- The Maya Vigesimal System of Numeration
- What is a Vigesimal System of Numeration?
- The Maya Place-Value System

- Maya Numerals
- The Bar-and-Dot Notation
- The Head-variant Numeral Glyphs

- The Maya Concept of Zero
- A Brief History of Zero
- The Maya Zero

- How to Write Maya numbers
- Positional Notation
- Additive Notation
- Multiplicative Notation

- How to Convert Decimal and Vigesimal Numbers
- How to Convert Maya Numbers into Decimal Numbers
- How to Convert Decimal Numbers into Maya Numbers

- Maya Arithmetic Calculations
- Adding Maya Numbers
- Subtracting Maya Numbers

- References

With its intriguing numerals, the Maya number system is remarkable in many ways. The use of the concept of zero is testimony to its sophistication while the contrast between the simplicity of the notation and the perceived complexity of the base-20 (vigesimal) system continues to captivate students and scholars alike.

The truth, though, is that the apparent difficulty lies mainly in the unfamiliarity of the signs and the system. Actually, it is not difficult to understand and use the Maya number system, but it requires some method and explanation.

But what is a vigesimal system of notation?

## 1- The Maya vigesimal system of numeration

For many people, one of the most perplexing aspects of the Maya number system is its vigesimal nature.

### What is a vigesimal system of numeration?

To explain what a vigesimal system is, it is useful to start from something more familiar. The most commonly used number system is the Hindu-Arabic notation which reached Europe in the 11th century AD.

In the Hindu-Arabic numeral system, the value of a digit (i.e. 0-9) depends partly on its position. Take the number “200”, for example: “2” actually represents “two counts of one hundred” because the digit “2” is located in the “hundreds place”. The value of the numeral is linked to its position. That’s why it is called a place-value (or “positional”) notation.

In that system, ten digits (0-9) are used to write numerals. After “9”, we need to move across to the next position on the left where one unit is added. So after “9” comes “10”, after “19” comes “20” and after “199” comes “200”.

In other words, the value of the position increases from right to left in powers of ten: 1, 10, 100, 1000, 10000, etc. That’s why it is called a Base-10 (or “decimal”) numeral system. What “199” really stands for is: “1×100 + 9×10 + 9×1” (or “1×10^2 + 9×10^1 + 9×10^0”):

By contrast, the ancient Maya used a Base-20 positional system, also called “vigesimal” notation.

### The Maya Vigesimal System

The Maya system of numeration was first described by the Franciscan Diego de Landa (1524-1579)^{1}Tozzer 1941 shortly after the Spanish conquest of Yucatan:

“Their count is by five up to twenty, and by twenties up to one hundred and by hundreds up to four hundred, and by four hundreds up to eight thousand; and they used this method of counting very often in the cacao trading. They have other very long counts and they extend them ‘ad infinitum’, counting the number 8000 twenty times, which makes 160,000; then again this 160,000 by twenty, and so on multipliying by 20 […]”

It’s quite clear from Landa’s description that the Maya number system was based on twenty instead of ten like the Hindu-Arabic notation.

This vigesimal structure is still noticeable in Mayan languages. Like in all Mesoamerican languages, numerals have a base 20 structure^{2}Justeson 1986:440. The Maya have non-composite words for 1, 20, 400 (i.e. 20^2), 8000 (i.e. 20^3), 160000 (i.e. 20^4), 3200000 (i.e. 20^5) and 64000000 (i.e. 20^6)^{3}Seidenberg 1986:380. In Mayan Yucatec, for instance, you would use:

*hun*for 1*k’al*for 20*bak’*for 400*pik*for 8,000

What this means, is that in a vigesimal notation the number “2,000”, for example, would be “five counts of four-hundred” whereas it is literally “two counts of one thousand” in the decimal system. The Maya would say *ho’ bak’* (“five four-hundred”). The result, though, is the same: 2×1000 = 5×400 = 2000. Only the mental paths to get there differ.

In essence, in a Base-20 number system, the value of each position increases in powers of twenty (1, 20, 400, 8000, 160000, etc) instead of ten. Consequently, in a vigesimal number system, we move over to the next position after 19 (instead of 9 like in a Base-10 system).

Let us consider the following examples:

The number “three hundred” is written “300” in a decimal system and it means “3×100 + 0x10 + 0x1”. In a vigesimal system, however, “three hundred” is “15×20 + 0x1” because numbers progress in powers of twenty. In other words, we would need to reach “400” (20×20) before adding a unit to the next position (i.e. 400s). This is what happens with “seven hundred and fifteen” because, in a vigesimal system, 715 is “1×400 + 15×20 + 15×1”.

Admittedly, using Arabic digits (i.e. 0-9) to write numbers in a Base-20 system doesn’t make calculus any easier and can even create unfortunate situations. Imagine a Maya employer offering you USD1915. You happily sign the contract thinking you are going to get “19×400 + 1×20 + 5×1” (i.e 7225 in decimal notation). But what a huge deception when you realise that your salary is actually “1×400 + 9×20 + 15×1” (i.e. 595 in decimal notation).

The ancient Maya who, of course, wanted to avoid this kind of regrettable confusion, came up with a clever numeral system of their own as we are going to see now.

## 2- Maya numerals

While a vigesimal system of notation such as the one used by the ancient Maya requires twenty signs for the numerals 0-19, it doesn’t requires twenty different digits. The digits are the individual signs that represent the basic numerical values. In the Hindu-Arabic number system, ten different digits (0-9) are used to represent the numerical values 0 through 9.

Similarly, to avoid confusion, an ideal vigesimal system would have twenty symbols such as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, G, H, I, J. The problem, though, is that the more symbols you use, the more complex your system becomes. The ancient Maya wrote all the numbers with only three signs.

But how do you represent any number with only three symbols?

### The bar-and-dot notation

About three thousand years ago, the people of Mesoamerica came up with a clever and quite unique solution: the bar-and-dot notation^{4}Coe 1965, Marcus 1976, Justeson 1986:440

This system has been described in a variety of ethno-historical sources and chronicles from the time following the conquest. Here’s how a 17th century native Maya put it into words^{5}Brinton 1882:48:

“They (our ancestors) used (for numerals in their calendars) dots and lines back of them; one dot for one year, two dots for two years, three dots for three years, four dots for four years, and so on ; in addition to these they used a line; one line meant five years, two lines ten years ; if one line and above it one dot, six years ; if two dots above the line, seven years ; if three dots above, eight years ; if four dots above the line, nine ; a dot above two lines, eleven; if two dots, twelve ; if three dots thirteen.”

In summary, a dot has a numerical value of 1, and a bar (or a line) stands for 5. A third symbol such as a seashell was used to represent zero^{6}Morley 1915:87-91.

So, in the bar-and-dot system, you just need pebbles, twigs and shells to do your accounting!

We’ve already seen that the Maya also made use of a Base-20 or vigesimal system. This means that all numbers up to 19 were written in a non-place-value notation. To put it differently, the values of the first twenty numerals (0-19) reside in the signs themselves not their positions (fig. 3). Only larger numbers have a place-value component.

That is quite similar to the Hindu-Arabic numbering system where the numerical values 0 to 9 are denoted by the corresponding 0-9 digits. Only larger numbers (e.g. 11, 12, 13, etc) have a place-value component increasing in value from right to left.

Here are the Maya numerals 0-19 in the bar-and-dot notation:

Maya scribes and artists did not like to leave voids and blank spaces. They were quite punctilious in their approach to aesthetic norms. So, in numerals such as 8 or 13, they spread out the dots evenly to cover the full width of the bar and, where dots were missing (in numerals like 11 or 12), they filled the spaces with short, curved elements that can be misleading:

Anyone who had some interest in the Maya civilisation knows about the bar-and-dot system. What most people ignore, though, is that there was another set of signs that were used to represent the twenty numerals 0-19. This set is known as the head-variant numeral glyphs.

### The Head-variant numeral glyphs

The Maya scribes had to compose with a strict writing system but they were able to express their creativity in the design of the hieroglyphs. They clearly enjoyed the art of calligraphy. To bring variety to their inscriptions which were heavy with dates, they designed specific and elaborate glyphs to represent the numerical values 0 to 19:

It is noteworthy that the head variants 13 through 19 are composite portraits of the heads for the numerals 3-9 with attributes from the glyph for 10. This practice is consistent with Mayan languages such as Yucatec. In this language the words for the numerals 1 to 12 are distinct but the words for the numerals 13 through 19 are composed of the word for 10 (*lahun*) and the words for the numerals 3 to 9:

- hun
- ca
- ox
- can
- hoo
- uax
- uuc
- uaxac
- bolon
- lahun

- buluc
- lahca
- ox
*lahun* - can
*lahun* - hoo
*lahun* - uax
*lahun* - uuc
*lahun* - uaxac
*lahun* - bolon
*lahun*

In other words, the head variants for the numerals 13 through 19 are hybrid signs with elements of the glyph for 10 superimposed to the glyphs 3-9:

The head variants are not very common in the corpus of Maya inscriptions. Possibly due to the amount of work required to draw and carve them, these glyphs were used only in the texts associated with monumental sculptures depicting kings and queens and are only found in the introductory calendric sequence of these inscriptions.

The amazing graphic quality of the head-variant numerals led us to overlook one crucial point: not only there is a specific symbol to denote zero but, in the Maya number system, zero is also a numeral. And this entails a lot more than you can imagine.

## 3- The Maya Concept of Zero

Thanks to our education we are familiar with the number zero; what it means and how to use it. But have you ever wondered why scientists and historians make such a big deal of it?

Well, it is because zero is not quite like any other number.

### A Brief History of Zero^{1}For a more comprehensive account, see Kaplan 1999

Natural numbers such as one, two, or three have a sensory equivalent: 1 cow, 2 pigs, 3 hens, etc. There is something to be counted. With zero it’s not the case. Zero doesn’t exist in the sensory world. It’s an abstraction. And it had to be invented.

As a matter of fact, for the vast majority of our history, humans didn’t understand or use the number zero.

For basic mathematical tasks such as counting cattle, you would mainly need an easy and convenient way to record numbers and operate additions and subtractions. One of the most intuitive solutions is to make use of pebbles (or dots when writing) or twigs (bars): one vertical bar to represent one, four bars for four, ten bar bars for ten, etc. And to make it even easier, you can make groups of five bars, four vertical and one across, to easily spot and tally groupings of five.

It’s all well and good with small numbers but when the quantities you are recording become larger, you end up with an awful lot of bars (imagine having to write “1000” only with bars!). Faced with these issues many early civilisations did experiment with various grouping solutions^{7}Ifrah 2000:7-8, but bureaucratic needs required more efficient number systems. Starting in the 4th millenium BC, the Sumerians developed more sophisticated number systems along with cuneiform writing^{8}Ifrah 2000:77-133. By 2000 BC, the Babylonians were using a powerful Base-60 positional notation with different symbols to represent specific quantities^{9}Ifrah 2000:146. This is a critical element in the development of zero and we will get back to it.

Outside Mesopotamia, the Egyptians, the Phoenicians, the Greek, the Etruscans, the Romans, among others, went a different route and made use of additive systems which were sufficient for basic operations (i.e. additions and subtractions) and the tallying of goods^{10}Anderson 1971:62.

In additive systems, symbols are given for units and groups of units (e.g. 1, 10, 100, 1000, 10000, etc). To write a number you just add all the symbols needed. In other words, the value of a number is, basically, the addition of the individual values of each numeral. For example, the Roman way to write “2013” is MMXIII where M is 1000, X is 10 and I is 1. So, MMXIII is literally 1000+1000+10+1+1+1.

These number systems still require an uncomfortable quantity of symbols and, although basic operations were not too difficult as you just have to add or take away symbols, representing large numbers or doing longer calculations was arduous and the risk of errors was important^{11}Ifrah 2000: 187; Kaplan 1999: 5-6.

“Roman numerals, in fact, were not signs which supported arithmetic operations, but simply abbrevations for writing down and recording numbers. This is why Roman acccountants, and the calculators of the Middle Ages after them, always used the abacus with counters for arithmetical work”, Georges Ifrah (2000: 187)

One remarkable thing you might have noticed in all these systems is that there is no numeral for zero. Additive systems of numeration don’t require the use of a sign to denote zero. People might need a symbol to express “nothingness”, but the number system itself doesn’t require a numeral zero to function. Place-value notations, on the other hand, do require a placeholder digit.

But what is a placeholder digit anyway?

Take the number “2019”, for example. As we’ve already seen, the value of “2” depends on its position. Here, in fourth position to the left, “2” actually means “2000”. Then there are “1” tens and “9” ones. But no hundreds. To denote this absence and to avoid any confusion, we use “0”. Zero is called a “placeholder”.

As already mentioned, people in Mesopotamia made use of a positional numeral system. The problem for the Sumerians and their followers, though, was to differentiate some numbers. Three wedges, for example, could mean “3” if they were all in the 60^0 position, or “180” (i.e. 3×60) is they were in the 60^1 position, or “3661” if they were each in a different position: “1×60^2 + 1×60^1 + 1×60^0” (i.e. “3600+60+1=3661”).

The risk of confusion was quite important. So, in the 8th century BC, the Babylonians started to use specific signs (three hooks or, later, two slanted wedges) to indicate the absence of a digit in a string of numbers^{12}Kaplan 1999:12. Very much like 0 in 2019.

The Babylonian’s zero placeholder was used, however, only in the middle of a number, never at its end as we do in numbers like 1200, and never on its own. It was not a true zero yet. It was not seen as a numeral or a number but rather the lack of. Still, it was the first instance of a sign specifically designed to denote zero in a string of numerals.

Although the systems used by the Egyptians, the Romans or the Greeks didn’t require a zero-marker, being able to express the absence of something (e.g. zero apple) can be handy. Particularly if your job is to raise taxes for a distrustful and ill- tempered king.

So, in the early part of the 2nd millennium BC, the Egyptians started to use a specific hieroglyph (NFR or F35 in Gardiner’s sign list) to indicate “nothing” or zero balance in their accounting texts. Similarly, although much later (ca 5th century AD), the Romans used the Latin word *nulla* (“nothing”) which was reduced to the initial “N” in later Medieval manuscripts^{13}The Roman’s numeration system continued to be used in Europe until the 16th century!. The Greek likely brought back zero from Babylon when Alexander The Great conquered the city in 331 BC.

These symbols were not real mathematical signs, though. They were more like linguistic or symbolic elements used to express the absence of an item. One of the obstacles in the understanding of zero is that, in mathematical terms, zero is actually a quantity.

But how can nothing *be* something, you might ask?

That’s exactly what bothered western philosophers for centuries. From the Greek Parmenides of Elea (5th century BC) who proclaimed that “*nothing* cannot exist”, to the religious thinkers of Medieval Europe who concluded that anything that represents nothing must be the work of Satan (and should consequently be banned), zero remained a controversial concept for quite some times.

It took people with a different approach to push the idea further.

The Babylonian idea slowly made its way to the Indus valley where local mathematicians brought together the different aspects of zero: the sign to denote “nothing”, the placeholder and the number.

The religions of ancient India had a different and more positive view of the concept of “nothingness”^{14}Joseph 2011:341. Indians also used a positional numeral system. So the conditions were ripe for the emergence of the true zero.

How it got there is still debated. There is some evidence of a Greek influence^{15}Kaplan 1999:17-18. After all, Alexander the Great conquered what was left of Babylon in 331 BC, before heading to India. But we can’t exclude that the Indians might have developed zero independently either^{16}Ifrah 2000:409; Joseph 2011: 301.

This being said, it is important to note that Mesopotamia (actual Iraq) was not too far away from India and that the Arabian sea has been an important and vibrant marine trade route since the 2nd millennium BC at least^{17}This hub of activity was described around the middle of the first century AD by a Greek speaking Egyptian merchant in a document called *Periplus Maris Erythraei* (“Voyage around the Erythraean See”). The coastal trade route linked many ports on the west coast of India to Mesopotamia via Persian Gulf ports. And via the Red sea, it went as far as Egypt where Pharaohs had several canals built to service the trade.

So, goods and ideas have been exchanged via the Arabian sea trade routes for centuries. The possibility of multiples back and forth waves of influence in the sphere of sciences and more particularly mathematics is a reasonable assumption. As a matter of fact, it is also via this same route that zero came back to us as we are going to see^{18}Joseph 2011:461-462.

What everybody agrees on, though, is the crucial role played by Indians mathematicians in the genesis of zero as we know it. The first documented description (i.e. definition) of the arithmetic of zero was given in 628 AD. The author was the great Indian scholar Brahmagupta (598-668 AD) who wrote in his book *Brahmasphutasiddhanta* (“Corrected Treatise of Brahma”)^{19}Plofker 2009:428-434:

“[The sum] of two positives is positives, of two negatives negative; of a positive and a negative [the sum] is their difference; if they are equal it is zero. The sum of a negative and zero is negative, [that] of a positive and zero positive, [and that] of two zeros zero […]”

What Brahmagupta is saying is that zero is obtained by adding a positive number to a negative number of equal magnitude^{20}The magnitude of a number is defined as the distance of that number from zero on the number line, for example 1 + (-1), and that when zero is added or subtracted from a number that number remains the same (e.g. 1-0=1).

It took several centuries for the concept to reach Europe but we have to thank Muslims mathematicians for that. The idea is said to have reached Baghdad around 773 AD^{21}Kaplan 1999:90; Joseph 2011:462 where the work of scholars such Al-Kindi (801-873 AD) and Al-Khwarizmi (780-850 AD, better known as “Algoritmi”), were instrumental in the diffusion of the Indian system of numeration.

Zero and the Hindu-Arabic number system traveled throughout the Middle East to Damascus, then the Mediterranean to Cordoba. The Muslims had conquered Spain in the 8th century AD. The territory they established there (Al-Andalus) was a beacon of science, art and erudition. The use of the new mathematical notation in Muslim Spain is firmly attested by the end of the 10th century^{22}Kaplan 1999:970; Joseph 2011:463.

The Hindu-Arabic number system really reached Christian Europe in the 12th century when the seminal work of Al-Khwarizmi was translated into Latin in a book titled *Algoritmi de numero Indorum* (i.e. “Al-Khwarizmi on the Numerals of the Indians”). But it was the book of Leonardo of Pisa (1170-1250 AD), better known as Fibonacci, *Liber Abaci* (“The Book of Calculation”) published in 1202 AD, which popularised the new numerals in Europe. Although the full adoption was a slow process that didn’t come to completion until the 17th century^{23}Kaplan 1999:97-103; Joseph 2011:466, it opened a world possibilities.

Without zero we wouldn’t have calculus, algebra, algorithms, Cartesian coordinate system, physics, engineering, computer science and many aspects of economics and finance. So, yes, zero is quite a big deal!

What the existence of a head-variant numeral glyph for zero suggests is that the Maya fully understood the mathematical concept of zero.

### The Maya Zero

The long and arduous history of the zero in the old world only further highlights the achievement of the ancient Maya mathematicians who independently developed the concept of zero in their number system.

The earliest known occurrences of zero in Classic Maya inscriptions are from the 4th-century AD. Two monuments (Stela 18 and 19 from Uaxactun) bear the Maya date 8.16.0.0.0 (357 AD) in the Long Count calendar. This makes it one of the oldest documented instance of zero in the world.

It is important, however, to keep in mind that older numerical inscriptions appear in Long Count dates on Preclassic monuments. All known Long Count dates are written in bar-and-dot place-value notation^{24}Blume 2011:57. This implies that zero as a place-holder was already in use as early as the 2nd century BC and might have developed from an earlier Olmec innovation.

In the Classic period (AD 150-950), the Maya mostly used three symbols to represent zero (fig. 6):

- (a) the Quatrefoil
- (b) the Shell-in-hand glyph
- (c) the Head variant glyph

Later, however, during what is called the Late Postclassic (AD 1200-1500) the way to write numbers became more streamlined and standardised. The sign changed to stylised sea shells. The most common representation in the Dresden Codex has been identified as the *Oliva* shell (*Americoliva,* fig. 6: d, e)^{25}they have also been wrongly described as “eye” or “seeds” but other types of sea shells such a bivalve or conch marine shells were also used (fig. 6: f)^{26}Blume 2011:78-82.

From a symbolic point of view, the Maya signs for zero clearly show a connection with the mythological dwelling place of deities and ancestors, the “watery Underworld”, and consequently, with death.

The Maya Underworld itself is linked with death because it is the place where the soul goes after one dies. The Maya saw death as the *completion* of a life cycle. We know that the Maya also understood zero as denoting “nothing”.

Deciphered in the 1990ies, the so-called Quatrefoil and Shell-in-hand hieroglyphs (fig. 6: a, b) are read mi/MIH which means “no” (as in “*no* crops”) in Mayan languages. The numerical value of the Maya zero is further attested by some instances of “Distance Numbers”.

Distance Numbers appear in calendrical inscriptions. They are intervals between dates. Instead of writing the full date of an event before or after a main event for which the date has already been given, Maya scribes would simply say “this happened x many days before/after the [main] event”. All this was highly standardised but, in some rare cases the scribes wrote the Distance Numbers in an unconventional way^{27}Piedras Negras, Lintel 3; Palenque, Temple of the Cross, West Panel:

“[they used] the moon glyphic variant for number 20 with numbers from 0-19 to represent numbers of days from 20 to 39. When three dots were placed in front of this moon glyph, the 3 was understood as being added to the 20 to equal the quantity 23.” Anna Blum (2011:67)

Zero in front of the moon glyph cannot mean “no twenty” or “no day” as it wouldn’t make sense to add a sentence just to say “this happened *zero day* after/before the [main] event”. The Maya scribe would simply say “on this date this and that happened”. So “zero moon” (fig 7a) should rather be understood as 0 + 20 = 20 days^{28}Blum 2011:67; Stuart 2006:118. In this case, the sign for zero was probably used to maintain visual consistency with the rest of the inscription.

Here are some examples of the use of zero in Maya inscriptions:

Binding together the concepts of “nothingness” and “completion” brings to mind the idea of zero as the end point of a countdown. But the Maya also believed that the end of one cycle led to the start of another cycle. A new start from scratch. And the idea of never-ending cycles leads to the notion of “infinity”.

The shells used to denote zero are not just any shells. They are marine shells. So there is a clear symbolic connection with the sea. The *endless* sea as far as the Maya could tell.

“When the Maya scribes wrote, on some level they engaged in a willful act of expression that ranged from vast quantities of time to an expense of nothingness” Anna Blum (2011:84)

So the Maya zero brought together the ideas of “nothingness”, “completion” and “infinity”. The end and the beginning merged into one symbol. This, in a way, sums up the magic of the number zero.

Now that we have seen the first nineteen numerals and the three digits of the bar-and-dot notation, we shall learn how to write bigger numbers.

## 4- How to Write Maya numbers

For most of us, the unfamiliarity with the vigesimal system makes writing numerals in the Maya notation a bit laborious. This being said and putting aside the Base-10/Base-20 difference, the Maya numerical system worked very much like ours.

To represent large numbers (i.e. greater than nineteen), the ancient Maya made use of several systems:

- a positional notation which functions quite like our Hindu-Arabic system
- an additive notation similar to the Aztec system
- a multiplicative notation which is similar to the Chinese system

### The Positional Notation

The study of the Dresden Codex^{29}This document is a Maya record of astronomical notations which gives the synodic revolutions of Venus shows that the Maya not only developed the concept of zero but they also figured out the place-value system^{30}Ifrah 2000:311.

“These [place-value system and zero] are two fundamental discoveries that most civilisations failed to make, including especially Western European civilisation, which had to wait until the Middle Ages for these ideas to reach it from the Arabic World, which had itself acquired them from India”, Georges Ifrah (2000: 311).

As we’ve seen earlier, in a positional notation, the value of a numeral depends partly on its position: “2” in the decimal number “200” means “two counts of hundred” whereas it represents only “two counts of ten” in the number “20”. The Maya made use of a similar system but the value of each position increases in powers of twenty (i.e. 1, 20, 400, 8000, 160000, etc) instead of ten.

So, very much like we add one unit to the next position when we move over 9, Maya scribes would add one unit to the next position when one position reached 20.

The Maya, though, preferred columns instead of rows and worked upward starting from the bottom:

Interestingly, and as far as epigraphers can tell, the positional column notation was always reserved for the count of time (days, months, etc) and astronomical observations^{31}Stuart 2012 which require the computation of very large numbers.

In other words, no positional counting of large numbers of anything other than time has been found so far. For more mundane matters and smaller quantites, the Maya preferred a simpler system more adapted to everyday life numbers.

### The Additive Notation

The ancient Maya kept records of food offerings and tribute tallies. While there are few Maya documents available, this practice is well documented in Aztec codices^{32}i.e. books written by native Mesoamericans before or after the conquest.

To keep track of tribute tallies, the Aztecs used pictograms to represent items such as jaguar hides and added symbols to indicate the quantity of each items^{33}Vaillant 1941:206-209; Payne and Closs 1986:226-229. For example, the drawing of a jaguar hide with four dots would simply mean “four jaguar hides”.

Groupings of 20 were denoted by a *flag*, quantities of 400 by a stylised* fir tree*, and groupings of 8,000 by an *incense bag*. Five flags would thus mean a quantity of one hundred (i.e. 5×20). Examples from a page of the Codex Mendoza can be seen below. Notes written in Spanish are visible and include roman numbers which confirm the numerical values of the different symbols:

Due to the scarcity of appropriate sources^{34}whilst there are over 40 known Aztec codices, only four Maya codices have survived, our knowledge of the matter is more limited for the Maya. Nonetheless, around the same time the Aztecs were recording tributes in the Codex Mendoza (created ca A.D. 1541), the Maya were keeping track of goods in a similar fashion.

In the Dresden (ca A.D. 1450) and Madrid (ca A.D. 1350-1697) codices, counts of food offerings are given with simple bar-and-dot numerals (for 1-19) or as groupings of 20 with accompanying bars and dots^{35}(Love 1994: 56-63, Stuart 2012). In these documents, groupings of 20 are represented by the so-called WINIK signs or by the phonogram *k’al*.

It wouldn’t make sense to represent a number with several groupings of 20 if the notation was positional or multiplicative. So the system had to be additive like the Aztec’s. In other words, all numerals must be added to obtain the total. For example, three WINIK signs and the numeral 13 written with bar-and-dots would make 73 (i.e 3×20 + 13).

Interestingly, on earlier documents from the Classic period, the Maya made use of a multiplicative notation.

### The Multiplicative Notation

In the multiplicative notation, the quantities are given as multiples of vigesimal values such as *k’al* (20), *bak’ *(400), or* pik* (8000), etc. Maya scribes would use the glyphs for *k’al*, *bak’* or *pik* and add bars and dots to indicate the quantity of each units. In other words, the bar-and-dots numerals act as coefficients (i.e. multipliers) of the vigesimal expressions. For example, two dots with the glyph for twenty (*k’al*) means forty. Only some of these glyphs have been identified in the corpus of Classic Maya inscriptions so far:

There are very few known examples of such inscriptions, unfortunately. Mentions of quantities of goods are most commonly found on painted ceramics with court scenes.

In one depiction (Fig. )^{36}this cylinder vase is part of the collections of the Boston Museum of Fine Arts. Accession number: 2004.2204, an emissary of the powerful king of Calakmul is seen greeting an otherwise unknown lord named *Ch’ok Wayis*. A variety of goods is on display but the text does not specify if it is a tribute payment or a diplomatic gift.

While the emissary carries a backpack marked with the numeral “5” (or “15”), one bag in front of the throne is tagged with the inscription^{37}Tokovinine and Beliaev 2013:174-177:

“**huux**pik,**three**[counts of] eight thousand” [i.e. 24,000]

The famous wall-paintings in Room 1 of Structure 1 at Bonampak depict a similar, but much larger, scene. Bags of cacao beans are lined up in front the throne. One inscription is can be read^{38}Miller 1997:40:

, “**ho**pik kakaw**five**[counts of] eight thousand cacao beans” [i.e. 40,000]

With the exception of Naranjo Stela 32, there is barely any mention of tribute in the corpus of monumental inscriptions. Stela 32 is a large slab of limestone. It was carved in low relief around AD 820. The sculpture portrays the king of Naranjo, *Waxaklahun Ubah K’awil*, and the associated text recounts a number of events in his life with two interesting mentions of tribute^{39}LeFort and Wald 1995:

, “**ka**k’al ti ikatz**two**[counts of] twenty bundles of …” [i.e. 40]“**ho**k’al ti ikatz**five**[counts of] twenty bundles of …” [i.e. 100]

Although there are only a few instances of counts of food offerings or tribute tallies in the corpus of Maya inscriptions, they display an interesting variety of forms.

Whilst a positional notation was used for matters related to the count of time (i.e. calendar and astronomical cycles), the Maya preferred different notations to deal with smaller quantities: a multiplicative notation during the Classic period and an additive notation during the Postclassic period.

All these notations follow the vigesimal structure of the Maya system of numeration.

But how do you convert Maya numbers into decimal numbers and vice versa?

## 5- How to Convert Decimal and Vigesimal Numbers

For most of us, the unfamiliarity with the vigesimal system makes writing numbers in the Maya notation a bit laborious. This being said and putting aside the Base-10/Base-20 difference, the Maya numerical system worked very much like ours.

### How to easily convert Maya numbers into decimal numbers

Converting Base-20 numbers into Base-10 notation is quite easy. The best way to proceed is to use a vigesimal grid:

Start with the bottom numeral and work your way up. Remember, a dot stands for 1, a bar for 5 and the shell for 0.

### How to easily convert decimal numbers into Maya numbers

Converting Base-10 numbers into Maya notation is not as straightforward as going from a vigesimal system to a decimal system. First you need to convert the decimal numbers into Base-20, then you need to convert each numeral into the Maya bar-and-dot notation.

There are several ways to do it but the easiest one is the following; let’s take the number 2313 for example (fig. 14):

- In the context of natural numbers, 2313 can’t be divided by 20 as it is not a multiple of 20. More precisely, though, the division 2313 ÷ 20 leaves a remainder of 13 with a quotient of 115 (2313 ÷ 20 = 115 R 13). So, let’s bring down 13 and move 115 over to the next position (i.e. 20^1=20s).
- 115 divided by 20 equals 5 with a remainder of 15. Let’s bring 15 down and move 5 over to the next position (i.e. 20^2=400s).
- 5 is smaller than 20 so we can stop here.

The result is: 13×20^0, 15 x 20^1, 5 x 20^2. Or, in other words, 13x1s, 15x20s, 5x400s.

Some numbers can be a bit tricky. Take 2013 for example (fig. 15):

- 2013 divided by 20 equals 100 and leaves a remainder of 13. So, let’s bring down 13 and move 100 over to the next position (i.e. 20^1=20s).
- 100 divided by 20 equals 5 with no remainder. But, remember the chapter about zero, “no remainder” actually means that the remainder is 0. Let’s bring 0 down and move 5 over to the next position (i.e. 20^2=400s).
- 5 is smaller than 20 so we can stop here.

The result is: 13×20^0, 0x20^1, 5 x 20^2

The last aspect I’d like to cover is arithmetic. How do you do additions and subtractions using Maya numbers?

## 6- Maya Arithmetic Calculations

No indication that Maya multiplied or divided. Adding and subtracting with dot and bars is actually quite easy.

### Adding Maya numbers

Additions are easy. You just have to add the symbols, summing bars and dots in parallel registers:

But keep in mind that 5 dots become a bar:

When the sum of the numbers is bigger than 19, it is useful to be organised and work with a vigesimal grid. Start at the bottom and work your way up. Bear in mind that 4 bars make 20 (i.e. 4×5) which becomes a unit (i.e. a dot) in the next position up:

Promoting 4 bars to a dot in the next register is analogous to carrying in the addition of Arabic numerals.

Remember to mark zero with the shell where necessary:

Proceed similarly with bigger numbers. Just add dots and bars in each position starting with the 1’s at the bottom. Don’t forget that the accumulation of 5 dots is transcribed as a bar:

and that the accumulation of 4 bars in one register (i.e. 20) becomes a dot (i.e. a unit) in the next position up:

Adding Maya numerals is quite easy and necessitate only three rules:

- a dot represents 1 unit
- an accumulation of 5 dots becomes one bar
- an accumulation of 4 bars in one register is transcribed as a dot in the next higher register

“Maya addition is far simpler than Arabic addition, which requires ten symbols and the memorization of a host of rules such as 2 + 3 = 5 and 4 + 5 = 9”, (Lambert & al. 1980:249)

Now let’s move on to subtractions.

### Subtracting Maya numbers

Subtractions are quite easy too. You just remove symbols, taking differences of dots and bars in equal registers:

Bear in mind that 1 bar is equivalent to 5 dots. You might need to “convert” one bar into five dots to carry out the subtraction:

With bigger numbers, it helps to use the vigesimal grid. Remember that 1 dot in a given position equals 4 bars in the previous. Borrow from the next higher register when necessary:

Some numbers can be trickier but with a good method, there is no problem. Start from the bottom and work you way up:

Subtractions are easy to carry out and necessitate only the cancelling of symbols and to follow three rules:

- a dot represents one unit
- if there are insufficient dots, then a bar in the same level is converted to five dots
- if there are insufficient bars, then one dot from the next higher level is converted to four bars in the lower register.

## References

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## Credits

**Author:** Dr Patrice Bonnafoux

**Illustrations** (unless otherwise stated): Patrice Bonnafoux

**INTRODUCTION**

**Photo:** Vessel with Two Scenes of Pawahtun Instructing Scribes. Late Classic period, A.D. 700-750. Ceramic with monochrome decoration. height 8.9 cm. Kimbell Art Museum, Fort Worth (AP 2004.04). Photograph © User:FA2010 / Wikimedia Commons / Public Domain.

**SECTION 1: The Maya vigesimal system of numeration**

**Photo:** Wooden lintel. Tikal, Temple 4, Lintel 3. Late Classic period, after A.D. 746. Ethnographic Museum, Basel, Switzerland. Photograph © Sven Gronemeyer, 2016.

**SECTION 2: Maya Numerals**

**Photo:** Dresden codex, 43-44, frame 2. Förstemann version, 1880. FAMSI.

**Fig. 2.3:** Adapted from Kettunen & Helmke 2014, Table VI. Drawing of the head variants by John Montgomery.

**SECTION 3: The Maya Concept of Zero**

**Photo:** Full-Figure Zero, carved tuft, 53.3 × 52.1 × 35.6 cm, originally from the Façade of Structure 10L-22A at Copan. Courtesy of The Cleveland Museum of Art. Gift of the Hanna Fund (1953.154).

**Fig. 3.1: **Babylonian cuneiform numerals. User:Josell7 / Wikimedia Commons / Creative Commons License.

**SECTION 4: How to Write Maya Numbers**

**Photo:** Cylinder vase. Tribute presentation scene. Late Classic period, A.D. 691. Ceramic with polychrome decoration. height 21.3 cm. Museum of Fine Arts, Boston (2004.2204). Photograph © MFA / Fair use for educational purposes.

**Fig. 4.2:** Codex Mendoza, Folio 37 recto. Bodleian Libraries, Oxford University. Photograph © User:Ptcamn~commonswiki / Wikimedia Commons / Public Domain.

**Fig. 4.3:** Dresden codex, 25-28, frame 4-6. Förstemann version, 1880. FAMSI.

**Fig. 4.5:** Court scene – ceramic, AD 691 (K5453). Adapted from Tokovinine and Beliaev 2013, Fig. 7.3. Drawing by A. Tokovinine.

**Fig. 4.7:** Naranjo, Stela 32. Drawing by Ian Graham, 1978. Corpus of Maya Hieroglyphic Inscriptions, Vol. 2, Part 2. Peabody Museum of Archaeology and Ethnology. Harvard University, Cambridge.

**SECTION 5: How to Convert Decimal and Vigesimal Numbers**

**Photo:** Lady of Ik’ polity cylinder vase. Late Classic period, A.D. 750–780. Ceramic with monochrome decoration. height 14.8 cm. Museum of Fine Arts, Boston (1988.1175). Photograph © MFA / Fair use for educational purposes.

**SECTION 6: Maya Arithmetic Calculations**

**Photo:** Dresden codex, 9-10, frame 2 and 3. Förstemann version, 1880. FAMSI.