Mexico in the 1300s through the
early 1500s.

Geographer Barbara Williams of the
University of Wisconsin, Rock County, and mathematician
Maria del Carmen Jorge y Jorge of the Universidad Nacional Autónoma de
Mexico analyzed two documents that describe the agricultural
properties owned by households in the city-state of Tepetlaoztoc from about
1540 to 1544.

"These are the only prehispanic documents that contain
perimeter and area data, as far as I know," del Carmen Jorge y Jorge said.
"Most of the documents from this time were lost."

The researchers tried to reproduce the area calculations and
initially had trouble. When they realized that the arrow, hand, and heart
drawings represented ground distances, they were finally able to arrive at the
same calculations as the

"The system they use to record the areas is
puzzling," del Carmen Jorge y Jorge told LiveScience. "We were very excited to see that actually using
the [symbols] we were able to recover many of the areas."

Each symbol stood for a distance that was less than the
standard distance unit, called a land rod.

"These arrow, heart, and hands were similar to what we
would now call fractions," del Carmen Jorge y Jorge said. "We call
them units of measure smaller than the length unit. Suppose if you have inches.
An inch isn't just part of a foot, but a measure unit itself. Think of these
heart, hand, and arrow as individual units."

The scientists describe their findings in the April 4 issue
of the journal Science. The research was supported by
the Departamento de Matemáticas y Mecánica at Instituto de Investigaciones en
Matemáticas Aplicadas y en Sistemas-UNAM.

The Aztecs had their own form of arithmetic. They used a
base-20 number system, and designated ones with lines and 20s with dots. For
example, 23 would be symbolized by one dot and three lines.

The land holding documents were originally written for tax
purposes, the researchers think. The amount of money the indigenous people were
asked to pay to the Spanish conquerors was most likely determined by how much
property they had.

"The Spanish conqueror who was in charge of the town
was asking a lot of tribute," del Carmen Jorge y Jorge said. "The
Indians wanted to prove in Spanish court that they were not able to pay the
tribute the Spanish were asking."

The type of mathematics the Aztecs used to calculate land
holdings seems to be consistent with their calendar mathematics, which are more
well-known, Williams said.

"This fits very nicely with the mathematics
we are more familiar with, which is the mathematics of calendrics,"
Williams said. "They coordinated the solar calendar with the lunar
calendar with Venus cycles. They did it proportionally, one cycle with

The ability to make calculations using proportions was
widespread across cultures by that time.

"The Maya
had this also," Williams said. "Also we see it in the counting of
people and objects. This all fits in very nicely."