The ancient Aztec civilization employed unique units of measurement that were integral to their daily life, surveying, and tribute systems. These units—such as arrow, heart, hand, bone, and arm—were considered indivisible and served as fundamental measures in their mathematical calculations. Modern scholars speculate that an 'arrow' might represent the distance from the shoulder to the hand, akin to the length of a taut bow. A 'heart' could denote the measurement from that organ to the tip of the hand, while a 'hand' likely refers to the span from outstretched hand to outstretched hand, similar to what we recognize today as an 'arm's length.' These units formed the basis of Aztec measurements, especially in land surveying and resource allocation.
The Aztec surveyors and mathematicians developed at least five distinct formulas to ascertain the areas of irregularly shaped plots of land. Their methods included averaging the lengths of opposite sides and then multiplying these averages to estimate the area, a technique that showcases their understanding of geometric principles. In more complex cases, they would divide irregular fields into triangles, measure each segment precisely, and sum these measurements to calculate total areas. These sophisticated strategies highlight the Aztecs' practical approach to geometry, essential for agriculture, construction, and taxation.
Beyond measurements, the Aztecs also had a rich symbolic system to represent numbers, crucial for recording tribute payments and administrative purposes. For example, a bundle of canes tied with an incense bag was used as a glyph symbolizing the number 8,000, indicating the symbolic importance of quantities and the use of everyday objects for numerical representation. Other symbols included a banner or flag to denote twenty, and dots or circles to represent single units—either as fingers or marks—highlighting their visual approach to counting.
Their numerical language, called Nahuatl, included simple names — 'ce' for 1, 'ome' for 2, 'yei' for 3, and 'nahui' for 4. Numbers up to twenty could be depicted by dots or fingers, providing a straightforward visual counting system. The Aztecs also used bars, resembling a line, to indicate the value of five. This simple yet effective method facilitated counting in daily life and official records.
Larger numbers involved more complex symbols. For instance, a feather represented 400, emphasizing symbolic significance over mere quantity. From 20 to 400, Aztecs combined symbols—such as flags or banners, each representing twenty units—and added dots to fine-tune their totals. To express larger figures, they multiplied these units; notably, 20 times 400 equaled 8,000, symbolized by an incense bag or pouch called 'xiquipilli' in Nahuatl. The incense bag symbolized the vast quantities of resources, such as cacao beans, which the Aztecs counted by number and volume, not weight. This system underpins the Aztecs’ advanced understanding of large numbers and their practical application in tribute collection and resource management.
Overall, Aztec mathematics was deeply intertwined with their daily life, religious practices, and governance, demonstrating an innovative approach to measurement, calculation, and symbolic representation that continues to fascinate scholars today. Their methods reflect a practical, visually driven numerical system that supported their expansive empire and complex societal structure.