Vigesimal numeration is a numeral system in which all derived units are based on the number 20 and the powers of 20. Vigesimal is derived from Latin word vicesimus (twentieth), based on viginti (twenty) that itself descended from a Sanskrit word, vimsatih (twenty). Other related words are vicennial (once every 20 years) and vicenary that has the same meaning as vigesimal. The use of 20 as a grouping (or base) number was used by many cultures throughout our human history (most likely) because people have twenty digits, or the number of fingers and toes.
The Aztecs and Mayans used a base 20 number system as did almost all Eskimo tribes, some native North American societies, almost all peoples native to Central and South America, and some cultures in northern Siberia and Africa. Mayan mathematics brought about the most sophisticated counting system ever developed in the Americas. Mayan numerical systems originated in concepts inherited from the Olmecs, an ancient civilization from Mexico and the adjacent Central American region. The Mayan system was a positional number system with a quasi-vigesimal base. It was "quasi-vigesimal" because successive powers of the base number (i.e., "b") had the values: 200 = 1, 201 = 20, 18 x 201 = 360, 18 x 202 = 7200, 18 x 203 = 144000, and so forth, rather than strict powers of 20 (200 = 1, 201 = 20, 202 = 400, 203 = 8000, 204 = 160000, and so forth). The reason for the curious use of "18 x 20" as a base in the Mayan number system was most likely due to the official Mayan year consisting of 360 days (18 x 20 = 360). The Mayan Indians compiled extensive observations of planetary positions in base 20 notation. The recording of time was an extremely complicated affair. It was based on the superimposition of two calendar cycles called the "TZOLKIN", or Sacred Almanac, and the "HAAB". The TZOLKIN cycle consists of 20 periods of 13 days each, while the HAAB consisted of 18 months of 20 days each, plus 5 "unlucky" days. The Mayan number system was notable in its development of the zero as a placeholder hundreds of years before AD 876, its earliest known use in India.
The Maya used a system of dots and bars (most likely representing pebbles and sticks, respectively) as a way to count. The symbol of a "dot" stood for one and a symbol of a "bar" stood for five, while an ovular shell with vertical bars inside stood for zero. For example, one dot represented "1", two dots for "2", ... , one bar for "5", one dot and one bar for "6", ... , two bars for "10", one dot and two bars for "11", ... , three bars for "15", one dot and three bars for "16", ... , four dots and three bars for "19", and so forth. Mayan numbers were written from bottom to top, rather than horizontally. In writing these symbols, the bars are placed horizontally and the dots placed on top of them, and the vigesimal positions develop upward from the base. As an example of representing a number in the Mayan vigesimal number system, the number 34 would be stated as "20 + 14", or two bars and four dots [on the bottom to represent 14 as "(2 x 5) + (4 x 1)"] followed by one dot [on the top to represent 20 as "1 x 20"].
The Maya considered some numbers more sacred than others. One of these special numbers was 20, as it represented the number of fingers and toes a human being could count on. Another special number was five, as this represented the number of digits on a hand or foot. Thirteen was religiously sacred as the number of original Maya gods. Another sacred number was 52, representing the number of years in a "bundle", a unit similar in concept to a century. Another number, 400, had sacred meaning as the number of Maya gods of the night.
The modern-day vigesimal number system is based on the powers of the base 20, that is (..., 160000, 8000, 400, 20, 1, 1/20, 1/400,...) = (..., 204, 203, 202, 201, 200, 20-1, 20-2,...). Any real number can be represented in a positional number system of base "b". For the base 20 (i.e., "b = 20") numbering system a real number x can be represented as x = anbn + an-1bn-1 + ... + a1b1 + a0b0 + a-1b-1 + ... + a-(n-1)b-(n-1) + a-nb-n. The equation of "x" must satisfy the requirements on ai, where i = {0, 1, 2, ... , n} and 0 ai < b.
The (modern) base 20 notational system uses the twenty digits of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, G, H, I, and J. This notation is to be understood as follows: A represents ten, B represents eleven, C represents twelve, and so forth in order up to the letter J which represents nineteen. As an example of expressing a number in vigesimal form, consider the number (in decimal form) "x = 4,265". In the vigesimal system it is represented as "AD5". Using the vigesimal numeric scheme described previously we see that AD5 = (A x 202) + (D x 201) + (5 x 200) = (10 x 400) + (13 x 20) + (5 x 1) = 4,265.
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