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Jan 16, 05:08 PM: Mayan mathematics. Part 3.
Mayan Numerals.
Almost certainly the reason for base 20 arose from ancient people who counted on both their fingers and their toes. Although it was a base 20 system, called a vigesimal system, one can see how five plays a major role, again clearly relating to five fingers and toes. In fact it is worth noting that although the system is base 20 it only has three number symbols (perhaps the unit symbol arising from a pebble and the line symbol from a stick used in counting). Often people say how impossible it would be to have a number system to a large base since it would involve remembering so many special symbols. This shows how people are conditioned by the system they use and can only see variants of the number system in close analogy with the one with which they are familiar. Surprising and advanced features of the Mayan number system are the zero, denoted by a shell for reasons we cannot explain, and the positional nature of the system. However, the system was not a truly positional system as we shall now explain.
In a true base twenty system the first number would denote the number of units up to 19, the next would denote the number of 20’s up to 19, the next the number of 400’s up to 19, etc. However although the Maya number system starts this way with the units up to 19 and the 20’s up to 19, it changes in the third place and this denotes the number of 360’s up to 19 instead of the number of 400’s. After this the system reverts to multiples of 20 so the fourth place is the number of 18×20 2, the next the number of 18×20 3 and so on. For example [8;14;3;1;12] represents
12 + 1×20 + 3×18 x 20 + 14×18 x 20 2 + 8×18 20 3 = 1253912.
As a second example [9;8;9;13;0] represents
0 + 13×20 + 9×18 x 20 + 8×18 x 20 2 + 9×18 x 20 3 =1357100.
Both these examples are found in the ruins of Mayan towns and we shall explain their significance below.
Now the system we have just described is used in the Dresden Codex and it is the only system for which we have any written evidence. In [4] Ifrah argues that the number system we have just introduced was the system of the Mayan priests and astronomers which they used for astronomical and calendar calculations. This is undoubtedly the case and that it was used in this way explains some of the irregularities in the system as we shall see below. It was the system used for calendars. However Ifrah also argues for a second truly base 20 system which would have been used by the merchants and was the number system which would also have been used in speech. This, he claims had a circle or dot (coming from a cocoa bean currency according to some, or a pebble used for counting according to others) as its unity, a horizontal bar for 5 and special symbols for 20, 400, 8000 etc. Ifrah writes [4]:
Even though no trace of it remains, we can reasonably assume that the Maya had a number system of this kind, and that intermediate numbers were figured by repeating the signs as many times as was needed.
Let us say a little about the Maya calendar before returning to their number systems, for the calendar was behind the structure of the number system. Of course, there was also an influence in the other direction, and the base of the number system 20 played a major role in the structure of the calendar.
The Maya had two calendars. One of these was a ritual calendar, known as the Tzolkin, composed of 260 days. It contained 13 “months” of 20 days each, the months being named after 13 gods while the twenty days were numbered from 0 to 19. The second calendar was a 365-day civil calendar called the Haab. This calendar consisted of 18 months, named after agricultural or religious events, each with 20 days (again numbered 0 to 19) and a short “month” of only 5 days that was called the Wayeb. The Wayeb was considered an unlucky period and Landa wrote in his classic text that the Maya did not wash, comb their hair or do any hard work during these five days. Anyone born during these days would have bad luck and remain poor and unhappy all their lives.
Why then was the ritual calendar based on 260 days? This is a question to which we have no satisfactory answer. One suggestion is that since the Maya lived in the tropics the sun was directly overhead twice every year. Perhaps they measured 260 days and 105 days as the successive periods between the sun being directly overhead (the fact that this is true for the Yucatán peninsular cannot be taken to prove this theory). A second theory is that the Maya had 13 gods of the “upper world”, and 20 was the number of a man, so giving each god a 20 day month gave a ritual calendar of 260 days.
At any rate having two calendars, one with 260 days and the other with 365 days, meant that the two would calendars would return to the same cycle after lcm(260, 365) = 18980 days. Now this is after 52 civil years (or 73 ritual years) and indeed the Maya had a sacred cycle consisting of 52 years. Another major player in the calendar was the planet Venus. The Mayan astronomers calculated its synodic period (after which it has returned to the same position) as 584 days. Now after only two of the 52 years cycles Venus will have made 65 revolutions and also be back to the same position. This remarkable coincidence would have meant great celebrations by the Maya every 104 years.
Now there was a third way that the Mayan people had of measuring time which was not strictly a calendar. It was an absolute timescale which was based on a creation date and time was measured forward from this. What date was the Mayan creation date? The date most often taken is 12 August 3113 BC but we should say straightaway that not all historians agree that this was the zero of this so-called “Long Count”. Now one might expect that this measurement of time would either give the number of ritual calendar years since creation or the number of civil calendar years since creation. However it does neither.
The Long Count is based on a year of 360 days, or perhaps it is more accurate to say that it is just a count of days with then numbers represented in the Mayan number system. Now we see the probable reason for the departure of the number system from a true base 20 system. It was so that the system approximately represented years. Many inscriptions are found in the Mayan towns which give the date of erection in terms of this long count. Consider the two examples of Mayan numbers given above. The first
[8;14;3;1;12]
is the date given on a plate which came from the town of Tikal. It translates to
12 + 1×20 + 3×18 x 20 + 14×18 x 20 2 + 8×18 x 20 3
which is 1253912 days from the creation date of 12 August 3113 BC so the plate was carved in 320 AD.
The second example
[9;8;9;13;0]
is the completion date on a building in Palenque in Tabasco, near the landing site of Cortés. It translates to
0 + 13×20 + 9×18 x 20 + 8×18 x 20 2 + 9×18 x 20 3
which is 1357100 days from the creation date of 12 August 3113 BC so the building was completed in 603 AD.
We should note some properties (or more strictly non-properties) of the Mayan number system. The Mayans appear to have had no concept of a fraction but, as we shall see below, they were still able to make remarkably accurate astronomical measurements. Also since the Mayan numbers were not a true positional base 20 system, it fails to have the nice mathematical properties that we expect of a positional system. For example
[9;8;9;13;0] = 0 + 13×20 + 9×18 x 20 + 8×18 x 20 2 + 9×18 x 20 3 = 1357100
yet
[9;8;9;13] = 13 + 9×20 + 8×18 x 20 + 9×18 x 20 2 = 67873.
Moving all the numbers one place left would multiply the number by 20 in a true base 20 positional system yet 20×67873 = 1357460 which is not equal to 1357100. For when we multiple [9;8;9;13] by 20 we get 9×400 where in [9;8;9;13;0] we have 9×360.
We should also note that the Mayans almost certainly did not have methods of multiplication for their numbers and definitely did not use division of numbers. Yet the Mayan number system is certainly capable of being used for the operations of multiplication and division as the authors of [15] demonstrate.
Finally we should say a little about the Mayan advances in astronomy. Rodriguez writes in [19]:-
The Mayan concern for understanding the cycles of celestial bodies, particularly the Sun, the Moon and Venus, led them to accumulate a large set of highly accurate observations. An important aspect of their cosmology was the search for major cycles, in which the position of several objects repeated.
The Mayans carried out astronomical measurements with remarkable accuracy yet they had no instruments other than sticks. They used two sticks in the form of a cross, viewing astronomical objects through the right angle formed by the sticks. The Caracol building in Chichén Itza is thought by many to be a Mayan observatory. Many of the windows of the building are positioned to line up with significant lines of sight such as that of the setting sun on the spring equinox of 21 March and also certain lines of sight relating to the moon.
Article by: J J O’Connor and E F Robertson
Technorati tags: Maya, Civilization, Mathematics, Mayan Numerals