The number system used throughout the modern world is a fully place-valued decimal system. This means that we count in groups of ten using nine digits, \(1, 2, 3, 4, 5, 6, 7, 8, 9,\) and a symbol zero, \(0,\) that plays double duty as both a place holder and a number representing "none". Our ten numerals are referred to as the Indo-Arabic (or Hindu-Arabic) numbers since they were developed in India by the 9th century and then transmitted to the Western world via the Arabs. Most notably they appear in the 1202 *Liber Abaci* (or *Book of Calculation*) of Leonardo of Pisa (better known as Fibonacci).

In a place-value system, the value of each symbol used is determined by its location in the number. Examples of other place-value systems are the ancient Babylonian cuneiform, ancient Chinese rod numeral, and Mayan systems. The first two systems do not have a place holder; however the Chinese rod numerals are unambiguous due to a very structured style of writing as well as the clever alternating of the horizontal/vertical alignment of the numbers. Examples of non-place-value systems are those of ancient Egypt and Rome. For example, an \({\rm X}\) in Roman numerals represents ten, no matter its location. (Numerous websites are available to learn more about any of the above systems.)

The Babylonian, ancient Chinese, and Maya number systems are additive, meaning that they repeat a symbol to represent larger numbers. For example, the Babylonians used a base 60 number system but with only two symbols, a vertical line for one and a wedge shape for ten that were made with the end of a stylus impressed into wet clay. In the Babylonian system, the value of each place had to be inferred from context since spacing between places was unclear. For example,

may represent \(13\times 1 = 13\) or \(13\times 60 =780\) or \(10\times 60 + 3\times 1 =603\) or \(10\times 3600 + 0\times 60 + 3\times 1 = 36003\) or .... All three of these systems use abstract or stylized symbols based on a single line segment or dot to represent one. In contrast, the Egyptian system is pictographic. Our favorite example is the glyph of an astonished man to represent one million:

In order to depict larger and larger numbers, the Egyptians and the Romans needed more and more symbols. Any base system having as many numerals as its base does not need to use the additive property to depict all of its numbers.

The number system used throughout the modern world is a fully place-valued decimal system. This means that we count in groups of ten using nine digits, \(1, 2, 3, 4, 5, 6, 7, 8, 9,\) and a symbol zero, \(0,\) that plays double duty as both a place holder and a number representing "none". Our ten numerals are referred to as the Indo-Arabic (or Hindu-Arabic) numbers since they were developed in India by the 9th century and then transmitted to the Western world via the Arabs. Most notably they appear in the 1202 *Liber Abaci* (or *Book of Calculation*) of Leonardo of Pisa (better known as Fibonacci).

In a place-value system, the value of each symbol used is determined by its location in the number. Examples of other place-value systems are the ancient Babylonian cuneiform, ancient Chinese rod numeral, and Mayan systems. The first two systems do not have a place holder; however the Chinese rod numerals are unambiguous due to a very structured style of writing as well as the clever alternating of the horizontal/vertical alignment of the numbers. Examples of non-place-value systems are those of ancient Egypt and Rome. For example, an \({\rm X}\) in Roman numerals represents ten, no matter its location. (Numerous websites are available to learn more about any of the above systems.)

The Babylonian, ancient Chinese, and Maya number systems are additive, meaning that they repeat a symbol to represent larger numbers. For example, the Babylonians used a base 60 number system but with only two symbols, a vertical line for one and a wedge shape for ten that were made with the end of a stylus impressed into wet clay. In the Babylonian system, the value of each place had to be inferred from context since spacing between places was unclear. For example,

may represent \(13\times 1 = 13\) or \(13\times 60 =780\) or \(10\times 60 + 3\times 1 =603\) or \(10\times 3600 + 0\times 60 + 3\times 1 = 36003\) or .... All three of these systems use abstract or stylized symbols based on a single line segment or dot to represent one. In contrast, the Egyptian system is pictographic. Our favorite example is the glyph of an astonished man to represent one million:

In order to depict larger and larger numbers, the Egyptians and the Romans needed more and more symbols. Any base system having as many numerals as its base does not need to use the additive property to depict all of its numbers.

Our modern Indo-Arabic base \(10\) place-value system eventually became the predominant system of all the various systems used throughout history (although it can be argued that a base \(12\) system would be more user-friendly given the larger number of divisors of \(12\) compared to \(10\)). Encoded in our system is the value of the number decomposed as a sum of multiples of powers of ten. Any whole number can be written as a sum of powers of \(10\) in the following way: \[a_0+a_1\,10+a_2\,10^2+\cdots+a_k\,10^k,\quad\quad\quad\quad (1)\] where each \(a_i\) is one of \(0,1,2,3,4,5,6,7,8\) or \(9.\) (The system can be extended to represent any decimal fraction, rational or irrational. Decimal fractions did not appear until the 16^{th} century.)

To take a completely arbitrary example, Amy's dog Isabel was born on May 4th, 2006.

**Figure 1.** Isabel is a five-year-old Keeshond.

If we encode Isabel’s birth date as \(5042006,\) it would be represented as: \[5(10^6 )+0(10^5 )+4(10^4 )+2(10^3 )+0(10^2 )+0(10^1 )+6(10^0 ).\] Note that in keeping with writing the most signficant digit on the left, we have reversed the order of the powers to align with the written number. Notice that, when writing a generalized expansion as in expression (1) above, we write in ascending powers of ten, but when we expand an actual number, we write in descending powers so as not to confuse the numerical order of the digits as compared to the actual written number.

Likewise, given any natural number \(b\,\,(> 1)\) as a base, any integer can be written as sums of multiples of powers of \(b\) in the following manner: \[a_0+a_1\,b+a_2\,b^2+\cdots+a_k\,b^k,\quad\quad\quad\quad (2)\] with all coefficients \(a_i\) satisfying \(0\le{a_i}\le{b-1}.\)

In addition to base \(10,\) two of the most ubiquitous number systems in use today are base \(2\) and base \(16,\) binary and hexadecimal respectively, used in computing. For example, in ASCII, every key stroke has one 8-bit binary representation and every 8-bit binary number represents a key stroke. To continue Amy's dog example (have to keep up with Elvis, the dog who does calculus, you know), suppose we wanted to convert Isabel's birth date into a hex number to see what her “true color” was. To express any integer \(n\) in a certain base, one of two iterative division algorithms can be used.

The “top-down” division algorithm works as follows. Divide \(n\) by the largest power of the base smaller than or equal to \(n.\) That quotient is the most significant, or leading, digit. The remainder is then divided by the next highest power of the base and so on. So Isabel's hex number is found as follows:

\(5042006\) | \(=\) | \(4(16^5)+847702\) |

\(=\) | \(4(16^5)+12(16^4)+612704\) | |

\(=\) | \(4(16^5)+12(16^4)+14(16^3)+3926\) | |

\(=\) | \(4(16^5)+12(16^4)+14(16^3)+15(16^2)+86\) | |

\(=\) | \(4(16^5)+12(16^4)+14(16^3)+15(16^2)+5(16^1)+6\) |

So we can write \({5042006}_{10} = {4(12)(14)(15)56}_{16}.\) Since hex needs more than ten digits (16 to be exact), the first six capital letters are used to denote ten through fifteen respectively. So Isabel's hex number is \(4CEF56.\)

It is important to note that this division algorithm is a *greedy* algorithm in that it fills the largest place value first by taking as large a "bite" as possible at each stage.

In the "bottom-up" algorithm, based on the Euclidean algorithm, we divide \(n\) by \(b,\) and take the remainder as the least significant digit. Then we consider the quotient, and iterate the procedure. In our case, we get

\(5042006\) | \(=\) | \(6+315125\times{16}\) |

\(=\) | \(6+(5+19695\times{16})\times{16}\) | |

\(=\) | \(6+5\times{16^1}+19695\times{16^2}\) | |

\(=\) | \(6+5\times{16^1}+15\times{16^2}+1230\times{16^3}\) | |

\(=\) | \(6+5\times{16^1}+15\times{16^2}+14\times{16^3}+76\times{16^4}\) | |

\(=\) | \(6+5\times{16^1}+15\times{16^2}+14\times{16^3}+12\times{16^4}+4\times{16^5}\) |

Hexadecimal numbers are currently used for encoding colors. Inputting Isabel's birth date in hex into an online RGB calculator, we found her RGB value to be \((76, 239, 86),\) which is a very bright green. We could not find an exact name for this color, but it is very close to one called "medium spring green." (Somehow we thought of her as more of an earth tone.) To convert back to base ten, simply multiply each base \(b\) digit by its respective power of \(b\) and add.

All place-value number systems inherently use the above expansion. This is precisely what makes a place-value system a place-value system and why these systems are so powerful. That the above expansion is well-defined is essential to the working of a place-value system. The existence and uniqueness of the above expansion have been attested to widely and need not be fully repeated here. The general proof of existence of such an expression is nothing but a generalization, using induction, of the procedure we used for Isabel's birth date. We will outline the proof of uniqueness on the next page, because it is exactly the issue we are going to address.

We now outline the proof of uniqueness of representation in a pure place-value system because this is exactly the issue we are going to address in the Maya number system. To establish uniqueness of the coefficients of the expression \[a_0+a_1\,b+a_2\,b^2+\cdots+a_k\,b^k,\quad\quad\quad\quad (2)\] in which all coefficients \(a_i\) satisfy \(0\le{a_i}\le{b-1},\) we can use a strong induction argument. The result is true for zero. (If you want the sum (2) to be zero, you need all coefficients to be zero.)

Now take a natural number \(n\) and assume uniqueness holds for every positive integer smaller than \(n.\) We wish to prove that \(n\) has a unique expression in base \(b.\) Suppose \[n=a_0+a_1\,b+a_2\,b^2+\cdots+a_k\,b^k=a^{\prime}_0+a^{\prime}_1\,b+a^{\prime}_2\,b^2+\cdots+a^{\prime}_t\,b^t,\] with, say, \(t\ge{k}.\) We want to prove that \(k=t\) and \({a_i}={a^{\prime}_i}\) for \(0\le{i}\le{k}.\) One can rearrange the equation to get \[a_0-a^{\prime}_0=(a^{\prime}_1-a_1)b+(a^{\prime}_2-a_2)b^2+\cdots\] where the left side of the equation is a number between \(-b+1\) and \(b-1\) and the right side is a multiple of \(b.\) Therefore both sides must be \(0,\) and \(a_0=a^{\prime}_0.\) Now we take, as was done in the example on page 2, the integer \(n_1\) such that \[n=a_0+bn_1\quad{\rm{or}}\quad{n_1}=\frac{n-a_0}{b},\] which is strictly smaller than \(n,\) and is expressed as \[n_1=a_1+a_2\,b+a_3\,b^2+\cdots+a_k\,b^{k-1}=a^{\prime}_1+a^{\prime}_2\,b+a^{\prime}_3\,b^2+\cdots+a^{\prime}_t\,b^{t-1}.\] By the hypothesis of strong induction, the result is true for \(n_1\) and we get that \(k=t\) and \({a_i}={a^{\prime}_i}\) for \(1\le{i}\le{k}.\)

If the encoding of a number were not well-defined, we would venture to say that our modern society, so reliant on technology that is driven by numbers (in particular base \(2\)), would come to a grinding halt. It should also be said that we are referring to the uniqueness of a finite integer, regardless of the base. But it also applies to terminating fractional expressions. The well known case of the non-uniqueness of decimal fractions ending in repeating \(9\)'s (for example, \(1.999\dots=2\)) does not fall under this definition. However, since no computer system uses an infinite expansion, the dire consequences of non-uniqueness are avoided. But there was one highly advanced civilization whose place-value system did not maintain the uniqueness requirement. Although this civilization did not come to a grinding halt, it did mysteriously decline ....

The Maya of the Yucatan were a vibrant and sophisticated people. They flourished in the jungles of what is now central and southern Mexico and Guatemala from approximately 1500 BCE to 1500 CE. During their Classical Period (250-900 CE) they created magnificent buildings and monuments to their kings and gods, elaborate works of art, and a sophisticated hieroglyphic language. As with any civilization with grand buildings, maintenance of such an advanced society had its costs on its citizens in the form of taxes and labor, as well as exhaustion of resources. For unknown reasons, but perhaps due to the latter, the Mayan civilization went into decline about the same time Europe was waking from its dark period. In the early 16th century, it was all but stamped out by the Spanish conquistadors.

Living in the thick jungles of Mesoamerica was difficult. The Mayan culture was agricultural, which required accurate predictions of seasonal changes. Their society revolved around a pantheon of gods and the rituals required for them. Intimately tied to their religion was their calendar with its notion of cycles, while central to their astrology was the planet Venus and its cyclical movements. Over time they became expert astronomers and developed a very accurate, though extremely complicated, system of calendrics.

The Maya had an elaborate hieroglyphic written language, with over 700 glyphs, only a portion of which have been translated to date. The vast majority of Mayan writing was on civic monuments or in books made of deer skin, cotton cloth or maguey paper, folded like a fan and now called codices [6]. The primary purpose of these writings was to extol their gods, record the reigns of their kings, commemorate important events, or retell epic battles. Given the often harsh payment in human blood required by their gods, many of the artistic depictions found in the codices are graphic and gory. When the Spanish found these books, they naturally (given their background) viewed them as works of the Devil. Countless codices were destroyed by fire at the hands of mainly one man, Friar Diego de Landa (1524-1579). Sadly or luckily, depending on your outlook, only four intact codices are known to have survived (possibly because they had already been removed from Mesoamerica). The Dresden, Paris, and Madrid Codices are named after the locations of the museum at which they are housed, while the Grolier Codex is named after its New York publisher. In an attempt to save some books, the Maya buried them. But of the ones that have been found, the damp conditions took their toll and only scraps remain [6]. So the vast majority of the Mayan writings and numbers that remain are found on carvings on stelae and buildings, and they include many dates. These dates are given in the context of the Mayan Long Count calendar.

From the cycles of the seasons to the vast cycles of the heavens, the Mayan world view literally *revolved* around cycles. The Maya believed that the world went through cycles of rebirth and destruction, and that the current world was the fifth such manifestation or "Great Cycle". The current Great Cycle is now over 5000 years old and started on August 12, 3113 BCE in the proleptic Julian calendar, according to Cooke [2] (though other scholars have calculated a slightly different start date), and will end, possibly cataclysmically, on December 21, 2012. The Long Count gives the time in days since the start of the current Great Cycle. But the Long Count was just one record of time for the Maya, who had at least three other calendars. The Maya calendar system is a marvel of modular arithmetic that accounts for the cyclical nature of the seasons and the motions of the moon, the sun, Venus, and all of the heavens.

Briefly, there were four calendars in use, three for time periods on a human scale and one for the Long Counts. The *Tzolkin,* which means "count of days," was a 260-day year composed of two independent cycles made up of 13 day-numbers and 20 day-names [3]. The Maya also employed a 365-day solar year known as the *Haab* or "vague" year because it slowly became out of sync with the seasons. In the solar calendar, the Haab is made up of 18 months of 20 days, plus 5 dangerous "unnamed" days, called the *Wayeb.* These two calendars align every 52 Haab years or 73 Tzolkin years and together make up a *Calendar Round* [1, 4]. For much longer periods of time, needed to record historical events, the Long Count was used.

*Editor's note:* More information about Maya calendars can be found in the following articles here in *Convergence*.

Maya Calendar Conversions, by Ximena Catepillan and Waclaw Szymanski (2010)

Maya Cycles of Time, by Sandra Monteferrante (2007, reprinted 2012)

The Mayan culture used a base \(20\) number system. It was an additive positional system that used two symbols, a dot for one, a horizontal bar for five, and a cowry shell for a place holder (it is unclear whether they also considered it a true numeric "zero''). Numbers were written vertically with the most significant digit at the top. They used a purely base \(20\) system for simple recording of commodities. However, their much more prevalent and culturally important calendar system used a modified base \(20\) system. The place values employed in this number system are tied to the Calendar Round and Long Count calendars. Each place represents the next order of numbers of days. So the first place, \(20^0,\) is days. The next, \(20^1,\) is months. The third would be years made up of \(20^2=400\) days. However, \(18\times20\) more accurately represents the year of 360 days (plus 5 Wayeb days) than does \(400,\) so here they used \(18\times20=360\) rather than \(20^2=400.\) Thus the Maya developed a modified base \(20\) system in which a date is represented by a number written (in modern Indo-Arabic notation) as: \[a.b.c.d.e=a(18\times20^3)+b(18\times20^2)+c(18\times20^1)+d(20^1)+e(20^0),\] where the third place value is not \(20^2\) but \(18\times20.\) After the third place, each higher place is \(20\) times the previous place value. So the system breaks only in the third place.

Since the Mayan system is base \(20,\) as in hex (base \(16\)), we need more than our ten digits to write their numbers. Standard modern practice for writing Mayan numbers is to write left to right with the most significant digit on the left, as in modern numeration, separating each place by a period. As an example using Indo-Arabic numerals for simplicity, we return to Isabel's birth date of \(5042006,\) which converts via the division algorithm to the Mayan numeral \(1.15.0.10.6\). This would be expanded as: \[1 (18 \times 20^3) + 15 (18 \times 20^2) + 0 (18 \times 20^1) + 10 (20^1) + 6 (20^0).\] This expansion shows clearly the modified nature of the Mayan calendric number system. Each place is a higher power of \(20\) except in the third place (what we would think of as the hundreds place in base \(10\) and should be \(20^2\) in base \(20\)) where a \(20\) has been replaced by an \(18.\)

Please note that the calculation above is simply the modern Gregorian date for Isabel's birthday converted into Mayan numerals; it is not the date in the Mayan Long Count. Since the focus of this article is the uniqueness of the Mayan number system, and not their dating system, we preferred to focus on the conversion of a number and not how to convert dates. However, there are various websites that will convert a modern date into a Mayan Long Count date. Using the Maya Astronomy Page [7], the Mayan date of Isabel's birthday is \(12.19.13.3.8,\) or \[12 (18 \times 20^3) + 19 (18 \times 20^2) + 13 (18 \times 20^1) + 3 (20^1) + 8 (20^0) = 1,869,548\] days into the \(1,872,000\)-day Long Count cycle.

In general, a natural number \(n\) would be expanded in the Mayan modified base \(20\) system as: \[a_0 + a_1 (20) + a_2 (18 \times 20) + a_3 (18 \times 20^2) + a_4 (18 \times 20^3) + \cdots + a_k (18 \times 20^{k-1}),\] but would be encoded \(a_k. \cdots .a_3.a_2.a_1.a_0,\) again with the most significant digit on the left.

Below is an example of a *Serpent Date* found in the Dresden Codex.

**Figure 2.** Page from the Dresden Codex showing two Serpent numbers [5]

Two serpents are shown, each depicting two numbers or dates (one in black and one in outline). To read these "bar and dot" Maya numerals, we need to know only that bars represent \(5\)'s and dots \(1\)'s (and cowry shells \(0\)). Thus the numeral just to the right of the tail of the righthand serpent consisting of three bars and four dots is \(3\times5+4\times1=19,\) the largest Maya "digit". Taking the black number in the right serpent we have: \[4 (18 \times 20^4) + 6 (18 \times 20^3) + 9 (18 \times 20^2) + 15 (18 \times 20^1) + 10 (20^1) + 19 (20^0)\] \[= 12,449,559\] in base \(10.\) Figure 3 shows the Serpent number for Isabel's birth date, \(1.15.0.10.6\).

**Figure 3.** Isabel's birth date as a Mayan Serpent number. Amazing artwork by Amy and her son Brian, whose birth date is \(4.3.8.6.12.4\).

When converting from a base \(10\) number to a Mayan number using the greedy division algorithm, place values are filled from left to right, that is from largest place value down. As noted at the beginning, this process leads to a unique expression in a “pure'' base system such as ours. But the Mayan system is not pure; the \(18\) makes for some interesting choices, depending on whether you think of building a number from the top or from the bottom.

Sometimes it is easier to think of a base \(b\) place-value number as made up of bins. Each bin can have zero to \(b-1\) sets (or bags), each containing a full complement of the next lower place value. For example, in the base \(10\) number \(253,\) the left bin has \(2\) sacks, each with \(10\) bags of \(10\) counters; the next bin to the right contains \(5\) bags, each with \(10\) counters; and the last bin on the right contains \(3\) individual counters. So if a number is thought of as being filled from the top, as with the greedy algorithm, as many sets of the largest power of the base as possible are put in the left bin. Then from the remainder, as many complete sets of the next lower power as possible are put in the next bin, and so on.

On the other hand, if a number is thought of not in terms of sets of powers of the base, but as piles of individual counters, a natural method would be to start putting counters in a bin starting on the right with the units place. As soon as a set of size \(b\) is obtained, a single counter to represent a set of size \(b\) is put in the next bin to the left (the second bin from the right), the contents of the right bin are discarded, and one resumes putting counters in the right bin.

This is the practical procedure suggested by the second (bottom-up) algorithm we presented. This is also, in essence, how calculations are done on an abacus. When one column is full, a single counter on the next wire to the left is moved up and the previous column is cleared. In addition and multiplication we call this carrying (to the next highest power). For a pure base system, either method (filling from the top or the bottom) will give you the same encoded number.

But in the Mayan system, the third and higher place values are not pure. The third place is \(18\times20.\) If, in a certain expression, the number in the second place (the \(20\)'s place) is \(18\) or \(19,\) all or part of it could be carried over to the third (or \(18\times20\)’s) place. Or if the second place is only a \(0\) or a \(1,\) then an \(18\times20\) can be carried *backward* from the third place down to the second, making the second digit now \(18\) or \(19.\) Thus the same natural number can be encoded in two different ways in the Mayan system. We will make this explicit in a moment, but first some examples are in order. Please note the following. In a pure number system, when a place value is filled, \(1\) is carried to the next highest place and the lower bin is emptied. This is not the case in the second place of the Mayan numbers. When the second place is full, it has \(20\) sets of \(20,\) but the third place is \(18\times20.\) So the whole \(20\times20\) cannot be carried over to the third place; only \(18\times20\) can be carried, leaving \(2\times20\) in the \(20\)'s place. If the second place is \(18,\) then \(18\times20\) is carried and the place is emptied; if the second place has \(19\) or is full with \(20,\) then when \(18\times20\) is carried, a \(1\) or a \(2\) must be left behind in the second place.

Consider the Mayan number \(4.8.18.9\). In expanded form, this would be \[4(18\times20^2)+8(18\times20)+18(20)+9.\] Notice that the second and third places are both multiples of \(18\times20.\) So the whole \(18\times20\) in the second place can be carried over as one unit in the third place, resulting in the new, but equivalent Mayan number \(4.9.0.9\). Both of these representations equal \(32049\) in base \(10.\) See Figure 4 for two more (yet again amazing) serpent numbers depicting these two equivalent numbers. Likewise the Mayan number \(4.8.19.9\) could also be written as \(4.9.1.9\) by carrying from the second to the third place.

**Figure 4.** \(4.8.18.9\) on the left, \(4.9.0.9\) on the right, both equivalent to \(32049\) base \(10.\)

Working the same two examples the other way, if you have the Mayan numbers \(4.9.0.9\) or \(4.9.1.9,\) one unit of \(18\times20\) from the third place can be carried down to add \(18\) to the second place, giving \(4.8.18.9\) and \(4.8.19.9,\) respectively.

To summarize, given a Mayan number of the form \(a_k .\ldots.a_2.a_1.a_0,\) where \(a_1=18\) or \(19,\) an equivalent Mayan number would be \[a_k.\cdots.(a_2+1).(a_1-18).a_0,\] with a \(1\) carried to the fourth place as usual if \(a_2 + 1 = 20.\) Conversely, if a Mayan number is of the form \(a_k.\cdots.a_2.a_1.a_0,\) where \(a_2>0\) and \(a_1=0\) or \(1,\) then the equivalent Mayan number would be \[a_k.\cdots.(a_2-1).(a_1+18).a_0.\]

Outside of these two cases, any other combination of numerals in a Mayan number is unique. In other words, if we can find an expression for \(n\) in which \(a_1=18\) or \(19,\) we have two expressions for \(n.\) If this does not happen, the expression is unique. (Of course, any other non-pure place-value system would violate uniqueness. However, the Mayan system is the only non-pure system we are aware of.)

To check this, suppose \(n\) is a number that has no expression (in the Mayan system) in which the second place is \(18\) or \(19.\) Take two possible expressions:

\(n=\) | \(a_0+a_1\,(20)+a_2\,(18\times20)+a_3\,(18\times20^2)+\cdots+a_k\,(18\times20^{k-1})\) |

\(=\) | \(a^{\prime}_0+a^{\prime}_1\,(20)+a^{\prime}_2\,(18\times20)+a^{\prime}_3\,(18\times20^2)+\cdots+a^{\prime}_t\,(18\times20^{t-1}).\) |

According to our assumption, \(0\leq a_1,a^{\prime}_1\leq 17,\) and for \(i\neq 1,\) \(0\leq a_i\leq 19.\)

To start with, we have \(a_0=a^{\prime}_0,\) by the same argument used in the general proof. When we now take \(n_1=(n-a_0)/20,\) we get, using the very same argument, \(a_1=a^{\prime}_1,\) since \(a_1-a^{\prime}_1\) is both a multiple of \(18\) and a number in absolute value smaller than \(18.\)

We now take \(n_2=(n_1-a_1)/18\) and get equality of all other \(a_i\)'s, and \(k=t,\) since now we are dealing with a number in pure base \(20.\)

Thus for all cases when the second place number cannot be chosen to be \(18\) or \(19,\) the Mayan notation is unique.

So, for which base \(10\) natural numbers is the Mayan number not unique? As we said above, this happens when the second numeral can be chosen to be \(18\) or \(19.\) Then we can carry one \(18\times20\) to the next higher place. The smallest such numbers are found by assuming all higher place values are zero. Then we have the range of natural numbers \([360,399]\) corresponding to the Mayan numbers from \(18.0=1.0.0\) to \(19.19=1.1.19,\) for which the Mayan number is not unique. Other intervals of non-unique Mayan numbers would be found by filling in non-zero numerals in any and all of the higher places while retaining the restriction on the second place. The above ideas make for great open questions in many different classes, from elementary school through college courses, as long as the Mayan calendric number system has been introduced.

**Conclusion**

The natural question that Amy asked, being a math historian, is “Did the Maya have a preference for one encoding over the other? In fact, were they even aware of this duality of their number system?'' Given the limited number of Mayan texts remaining, this question may never be answered. A by no means exhaustive survey, as well as conversations with Dr. Ed Barnhart of the Maya Exploration Center (see [8]), indicate that no Mayan numbers with an \(18\) or \(19\) in the second place appear on known monuments or documents. This makes sense when you think again of the world view of the Mayans. As noted at the beginning, these numbers are Long Count dates used on stelae and in codices to record significant events. It would make sense to record dates using the largest units of time possible. For example, it would seem natural to record a date as one Haab (rounded to 360 days) as opposed to 18 months of 20 days.

Finally, Amy is happy to conclude that Isabel's birthday, written as a Mayan number, is \(1.15.0.10.6,\) which is a unique expression! Could you imagine the hassles with the IRS if the expressions of our birth dates were not unique?

In conclusion, we would just like to say: regardless of the number system you use,

**All Your Base Are Belong to Us.*******

**About the Authors**

**Pedro Freitas**, of Lisbon, Portugal, and **Amy Shell-Gellasch**, originally from Birmingham, Michigan, became dear friends during graduate school at the University of Illinois at Chicago in the late 1990s. Pedro studies matrix theory and is a professor at the FCUL Universidade de Lisboa. Amy is a historian of mathematics and will join the Hood College faculty in September of 2012. During a visit by Pedro from Lisbon to Amy’s home in Madison, Wisconsin in 2011, teaching math history came up in conversation. This led to the topic of different number bases, which in turn led to a comment by Pedro about the non-uniqueness of the Mayan numbers. This in turn led to several fun hours investigating all the implications of that and finally this article.

*****Google it!

[1] Ascher, M., Before the Conquest, *Mathematics Magazine,* Vol. 65, No. 4, Oct., 1992, pp. 211-218.

[2] Cooke, R., *The History of Mathematics: A Brief Course,* 2nd ed., Wiley-Interscience, John Wiley & Sons, 2005.

[3] Katz, V. J., *A History of Mathematics: An Introduction,* 3rd ed., Addison-Wesley, 2009, p. 270.

[4] Morley, S. G., *An Introduction to the Study of the Maya Hieroglyphs, *57, Smithsonian Institution Bureau of American Ethnology, Government Printing Office, 1915. Also available from Google Books.

[5] Maya Info: Ring and Serpent Dates in the Dresden Codex, www.mayainfo.org/works/ring/default.asp

[6] Mundo Maya: Mayan Codices, www.mayadiscovery.com/ing/history/codices.htm

[7] The Maya Astronomy Page: Maya Calendar, www.michielb.nl/maya/calendar.html

[8] Maya Exploration Center, http://mayaexploration.com/staff_barnhart.php

**Related articles in Convergence:**

Maya Calendar Conversions, by Ximena Catepillan and Waclaw Szymanski (2010)

Maya Cycles of Time, by Sandra Monteferrante (2007)