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

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. 1 Formulation of the Problem Traditional numerical systems use a single base. Most numerical systems are based on using a single number as a base. For example, in the decimal system, each natural number is represented by a sequence of decimal digits such as 2011, so that: • the last digit 1 means ones (i.e., multiples of 10); • the next digit means 10s (i.e., multiples of 10; • the next digit means 100s (i.e., multiples of 10); • the next digit means 1000s (i.e., multiples of 10),