Amazing Aztec Arithmetic

The Mayans can't take all the credit for zero. It turns out the Aztecs were math whizzes too.

Not only did the Aztecs have a symbol for zero, it appears that their mathematical understanding of sophisticated algorithms for calculating area, for example, was quite advanced. In addition to using both picture symbols and lines and dots for numerical notation, the Aztecs used a base-20 number system, with lines for ones and dots for 20s. Forty four, as such, would be symbolized by 2 dots and 4 lines (take a look at the image above and you'll notice that the plots have measurements indicated on each side with dots and lines).

"Positional line-and-dot notation was used to record areas of agricultural fields, and analysis of the documentary data suggests that areas were calculated arithmetically, " according to recent findings by mathematician Maria del Carmen Jorge y Jorge and geographer Barbara Williams in their study Aztec Arithmetic Revisited: Land-Area Algorithms and Acolhua Congruence Arithmetic.

The Aztecs even had units to represent fractions. "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."

According to Williams, "They used the four mathematical operations of addition, subtraction, multiplication and division. But in almost all of the early societies, they could do everything they needed to do, with just those four. They didn't need square roots. They didn't need trigonometry."

"We found these smaller units of measure that we call monads that have the role of a fraction," she says. "We don't like to call them fractions, though, because they were considered as unitary entities like inches, seconds or minutes."

To denote half the Aztec basic unit of measure—known by Aztec experts as tlalquahuitl or land rods—the surveyors used an arrow symbol. So for a field that measured 20 land rods by 10 land rods plus an arrow (or 20 multiplied by 10.5), the correct area was 210. "Two arrows is one unit, five hearts is two units, five hands is three units," del Carmen Jorge y Jorge notes.

The ability to accurately measure land was critical for levying the proper tax or tribute. They recorded only the total area of each parcel and the length of the four sides of its perimeter, Jorge y Jorge explained.

"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 ancient texts were extremely detailed and well organized, because landowners often had to pay tribute according to the value of their holdings." Officials calculated the size of each parcel using a series of five algorithms—including one also employed by the ancient Sumerians—she added.

Absolutely fascinating stuff.

Aztec Math Used Hearts and Arrows
How big is a heart? Two and a half land rods of course, according to the Aztecs
By David Biello

Aztec Arithmetic Revisited: Land-Area Algorithms and Acolhua Congruence Arithmetic.
B. J. Williams and M. d. C. Jorge y Jorge (2008)Science 320, 72-77

Aztec Math Decoded, Reveals Woes of Ancient Tax Time

Brian Handwerkfor National Geographic News
April 3, 2008

Aztecs were whizzes at math
Early civilization used hand, heart, arrow symbols to represent distances

By Clara Moskowitz

Aztecs devised sophisticated arithmetic system, Reuters

By Will Dunham

Landing in the garbage with a thud...

This week, after two years of deliberation, the National Mathematics Advisory Panel released their report aimed at improving math education in this country. And you could almost hear the sound of textbooks--that heavy one in your kid's backpack, and a stack of high-stakes math tests, the kind your kid take every year--landing in the garbage can with a thud.

Wow. That's a little harsh. I think that all those Everyday Math books and pitiful standardized tests that mirror them still have some value. They should at the very least be recycled into usable paper products. That way, trees would not have died in vain and all that garbage won't take up space in our nation's landfills.

*reduce * reuse * recycle

Calculating a New Approach
A report on math education fuels the debate about the Singapore model. What is it--and would it work here?

Peg Tyre Newsweek Web Exclusive
Mar 14, 2008 Updated: 4:33 p.m. ET Mar 14, 2008

Reading between the lines...

Any approach that continually revisits topics year after year without closure
is to be avoided.

really means...

Everyday Math
is to be avoided.