The use of Zero as a placeholder by the Maya

30th Nov 2019

Mexicolore contributor Dr Graham Atkinson

We are sincerely grateful to Dr. J. Graham Atkinson for this intriguing article on the use of zero by the ancient Maya, setting this into a global context. Graham Atkinson has a doctorate in mathematics from Oxford and a B.Sc. from St. Andrews University. As a member of the Pre-Columbian Society he runs a glyph group in Washington, DC which meets monthly to read Maya hieroglyphic inscriptions.

Summary
More than a millennium before it was introduced into Europe as part of the Hindu-Arabic number system the Maya were using a zero as a placeholder in their calendar system. Without trying to provide a comprehensive history of the use of zero as a placeholder in a positional number system this note explains how the Maya adopted the use of zero as a placeholder in their calendar system, and possibly their number system, and compares their use with the use of zero in other early number systems.

Introduction
A remarkable feature of the classic Maya culture that is not recognized as much as it should be is the very early use of a zero as a number and placeholder in their calendar and, presumably, number system. I say “presumably” because almost all the numbers we have from the classic Maya period are from calendrical situations. The Maya (and the earlier cultures from which they inherited their number and calendar systems) were using a zero in this way long before it came into use in European mathematics, and probably even before its use in South-East Asia. This note will attempt to place this use of zero in a temporal context to emphasize how remarkable this is.

Background
Before starting the historical discussion it would be worthwhile explaining what is meant by a positional number system. It is a system, like ours, in which the value of a symbol varies depending upon its position. For example, 1 can mean 1 or 10 or 100 depending upon where it occurs in a number. In contrast, in Roman numbers V means 5 wherever it occurs. Arithmetic is much easier in a positional number system.
The Egyptians used zero as a number in their accounting texts, but their number system was not positional, it was more like the Roman system, but using hieroglyphs rather than letters. The table above (pic 2) shows the Egyptian number system.

The first known use of zero as a placeholder in a positional or place number system was by the Babylonians in their Seleucid period (300 – 0 BCE). The Babylonians had a place number system using a base of 60. When they started using their number system they simply omitted zeros that we would consider to be needed, and the reader had to infer their absence, and where they should be, from the context. Later they started inserting spaces or symbols (e.g., two slanted wedges, which was a punctuation mark in the cuneiform script) to indicate where a zero should be. However, this sophisticated number system was forgotten for many centuries, and the Greeks and Romans regressed to cumbersome number naming systems using letters that made arithmetic extremely complex. Around the 7th century CE a zero started to be used as a place holder in South-East Asia. The first known example of this is found in an Old Khmer inscription, and consists of a dot as a placeholder in the tens position of the number 605 (a date that corresponds to 683 in our calendar) (pic 3).

Amir Aczel (2015) describes this as the first zero and documents its discovery in his book “Finding Zero: A Mathematician’s Odyssey to Uncover the Origins of Numbers”. The concept of zero and its use as a placeholder in a number system that uses the place position to indicate magnitudes probably spread from the old Khmer empire to India (see Diller 1995) and from there, probably, to the Arab world. It then made its way over to Europe. Fibonacci (1170-1250 CE) is credited with introducing the Arabic numbers to Europe.
At first the so-called Arabic numbers were considered suspect because they were so easy to modify and so to falsify in records, but their usefulness and ease of use in calculation eventually won everyone over, so they replaced the competing Roman number system for most practical purposes (pic 4).

The Maya “long count” calendar tells the amount of time that has passed between 11 August 3114 BCE* and the day being described – you can think of 11 August 3114 as being the equivalent of the start of our Current Era. The amount of time is expressed as the number of days or Kin, the number of 20 day months or Winal, the number of 360 day “years” or Tun, each consisting of 18 Winal, the number of 20 Tun periods or Katun, and the number of 400 Tun periods or Baktun**.
The Maya long count dates usually involve 5 numbers, representing the number of the Baktun (Pik), Katun (Winikhaab), Tuun (Haab), Winal (Winik), and Kin since the start of that count in 3114 BCE. Unfortunately we do not know when the long count came into use in Central America. The Maya long counts (but not some of the very early long counts, such as the one on the Mojarra Stela, drawing in pic 5, showing the number 8.5.16.9.9, and the Stela 2 from Chiapa de Corzo) almost always include glyphs indicating the calendrical unit to which the number applies.

Floyd Lounsbury (1981) states that the earliest reliable usage of day count chronology and place notation was 36/31 BCE (Stela 2 from Chiapa de Corzo has a long count date in 36 BCE - pic 6). Given that the usage of a place calendar system, and so a zero placeholder, almost certainly predates this, and earlier calendar round dates may yet be found, we can conclude that the use of zero as a placeholder in a positional calendar/number system was developed in Central America before the start of the current era, so within a couple of hundred years of such use in Babylonia, and more than a millenium before its introduction into Europe.

The long count date that starts the inscription on the east side of Quirigua stela C (pic 7) is in a fairly standard format for such Maya inscriptions, but is particularly relevant for this discussion because it includes 4 zeroes written in 2 different ways. The large top glyph is known as the Initial Series Introductory Glyph (ISIG) and tells us that what follows is going to be a date presented in several different calendars. The glyph on the left below the ISIG is 13 Baktun, and the right glyph 0 Katun. The next row down reads 0 Tun 0 Winal, but with the zero written in two different ways, then the row below 0 Kin 4 Ahau, and the bottom glyph is 8 Kumku. This date is the starting date for the Maya calendar and, as discussed earlier, is probably 11 August 3114 BCE in our calendar. 4 Ahau is the date in the ceremonial 260 day calendar, and 8 Kumku the date in their 365 day calendar.

As a footnote, it is interesting to notice that a few of the early known Maya inscriptions omit some zeroes in the long count. For example, Stela U from Quirigua has a starting long count date of 9.2.5.0.0 (480 AD), but the two zeros are omitted. This is in contrast to the earlier Stela C from Quirigua, which has a long count date of 9.1.0.0.0 (455 AD) but which does include all the zeros. An even earlier monument, Stela 19 from Uaxactun, has a long count date of 8.16.0.0.0 (357 AD) and has the zeros included. In a sense the zeros in this context are redundant because the time period involved (e.g., baktun) is specified after the number, so you do not need to know the place to know what the multiplier is.

* There is some debate about the exact starting date of the long count, but 11 August 3114 BCE is the one accepted by the majority of Mayanists.
** The terms Winal, Tun, Katun, and Baktun are Colonial Yucatec Mayan, and it is now thought that the classical period language used Winik, Haab, Winikhaab, and Pik. Kin is the word for day in both classic Maya and colonial Yucatec.

References/sources:-
• Amir Aczel, Finding Zero: A Mathematician’s Odyssey to uncover the Origins of Numbers, St. Martin’s Press, January 2015
• Anthony Diller, 1995, New Zeros and the old Khmer, Mon-Khmer Studies 25:125-132
• Floyd Lounsbury, Maya Numeration, Computation, and Calendrical Astronomy, in Dictionary of Scientific Biography, 15 & 16, supplement 1, Charles Coulston Gillispie, Editor in Chief, Charles Scribner’s Sons, New York, 1981
• Pic 1: from Mexicolore archives
• Pic 2: Egyptian number table: Wikipedia
• Pic 3: Khmer zero drawing: By Paxse - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=5689161
• Pic 4: Image from History of the Nordic peoples by Claus Magnus; Wikipedia (Arabic Numerals)
• Pic 5: Mojarra stela drawing and photograph: https://commons.wikimedia.org/wiki/File:La_Mojarra_Stela_1_Schematics.jpg
• Pic 6: from https://www.latinamericanstudies.org/chiapa-de-corzo.htm
• Pic 7: Quirigua Stela C, East side
https://creativecommons.org/licenses/by-sa/3.0/
• Pic 8: Dresden Codex page, from famsi.org.