A NOTE ON MAYA MATHEMATICS Sandeep Kumar Bhakat
Siksha-Satra, Visva-Bharati
P.O-Sriniketan-731236; Dist-Birbhum, WB
Email: [email protected]
Mob: 09475988544
INTRODUCTION:
Maya mathematics generated by including religion, social customs, barbarism and scientific ideas. The
Maya Civilization was one of the two ancient civilizations that discovered and used the concept of Zero
before any other cultures in the world. The other was the Hindu of the Indus Valley. The Hindu used zero
for astronomical calculations only. The Maya mathematics was very organized and interesting. The Maya
used place-value system to write numbers. In the stage of their civilization, Maya used a very logical and
scientific manner to write very large numbers. European cultures obtained the zero only because Arab
scholars in Bagdad in the seventh century translated a Hindu text on astronomy and thus rediscovered the
zero. Subsequently, an Arab mathematician translated into Latin. Our Culture gained this vital idea,
although it did not come into general use in the Western Civilization until many centuries later when
Fibonacci in 1202 introduced the decimal system of writing numbers in Western countries.
The Maya also use a positional system, making easier to calculate and write big numbers, long before any
other culture. A few Sumerian tablets show the faint beginnings of calculations based on a positional
system, but no more.
THE SYMBOLS:
Maya Bar-Dot Mathematical System:
Maya used three distinct signs in the bar-and-dot notation for numbers. A clam like symbol
stands
for zero, the solid dot
had the numerical value of one, and the bar
wais equivalent to
number five. By combining these bars and dots, the Maya could write almost any number. For example,
12 was equivalent to equal two bars and two dots.
( 2 X 5 + 2 x 1)
In the Hindu-Arabic numeral counting system, there are ten symbols that are used to create numbers:
1,2,3,4,5,6,7,8,9, and 0. The whole numbers larger than 9 is written using combinations of these symbols.
For example, the number forty-six (84) is represented by a "8" with a "4". The "8" is in the tens place and
represents four sets of ten. And the 4 is in the ones place, representing four sets of one. With this system,
we can create any whole number.
The Mayan system was similar, but instead of having only ten symbols, like there are today, there were
twenty symbols. So instead of combing two symbols together after reaching nine, they started after
reaching nineteen. Because the Mayan system is based on 20 and not 10 like the Hindu-Arabic system,
instead of a tens place, there is a twenties place.
For example the number 46. There are two dots in the twenties place, indicating two sets of twenty. And there is a bar with a dot in the ones place indicating six sets of one. The following diagram represents 46 in terms of Maya numeral system. Explanation: This base twenty system is still in use today by such tribes as the Hopi and the Inuits. We can also add
with these larger numbers almost as easily as the single digit ones. To do so, first add together the bars
and dots in the 1s place. If the sum is over twenty, there will be some carrying involved, but we'll talk
about that in just a little bit. Next, you add the bars and dots in together that are in the 20s place. This
gives us the number in Mayan numerals. To translate the number into the kind of numbers that are in
popular use today, there is just one trick to remember- Every dot in the 20s place represents one set of 20.
And each bar in the 20s place represents one set of one hundred (or 5 sets of 20). Look at this example.
We start with the 1s place. First
we add three to six to get nine.
Then we move on to the 20s
place. We add the twenty to
the sixty (3 x 20) add get eighty.
So our final answer is eightynine. Let's see if you can try one
on your own. It's a little bit
tricky, but I think you can get it
anyway.
The Maya numerals chart:
The Maya dealt with 20 essential digits instead of ten digits as it is done in base ten. In a vigesimal
system, the number in the second position is twenty times that of the numeral; the number in the third
position is 20^2 times that of the numeral; the number in the fourth position is 20^3 times that of the
numeral, e.t.c. The place values were 1s, 20s, 400s, 8,000s, 160,000s, and so on.. In the Mayan
language, 20 was called kal, 400 was called bak, 8000 was called pic, 160,000 was called Calab,
3,200,000 was called kinchil, and 64,000,000 was called alau , they used place values to expand this
system and to allow the expression of very large numbers. For example, to express 352,589, using base
ten, we would write 3x10^5+5x10^4+2x10^3+5x10^2+8x10+9x10^0. The Maya would express this
number in a similar fashion except that they would use base 20:
2x20^4+4x20^3+1x20^2+9x20+9x20^0. Moreover, instead of writing out 352,589, the Maya
2x20^4+4x20^3+1x20^2+9x20+9x20^0. Moreover, instead of writing out 352,589, the Maya
would use a shorthand notation and write it as 2.4.1.9.9, where the numbers 2, 4, 1, 9, and 9
represent the “coefficients” in front of the powers of 20. Thus, using a vigesimal system gave
the Maya great advantage because it facilitated the expression of very large numbers and having
the feasibility to do this would enable them to count time.
COMPARISON WITH DECIMAL SYSTEM: The decimal mathematical system (base 10) widely used today goes by 1, 10, 100, 1000, 10000, etc. The
Maya use a vigesimal number system (base 20) Below show the place value from a vigesimal system or
base 20 system.
12,800,000,000 = 207
64,000,000 = 206
3,200,000 = 205
160,000 = 204
8,000 = 203
400 = 202
20 = 201
2
3
4
5
The number is: 14 + 7 × 20 + 1 × 20 + 3 × 20 + 0 × 20 + 15 × 20 + 5 × 20
6
The number is 14 + 140 + 1 × 400 + 3 × 8,000 + 0 + 15 × 3,200,000 + 5× 64,000,000
The number is = 14 + 140 + 400 + 24,000 + 0 + 48,000,000 + 320,000,000 = 368024554
0
xix im
10
Lahun
Maya Names for their Numbers
1
hun
11
Buluc
20
hun kal
400
hun bak
2
caa
12
Lahca
40
ca kal
800
ca bak
3
ox
13
Oxlahun
60
ox kal
1200
ox bak
4
can
14
Canlahun
80
can kal
1600
can bak
5
hoo
15
Hoolahun
100
hoo kal
2000
hoo bak
6
uac
16
Uaclahun
120
uac kal
8,000
pic
7
uuc
17
Uuclahun
140
uuc kal
160,000
calab
8
uaxac
18
Uaxaclahun
200
ka hoo kal
3'200,000
kinchil
9
bolon
19
Bolonlahun
300
ox hoo kal
64,000,000
alau
Note: Since the Maya were very religious, they also expressed numbers using the heads of the
thirteen gods of the upper world. The primary numbers were zero through twelve. A distinct
head that represented one of the thirteen gods expressed each number. The head for ten, for
example, was the god of death. Each head also had distinct characteristics. For example, the
head for six had a “hatchet eye,” whereas the head for ten was known for its “fleshless lower
jaw” and “truncated nose” . The numbers thirteen through nineteen were
represented by the head used for the primary numbers and the fleshless lower jaw of the head for
ten. To represent sixteen, for example, the head would be that of six with the fleshless jaw of ten
added to it . The 52-year cycle
The cycle of fifty-two years was central to Mesoamerican cultures. The Nahua's religious beliefs were
based on a great fear that the universe would collapse after each cycle if the gods were not strong
enough. Every fifty-two years a special New Fire ceremony was performed. All fires were extinguished
and at midnight a human sacrifice was made. The Aztecs waited for the dawn. If the Sun appeared it
meant that the sacrifices for this cycle had been enough. A fire was ignited on the body of a victim, and
this new fire was taken to every house, city and town. Rejoicing was general: a new cycle of fifty-two
years was beginning, and the end of the world had been postponed, at least for another 52-year cycle. (A
similar ceremony is still practiced by small indigenous groups, but without human sacrifice.) The
ceremony was older than the Aztecs. While originally it was believed it was a matter of luck to survive, the
Aztecs thought that constant sacrifice through the fifty-two year cycle could postpone the end.
According to Miguel León-Portilla, Tlacaelel reformed the original Nahua religion and the Aztecs viewed
themselves as the main representatives for feeding the gods. This gave them a new sense of identity,
from "people without face" as they were called by hostile neighbours, to the people in charge of the
existence of the universe. Thus they began to call themselves "The people of the sun". Other researchers
dispute León-Portilla's perspective, pointing to the relative lack of primary sources.
A great deal of cosmological thought seems to have underlain each of the Aztec sacrificial rites. The
most common form of human sacrifice was heart-extraction. The Aztec believed that the heart (tona) was
both the seat of the individual and a fragment of the Sun's heat (istli). To this day, the Nahua consider the
Sun to be a heart-soul (tona-tiuh): "round, hot, pulsating" In the Aztec view, humanity's "divine sun
fragments" were considered "entrapped" by the body and its desires:
Where is your heart?
You give your heart to each thing in turn.
Carrying, you do not carry it...
You destroy your heart on earth
The table below shows the festivals of the 18-month year of the Aztec calendar and the deities
with which the festivals were associated. In History of the Things of New Spain Sahagún
confesses he was aghast at the fact that, during the first month of the year, the child sacrifices
were approved by their own parents, who also ate their children.
A picture of “Sacrifices”
NATIONAL MEET ON CELEBRATION OF NATIONAL YEAR OF MATHEMATICS – 2012 DECEMBER 20‐22, 2012 REGISTRATION FORM Name : SANDEEP KUMAR BHAKAT__________________________________________________ Designation : Assistant Lecturer ______________________________________________ Affiliation : Visva‐Bharati ______________________________________________________________ Mailing Address : Sukhnogarh, Bhubandanga, P.O‐ BOLPUR‐731204. Dist‐ Birbhum (WB) _________________________________________________________ _________________________________________________________ Phone : (03463)253117___________________________ __________________________ Mob. _09475988544 E‐mail: [email protected] _________________________________________________________________ Accommodation required : AC/Non AC/PG Hostel __ To be informed on 19/12/2012 ____________________________ From ____________________________ To ____________________________ Payment details :_Rs 500/‐ ________________________________________________ Category : University Amount _Rs 500/‐____________________ Demand Draft: No 32282____________________ Date _16/11/2012________________ Bank: Axis Bank_______________________________________ Travel schedule : ___ ____________________ Title of paper to be presented Departure __21/12/2012___________________ A note on Maya mathematics ___________________________________ _______________________________________________________________________ Place _Bolpur_____________ Date _16/11/2012_____________ skbhakat_____________ Signature Note: Demand draft, in favour of ‘Secretary, NCERT, New Delhi’ payable at New Delhi, should be sent to R.P. Maurya Convener Department of Education in Science and Mathematics NCERT, New Delhi‐110016 Contact No: 011‐26561742 Mob: 09910018150, 09868372214 e‐mail: [email protected]