Mayan and Babylonian Arithmetics Can Be Explained by the Need to Minimize Computations

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Most number systems use a single base – e.g., 10 or 2 – and represent each number as a combination of powers of the base. However, historically, there were two civilizations that used a more complex systems to represent numbers. They also used bases: Babylonians used 60 and Mayans used 20, but for each power, instead of a single digit, they used two. For example, a number 19 was represented by the Babylonians as 19B = 1 · 10 + 9 and by the Mayans as 34M = 3 · 5 + 4. In this paper, we show that such a representation is not just due to historic reasons: for the corresponding large bases, such a representation is actually optimal – in some reasonable sense.