# Maya numbers and arithmetic

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As early as the Late Preclassic (ca 400 BC), the ancient Maya used a positional, vigesimal (base 20) notation involving the use of the mathematical concept “zero”, a truly great intellectual accomplishment and one of the earliest known instance of this concept in the world (the Europeans never invented the “zero”, they borrowed it from the Arabs, who themselves borrowed it from Indus Valley Civilization around the 7th century AD).

The Maya system of numeration operates with only three symbols, alone or combined, rather than the 10 digits (0-9) of our decimal system:

• “one” is expressed by a dot,
• “five” by a bar,
• “zero” (or “completion”) by a variable symbol (commonly a shell or a flower).

In this bar-and-dot notation, three dots stand for “three”, a bar for “five”, and three dots with a bar stand for “eight”. “Nineteen” is represented by four dots and three bars. To write digits greater than “nineteen”, the ancient Maya used a positioning system.

In the decimal (base 10) numeric notation, numbers higher than 9 are written by adding digits in the next position to the left: 9 then 10, 99 then 100, 999 then 1000, etc.

Numbers in higher places grow multiplied by 10´s (possibly because we have 10 fingers): 1, 10, 100, 1000, 10000, etc.

“Zero” acts like a place holder, that is, in number “10” there are one Tens and zero Ones.

The number “3257” can be considered as 3 Thousands, 2 Hundreds, 5 Tens, 7 Ones, that is 3×1000 + 2×100 + 5×10 + 7×1 or 3×10^3 + 2×10^2 + 5×10^1 + 7×10^0.

In the vigesimal positional system, which was in use throughout Mesoamerica during Precolumbian times, the shift is made at 20 with the lowest to highest positions being placed vertically and bottom-to-top, rather than horizontally and right-to-left as in the decimal scheme.

Numbers in higher places grow multiplied by 20´s (probably from a total of 20 fingers and toes), that is: 1, 20, 400, 8000, 160000, etc.

In Maya numeric notation, “3257” would thus be: eight 400s, two 20s, and seventeen 1s, that is 8×400 (=3200) + 2×20 (=40) + 17×1 (=17) or 8×20^2 + 2×20^1 +17×20^0. For chronological count, however, the ancient Maya had to compromise the harmonic vigesimal rule to create a closer approximation to the length of the actual solar year which they knew was 365+ days. So, rather than having a year with twenty months each with twenty days, totaling 400 days, they used a 360-day year with eighteen months of twenty days.

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