436 UNIT 4 MESOAMERICAN CULTURAL FEATURES
Box 11.8 The origin of place-value numeration and the discovery of zero
(from Justeson 2001)Mesoamerican languages express higher numbers with expressions that parallel polynomial
equations in algebra: Each number is the sum of a series of terms, each of which consists of a unit
that is a power of a base b(20, in vigesimal systems) multiplied by a coefficientthat is a whole
number between 1 and b1; in the vigesimal system (b= 20). This yields expressions like “two
20s and fifteen” for 55, or “three 400s and six” for 1,206. Spoken languages have no term cor-
responding to zero, so 1,206 is not expressed as “three 400s, zero 20s and six.”
During the Late Preclassic period, epi-Olmecs and Mayans used a system of place value
notation that is effectively an abbreviation for these kinds of representation: the coefficients are
listed in order, without the units, yielding a sequence like “two fifteen” for “two 20s and fifteen.”
So far as we know, this system was used only for counting days, and an adjustment was made to
better fit this use. Instead of the expected unit of 400 days, an approximate year of 360 days was
used in its place, and higher units were vigesimal multiples of that period: a year (360 days), a unit
of 20 years (7,200 days), and a unit of 400 years (144,000 days). This was employed especially to
record dates in the long count,also known as the “count of katuns” (katu:nand maywere words
for the 20-year period), and the system may have been developed for that specific application.
Several long count dates in this system were inscribed on stone monuments between 36 B.C. and
A.D. 125; among epi-Olmecs, this practice continued until at least A.D. 533.
The structure of the place-value notation system is consistent with the structure of numer-
als in the epi-Olmec language, but it is inconsistent with the structure of Mayan numerals. Among
Mayans, the normal way to express the number 55 would be “fifteen in the third 20,” and its ex-
pected abbreviation would be “three fifteen” (or “fifteen three”) rather than the “two fifteen” of
the actual system. So the structure of this notational system is probably owed to the epi-Olmecs.
We know from its colonial survivals in the Yucatec Mayan Books of Chilam Balamthat Mayas read
these dates as written, and that this non-Mayan type of expression became a fixed formal style
for expressing large counts of time: They would say something like “eight pih,four ma:y,five
ha7band seventeen kin” (corresponding to 8 400 360 + 4 20 360 + 5360 + 17 days).
A native Mayan expression would have been “five in the sixth ha7bof the fifth ma:yof the ninth
pih.”
How did this system deal with the cases in which a mathematician might use a zero coeffi-
cient, when the spoken language would simply have omitted mention of the unit? We know the
answer from an early Mayan monument, Abaj Takalik Stela 5 (see Figure 11.4). It records the seat-
ing of the new year on a date that would be transcribed 8.4.5.0.17 by epigraphers (in A.D. 125).
The digits in the place-value record of this date are eight, four, five, seventeen. There is simply
a small extra space where the number of 20-day units would otherwise have been recorded. In
this respect, the notation remained a direct if reduced transcription of the spoken version of the
numeral.
By the beginning of the Classic period on, Mayans had begun using a fuller representation
of these dates, recording glyphs that expressed the units used in speech as well as the coefficients
of those units. In the formal speech that was used to read out these dates, the dates made use of
poetic repetition, going beyond the typical couplet structure in which two parallel phrases were
juxtaposed to intone five parallel phrases in a row: “There were eight pih;there were twelve may;
there were fourteen ha7b; there were eight winikil;there were fifteen ki:n.” Among the most for-
mal of these occasions were the ends of katuns, when most Mayan monuments were erected in
connection with katun-ending rituals. But poetically these would be the least imposing dates to
recite: “There were eight pih;there were fourteen ma:y”.
By A.D. 317, Mayans seem to have solved this problem by a poetic expedient: “There were
eight pih;there were fourteen katu:n;there were no ha7b; there were no winikil;there were no
ki:n.” There is a widespread Mayan root mihthat means “there aren’t any,” and a phonetic sign