Number Systems - Mesopotamian and Mayan

     Glenn Mason-Riseborough (26/9/1997)

Introduction
As a civilisation progresses, it encounters numerous problems 
which must be overcome in order to develop.  Initially problems 
are of a practical nature and are concerned with the basics such 
as food and shelter.  Very early on, people learned the 
importance of abstract representation of objects to obtain 
consistency.  For example, a common method of recording by 
illiterate and innumerate shepherds was to use the concept of 
one-to-one correspondence to count their sheep.  They did this 
by placing a pebble in a bowl as the sheep walked past, and 
taking the pebbles out when the sheep returned.  This method of 
representation continues in some societies even to this day (Tee, 
1997).  Writing developed as an economic necessity and tokens 
with symbols were used in many areas for the bookkeeping of 
expanding businesses (Nemet-Nejat, 1993).  Thus, 
comprehensive number systems often developed as an offshoot 
of writing and reflected the need to record such things as time, 
distance, and quantities of objects.  This included such things as 
stock, possessions, censuses, and eventually armies, taxes, dates, 
and construction measurements.
From a current perspective, it is nearly impossible to examine the 
number system of a pre-literate society simply because no 
evidence remains today of the way they represented numbers.  
Objects which contain markings or tallies such as the Ishango 
Bone (1) are of interest, but we can never be entirely sure about the 
meaning of such artefacts.  Thus, if we are to examine the 
emergence of number, we must start from already literate 
societies.  This essay will compare and contrast the number 
system of two such cultures—that of the Mesopotamian and 
Mayan civilisations.  These two cultures developed completely 
separately at different times on different continents, but despite 
this there are numerous similarities between them.  The emphasis 
of this essay will be on the method the Mesopotamians and 
Mayans used to represent numbers, and the various arithmetic 
operations that were performed.

Mesopotamian Number System
Mesopotamia (literally meaning “the land between two rivers”) 
was the Greek name for the area of land contained by the 
Euphrates and Tigris rivers in the Middle East.  However, 
modern usage of this term usually refers to most of present day 
Iraq.  This modern definition adds the land east of the Tigris 
(ancient Assyria) and the lower Tigris-Euphrates Valley from 
Baghdad to the Persian Gulf (ancient Babalonia) (Jones, 1993).  
Mesopotamia has a long history of inhabitation, covering several 
millennia and including many cultures up to the present day.  In 
the context of number systems, this essay will focus on 
Mesopotamia from around 2200 BCE to 539 BCE (2), however it 
will be necessary to briefly discuss events outside this time frame 
in order to get a perspective of the developments.

Historical background
Clay tokens containing markings have been found throughout 
Mesopotamia dating as far back as the eighth millennium BCE 
(Hoyrup, 1994), however it was not until around 3500 BCE that 
small Sumerian city-states (such as Ur, Nippur and Lagash) 
formed and the earliest written records begin (Joseph, 1991).  
Around 2400 BCE these city-states were formed into the first 
Mesopotamian empire by Akkadian invaders led by Sargon I, 
with centres of power at Sumer and Agade.  This empire lasted 
until around 2200 BCE when wars and conflict returned rule to 
the individual city-states (Joseph, 1991).  The First (or Old) 
Babylonian empire emerged around 1900 BCE with its centre of 
power at Babylon.  This empire successfully merged the 
Sumerian and Akkadian cultures, and was most notably ruled by 
King Hammurabi (1792 - 1750 BCE).  The First Babylonian 
empire collapsed around 1650 BCE, and invasions by people 
such as the Hittites (1600 BCE) and Hurrians and Mitanni (1000 
BCE) meant long periods of unrest (Joseph, 1991).  In 885 BCE 
the area was invaded by the Assyrians who established their 
capital at Nineveh.  Notable rulers during this period included 
Sennacherib (705 - 681 BCE) and Ashurbanipal (668 - 627 BCE) 
(Joseph, 1991).  The Assyrians were conquered in 612 BCE by 
the Chaldeans who established the Second (or New) Babylonian 
empire with its capital at Babylon.  Nebuchadnezzar (604 - 561 
BCE) was a notable ruler during this period (Joseph, 1991).  In 
539 BCE a Persian invasion led by Cyrus the Great destroyed the 
Second Babylonian empire and in 312 BCE the Greeks 
developed a presence in the area with the establishment of the 
Seleucid dynasty (Joseph, 1991).

Writing system
The Mesopotamians developed a cuneiform style of writing using 
wedges which evolved from stylised pictures of objects.  These 
wedge shaped symbols were pressed into tablets of wet clay 
using a stylus, before allowing the clay to harden.  The stylus 
used was a pencil shaped cylinder with an oblique cut at one end 
that could make three different kinds of marks—circles, semi-
ellipses, and semi-ellipses with a straight line.  Combinations of 
these three marks represented the syllables of the language 
(McLeish, 1991). Hundreds of thousands of clay tablets (ranging 
in size from postage stamps to pillows) have been found 
containing writing on various subjects however less than 500 of 
these directly relate to mathematical ideas (Joseph, 1991).

Representation of numbers
Mesopotamian tablets show that a sexagesimal (base 60) 
positional number system was developing as early as 3000 BCE 
however there are few records in existence until the First 
Babylonian empire (1900 BCE) (Joseph, 1991).  The few early 
tablets discovered show that around 3000 BCE the system was 
not yet fully positional for all powers of ten and their multiples.  
Hence, there were different symbols for 1, 10, 60, 600, and 
3600.  However, by 2000 BCE the system had evolved into a 
place-value system and contained just two symbols for 1 and 10 
respectively (Joseph, 1991).  The symbol T was used to 
represent one and the symbol < represented ten—collections of 
these symbols were used to write the numbers between 1 and 59.  
The symbol T also represented 601, 602, 603, ... (< represented 
each of these values multiplied by ten) and thus successive 
combinations of T and < could represent all possible positive 
integers (Hoffman, 1957).  For example <<<<TTT = 43 and 
<<T<<<TTTTT = 21 ? 60 + 35 = 1895.  This system was also 
expanded to enable fractions to be represented (Joseph, 1991).  
Thus, the symbol T also represented 60-1, 60-2, 60-3, ... (< 
represented each of these values multiplied by ten), and so 
TTTTTTT<<< = 7 ? 60-1 + 30 ? 60-2 = 0.125.
The most notable deficiencies of the system described above was 
the lack of a symbol for zero and no corresponding decimal point 
(Joseph, 1991).  Thus <TT may represent 12 or it may represent 
12 ? 60 = 720 or it may represent 12 ? 60-1 = 0.2.  Joseph 
(1991) explains that a possible reason for this deficiency was that 
it was not seen as a problem.  Numbers were only referred to in 
some context of everyday life, and thus in using a large base (60) 
was usually sufficient to cover all possibilities.  For example, 
Joseph suggests that prices of commodities were probably 
unlikely to be outside the range of 1 to 59 units.  Joseph also 
explains that the symbols were strictly written left to right, thus 
for example TT<<<< would always be interpreted as 160 
(multiplied by some multiple of 60) and not 42 (multiplied by 
some multiple of 60).  Having said this, Tee (1997) notes that 
there were cases in which zeros were represented by spaces (this 
would only show relative place values between each digit, but 
would not show how many zeros there were to the right of the 
least significant figure) or explicitly (after the 27th century BCE) 
by ?.  On the other hand, Joseph (1991) states that an explicit 
zero symbol (represented by     ) did not occur until during the 
Second Babylonian empire.
One clearly remarkable aspect of this system was the choice of 
base to use.  60 has a large number of factors (compared with 
other bases such as 5, 10, 12 or 20), these are 1, 2, 3, 4, 5, 6, 10, 
12, 15, 20, and 30.  Thus a large number of different fractions 
were able to be represented very easily with great precision and 
using only a few places.

Arithmetic operations
Mathematical problems in Mesopotamia were always of a 
practical nature, and reflected the need to find areas, lengths, 
weights, interest rates, and various measurements and 
calculations for everyday living.  The positional system of 
numeration allowed the Mesopotamians to solve many 
arithmetical problems that we solve today in a similar manner.  
Mathematical tables were used extensively to aid long 
calculations, and hundreds of tablets containing such tables have 
been found.  These tablets included tables for finding reciprocals, 
squares, cubes, square roots, cube roots, exponentials and even 
tables for n3 + n2, as well as tables for multiplication of numbers 
between 2 and 20 and for 30, 40 and 50 (Joseph, 1991).
Multiplication and division were performed in a similar manner as 
we would today.  Division was achieved by multiplying the 
dividend by the reciprocal of the divisor (obtained from a table of 
reciprocals) (Joseph, 1991).
Tables showing the upper and lower limits of fractions such as 
1/7 show that by the Seleucid period the Mesopotamians were 
aware of the incommensurability of certain numbers (Joseph, 
1991).  An exceptional example of their awareness of irrational 
numbers is from a tablet which dates at approximately 1800 BCE 
and was found in the ruins of Nippur (Tee, 1997).  This tablet 
contains a diagram of a square showing its diagonals.  Near one 
side the number <<< (30) is shown, while across the central 
diagonal is shown T<<TTTT<<<<<T< (1.41421297 or ?2) and 
<<<<TT<<TTTTT<<<TTTTT (42.4263891 or 30 ? ?2).  This 
is strong evidence that the Mesopotamians were aware of the 
Pythagorean theorem and there is evidence that they were aware 
of Pythagorean triples as well (3) (Joseph, 1991).

Mayan Number System
The ancient Maya were an American Indian people who 
inhabited central America.  At the time of the Spanish invasion in 
1519, their area of control covered over 300,000 km2 (Joseph, 
1991) including southern Mexico, Belize, Guatemala, El 
Salvador, and the western parts of Honduras (Closs, 1994).  The 
descendants of these people, the modern Maya, still live in this 
area today.

Historical background
Dates for Mesoamerican cultures are typically divided into a 
number of different periods and sub-periods.  The earliest Maya 
villages date back to the end of the Early Preclassic (about 1000 
BCE) (Closs, 1994).  Throughout most of Maya history the 
population was controlled by a small group of elite rulers and 
during the Classic period (250 - 900 CE) these rulers reigned as 
divine kings.  The Maya civilisation thrived during this period, 
however numerous ongoing problems such as invasion, famine 
and possibly disease contributed to its fall and by 1000 CE no 
lowland Classic Maya domain remained (Miller, 1993).  During 
the Postclassic period, numerous city-states held power briefly, 
however after the arrival of the Europeans in the early 16th 
century, disease and strife broke the Mayan civilisation into tiny 
states.  The Spanish took control in 1542 and built their own 
capital at Merida; the last Maya kingdom was taken by the 
Spanish in 1697 (Miller, 1993).

Writing system
Most Mayan writing found today consists of hieroglyphics 
written on stone pillars (stelae), walls of ruins and a few 
manuscripts made of paper from plant fibres.  Many more 
manuscripts were destroyed by the Spanish conquerors, most 
notably by Father Diego de Landa (1524 - 1579 CE) who 
considered them heretical.  As penance for his sins, Father Landa 
attempted to collect all Mayan knowledge by word-of-mouth, 
thus much of our knowledge comes from this source (Bidwell, 
1994).  Writing first developed in Mesoamerica during the 
Middle Preclassic (900 - 300 BCE) probably by the Zapotecs of 
Oaxica however the Mayans only began to use it during the Late 
Preclassic (300 BCE - 300 CE) (Miller, 1993).  Most writings 
occurred during the Classic period (250 - 900 CE) when the 
Maya civilisation was at its height.

Representation of numbers
The Mayan system of notation was very similar to that of the 
Mesopotamians in the sense that it only contained two numerical 
symbols plus a symbol for zero.  The symbols used were the dot, 
?, for 1 (which represented a pebble), the bar, |, for 5 (which 
represented a stick), and the zero symbol,      (which possibly 
represented a snail or clam shell) (Joseph, 1991).  Often non-
numerical symbols would be added to achieve a more aesthetic 
balance to the numeral (Closs, 1994). Unlike the 
Mesopotamians, the Mayans used a vigesimal (base 20) system 
of representation.  In some cases a positional notation was used 
and the numerals were set out vertically with the most significant 
figure at the top (Closs, 1994).  For example 240 may be 
represented as:

On dynastic monuments however, the Maya tended to use a non-
positional notation, using glyphs to represent chronological 
periods of different sizes.
In a similar fashion to the Mesopotamians, the Mayans only used 
zero in later texts.  A zero sign is found in numerical texts with 
positional notation by 665 CE (the Dresden Codex), however 
there is no zero sign for positional notation used on a monument 
dated at 36 BCE (Closs, 1994).  The zero symbol is used earlier 
for non-positional notation.  The oldest dated usage of this is 357 
CE (Stela 18 and Stela 19) while earlier monuments (e.g. Stela 
29, constructed 292 CE) do not contain the non-positional zero 
symbol (Closs, 1994).
Having said that the Maya used a vigesimal system with only two 
symbols, there are a number of exceptions and complications to 
this system.  Instead of bar and dot numerals, the Maya 
sometimes used glyphs of heads or even entire figures to 
represent numbers (Closs, 1994).  These heads represented the 
thirteen deities of the superior world with six variants (Joseph, 
1991).  This head-variant system often occurred alongside the 
dot and dash system (Joseph, 1991).  In addition, the Mayan 
numerical system was not completely vigesimal.  In order to get 
values similar to the number of days in the year and make their 
three calendars compatible, there is a departure from the normal 
representation in the second place-value position.  Instead of b2 = 
202 = 400, the system is 18b = 360 (Joseph, 1991).  All place-
values after this point reflect this irregularity and are of the form 
18bi, where i = 2, 3, ....  This irregularity introduces a significant 
inefficiency into the system such that it is no longer possible to 
multiply the number by 20 by adding a zero on the end.

Arithmetic operations
A large component of Mayan arithmetical operations was 
concerned with the calculation of calendars and dates.  The 
Mayans three calendars were calculated by the priests and used 
for religious or daily activities (Joseph, 1991).  As a 
consequence, the calculations could be relatively complex 
because they were only used by a small elite group.  
Unfortunately, we do not know a lot about the actual 
calculations performed because the dates or tables of data were 
only recorded in condensed format and did not show calculations 
(Bidwell, 1994).  The use of addition and subtraction is indicated 
in manuscripts and some manuscripts show tables of repeated 
additions which may be seen as multiplication in a limited sense 
(Bidwell, 1994).  No evidence indicates that there was 
multiplication or division in the modern sense, and there is no 
indication of any fractions in any manuscripts (Bidwell, 1994).  
Addition would have been performed by combining the bars and 
dots, carrying to the next higher place, then exchanging five dots 
for one bar and four bars for a dot in the next place.  Subtraction 
would have occurred by cancelling symbols and ‘borrowing’ 
where necessary.  It is possible that these operations were 
physically carried out using sticks and stones (Bidwell, 1994).  
Having said that there is no evidence of multiplication or division 
in Mayan numeration, the number system itself did not exclude 
the possibility of performing these operations.  Multiplication 
tables showing dots or dashes can easily be constructed and long 
division may be performed due to the positional nature of the 
number system.

Conclusions
Both the Mesopotamian and Mayan number systems developed 
out of a necessity to calculate and record practical problems.  
The Mesopotamians were often interested in problems which 
contained measures such as area, length and weight, while the 
Mayans concerned themselves primarily with dates, times and 
periodic cycles.  Both cultures developed a written system which 
utilised two numerical symbols plus a zero symbol, and in both 
cases their systems had subsidiary bases (10 for Mesopotamia 
and 5 for Maya) in addition to a main base (60 for Mesopotamia 
and 20 for Maya).  Another similarity between the two number 
systems was the development of the zero symbol which slowly 
came into usage over time.
One of the primary differences between the two cultures was the 
directions in which they were influenced and the consequence 
difference in emphasis in calculations.  The Mesopotamian 
history covered a greater time period and it included influences 
from the many different cultures of the surrounding and invading 
civilisations.  Their predominant calculation of areas and lengths 
drew their attention to multiplication, division, squares, cubes, 
roots and other more complex relationships.  Consequently, their 
number system included fractions, even though there was some 
ambiguity as to the place value of the numbers.  On the other 
hand, the Mayans developed complex and extremely accurate 
calendars which were able to calculate the year even more 
accurately than the Gregorian calendar.  They also calculated 
extremely precise accounts of numerous other time intervals and 
cycles.

References:
Bidwell, J. K. (1994). Mayan arithmetic. In F. J. Swetz (ed.) 
From five fingers to infinity: A journey through the history 
of mathematics. Chicago: Open Court.

Closs, M. P. (1994). Maya mathematics. In I Gratten-Guinness 
(ed.) Companion encyclopedia of the history and 
philosophy of the mathematical sciences. London: 
Routledge.

Hofmann, J. E. (1957). The history of mathematics. New York: 
Philosophical Library.

Hoyrup, J (1994). Babylonian mathematics. In I Gratten-
Guinness (ed.) Companion encyclopedia of the history and 
philosophy of the mathematical sciences. London: 
Routledge.

Jones, T. B. (1993). Mesopotamia. In New Grolier multimedia 
encyclopedia: Version 6.0 (Computer program). Grolier 
Electronic Publishing.

Joseph, G. G. (1991). The crest of the peacock: Non-European 
roots of mathematics. London: Penguin Books.

McLeish, J. (1991). Number. London: Bloomsbury.

Miller, M. E. (1993). Maya. In New Grolier multimedia 
encyclopedia: Version 6.0 (Computer program). Grolier 
Electronic Publishing.

Nemet-Nejat, K. R. (1993). Cuneiform mathematical texts as a 
reflection of everyday life in Mesopotamia. New Haven, 
Connecticut: American Oriental Society.

Tee, G. J. (1997). History of number systems. Unpublished 
Manuscript, University of Auckland.

Endnotes:
1 The Ishango bone is bone tool handle that was found in 
Central Africa and thought to have been made approximately 
20,000 years ago.  The bone contains a series of notches grouped 
together in different rows.  Some interpretations indicate that 
it may be more than a simple tally as some of the rows indicate 
addition and groupings of prime numbers (Joseph, 1991).
2  2200 BCE refers to the collapse of the Akkadian empire.  
Mesopotamians are generally considered to be the successor 
peoples of Mesopotamia who continued the Sumerian culture after 
the Akkadian empire collapsed (Tee, 1997).  539 BCE refers to 
the conquest by the Persians.  Soon after this, the area came 
under the influence of the Greeks and thus Mesopotamian culture 
(and mathematics) was merged with the expanding Greek influence.
3 This evidence comes from a tablet known as Plimpton 322.  This 
tablet contains four columns (possibly more columns have broken 
off) of numbers which show relationships indicating it may have 
served some ‘proto-trigonometric’ function (Joseph, 1991).