V. A Counting System on fingers and toes.

V.    A Counting System on fingers and toes 

[Contd. A Journey to the Wonderland of Math. by Ajay Kumar Chaudhuri]

Let us explore counting system and numerals of another civilization, flourished on a land where Christopher Columbus reached by mistake.It was August 3,1492, Columbus led his three  historic ships,namely, the Nina,the Pinta and the Santa Maria from the Spanish port of Palos.The objective was to find a route towards the west to reach Asia(Indies) where riches of gold ,precious pearls and variety of spices were awaiting.

Columbus first landed on a small island in the Bahamas, which he named San Salvador. Then with his crew sailed to a larger island Cuba. He did not stop here and continued to sail to reach Hispaniola. Columbus thought all these islands in the Caribbean Sea were near Asia, his desired destination. But he was surprised to know that none of the native people spoke Arabic.

Yet, the island of Hispaniola, the second largest in the Caribbean, attracted him most. He exclaimed, “The most beautiful island in the world”.Of late, this island became the major launching base for conquest of the Caribbean as well as American main land.

The unique counting system and numerals of the ancient civilization flourished in this part of Americas is the Mesoamerican Civilisation, called Maya Civilisation. Mesoamerica is a region and cultural areas in the Americas, extending approximately from Central Mexico to Belize, Guatemala, El Salvador, Honduras, Nicaragua and northern Costa Rica within which pre-Columbian societies flourished before the Spanish colonization of the Americas in 15^th and 16^th centuries. It is one of the six areas in the world where ancient civilization arose independently, and second in the Americas along the Inca Civilisation, which has  already been mentioned.

The Maya civilization, developed by the Maya peoples, noted for its hieroglyphic script (a written character that represents a word or a phrase. e.g the Chinese, Japanese scripts), as well as for its art, architecture, mathematics, calendar and astronomical system.

To know and judge a civilization, it is important to know its history. It is applicable to Maya civilization as well. Their amazing achievement in mathematics opened the gateway of splendid excellence in other fields. So let us have a glance at the history of the Mayans.

The history of Maya civilization may broadly be divided into two principal era – Pre-Columbian and Post-Columbian. Pre-Columbian era literally means the time preceding Columbus’s voyages in 1492. It usually refers to denote the entire history of indigenous Americas’ culture, until those were significantly influenced by Europeans, even if this happened decades or even centuries after Columbus’s first landing.

For the sake of convenience the Pre-Columbian era may be divided into several periods.

The ancient period of Maya civilization, called the Paleo – Indian period (10,000 – 3500 BC) spans from first sign of human presence in the region to the establishment of agriculture and other practices.

The next phase was during which the development of agriculture and establishment of permanent villages, in the Archaic Era (3500 – 2000 BC). Late in this era use of pottery and loom weaving became common and class divisions began to crop up.

The succeeding era is the Preclassic or Formative period (1800 BC– 200 BC). Some very important developments occurred during this period. The most important of them is the ‘writing’ skill, for writing is a medium of human communication that represents language and emotion through the inscription or recording of signs and symbols. Writing is not a language but a form of technology that developed as tools developed with human society. Apart from writing large-scale ceremonial architectures, cities and states were developed. Actually many distinctive elements of Mesoamericans civilization can be traced back to this period.

The next era was the classic era (250 – 900 CE). It was dominated by numerous independent city – states in the Maya region. Highly sophisticated arts such as stucco work, architecture, sculptural reliefs, mural painting, pottery, lapidary etc. developed and spread during this period. Many scientific advances were also achieved during this era. The Maya refined their calendar, script and mathematics to their highest level of development. Astronomy remained a matter of vital significance because its importance for agriculture which was the economic basis of the Mesoamerican society. It was also applied to predict events in future such as solar and lunar eclipses which provided the common people their links with the heavenly world.

The last phase of the Pre-Columbian age in the Post-Classic Era was between 900 CE and the conquest of Mesoamerica by the Spaniards which occurred between 1521 and 1697 CE. In this era many of the great nations and cities of the Classic Era collapsed, although a few of them continued. It is often viewed as the period of cultural decline. However it was a time of technological advancement in architecture, engineering, weaponry and also in metallurgy.

Of all Pre-Columbian Mesoamerican cultures perhaps the best known is that of the Mexico, sometimes called the Aztec. Among other things, its fame was because the Mexico state was the richest and most powerful in the region, which came at the cost of the of exploiting the surrounding people.

Post Columbian era of the Mayan civilization begins after conquest of the Mesoamerican region by the Spanish in 1521. After hearing Columbus’ discovery of new lands in the Carribeans, Spaniards sailed to reach that wonderland. In a course of time, the visitors turned invader turned conqueror.

Shortly after the first expeditions to the region, the Spanish initiated a number of attempts to subjugate the Maya who were hostile towards the Spanish Crown and establish a colonial presence in the Maya territories of the Yucatan Peninsula and the Guatemalan high lands. It was lengthy and dangerous exercise for the conquistadors, the Spanish and Portuguese soldiers of the fifteenth to seventeenth century were known thus, from the outset. It took about 200 years for the Spanish to establish substantive control over all Maya lands.

The Spanish church and government officials destroyed Mayan texts written in hieroglyphics; abhorred their religious practices, smashed the Mayan idols to pieces.

But by chance three of the Pre-Columbian books dated to the post classic period have been preserved. The most important are: The Dresden Codex now kept in the Sachsische Landesbibliothek Dresden; The Madrid Codex now kept in the American museum in Madrid; and the Paris Codex now in Bibliotheque nationale in Paris. The Dresden Codex is a treatise on astronomy, thought to have been copied in the eleventh century AD from an original document dated seventh or eighth centuries AD.

The importance of astronomy and calendar calculations in Mayan Society required mathematics very much. So the Maya constructed quite early a very sophisticated number system. The Mayan number system dates back to the fourth century and were probably 1000 years more advanced than the Europeans of that time. This system somewhat resembles to our current decimal system, with a base 10, while the Mayan’s used vigesimal system, with a base 20, The probable reason for using the base 20, was Mayans lived in such a warm climate that they rarely need to wear shoes. So 20 were the total numbers of fingers and toes at their ready disposal and they used these for counting. Therefore two important markers in this system are 20, which relates to fingers and toes and five which relates to the number of digits on one hand or foot.

It is also interesting to know that Mayan’s were also the first to symbolize the concept of nothing, namely zero. The most common symbol for this was that of a shell (   🍮) but there were other symbols too (e.g  a  head). Interestingly, with all of the great mathematicians and scientists of ancient Greece and Rome, it was the Mayan Indians who independently come with this symbol for nothing (or zero). The pre-classic Maya and their neighbours developed this concept of zero as early as 36 BC.

The Mayan numerals consisted only of three symbols : Zero, as already mentioned, represented as a shell shape, a dot (●) for unit or one and  four horizontal dots (●●●●) to represent four, a bar ( ➖ ) for five. The Mayans wrote their numbers vertically as opposed to horizontally with the lowest denomination on the bottom. After the number 19, larger numbers were written in a kind of vertical place value format [Table no.1: Mayan numerals] using powers of 20, Such as 20⁰ or 1, 20¹ or 20, 20² or 400, 20³ or 8000, 20⁴ or 160000 etc. In the Arabic form, that is, the decimal base we use now is based on powers of 10, 1, 10, 100, 1000, etc.

         Table No.1

Now let us see how the Mayan’s wrote the numbers using only three numerals.

The number 241482 would be figured out and written as follows :

 
 Also the number    •     •       in Maya

                              •    •    • 
                            〓〓〓〓

                             
                                   •
                           ☰☰☰☰☰ 
                             

                          •    •     •     •

numerals ,the number 21524 in the decimal system.

Mayans knew the technique of simple addition and subtractions. The addition they performed by simply combining 2 or more sets of symbols (within their same set).

A.  Simple example as:

                                                         •             +       •  •  •        =     •  •  •  •
                                                     ➖➖➖             ➖➖➖             〓〓〓〓

      For more complicated arithmetic you must simple remember that you are  to borrow or carry only when you reach 20.
                                For example,

                         Whose decimal equivalence will be
                                                    13607+17214 = 3082

Likewise for subtraction, say

                                                    
                                            

                                          

                                            

which in decimal is

           10893 – 8964 = 1929

However, it is not known whether a system was developed for multiplication and division.

The Calendar played an important role in Maya civilization, in their everyday lives, social and religious rituals. This calendric system was behind the structure of number system. The Maya had two Calendars, one of them was a ritual calendar, known as Tzolkin, consisting of 260 days. It contained 13 “months” of 20 days each. The months were named after 13 gods while the 20 days were numbered from 0 to 19. The second one was 365 day Civil Calendar called Haab. It consisted of 18 “months” named after agricultural or religious events, each with 20 days, again numbering from 0 to 19 and short “month” of only 5 days, called Wayeb, which was considered an unlucky period. It was learnt that the Maya did not wash, comb their hair or do any hard work during these five days. A common belief was that any one born during this period would have bad luck and remain poor and unhappy all their lives.

Despite not possessing the concept of a fraction, they made extremely accurate astronomical observations using no instruments other than sticks. They were able to measure the length of the Solar year to a far degree of accuracy than that of used in Europe. Their calculations yielded the length of the solar year as 365.242 days, compared to the modern value of 365.242198 days. The length of the lunar month, according to their estimate, was 29.5308 days, compared to the modern value of 29.53059 days.

              However due to geographical disconnect and lack of communication Mayans in particular
and Mesoamerican mathematics in general had absolutely no influence on the European and Asian numbering system and mathematics.
                                                                                                                        [To continue]

   Reference: Internet.

                    Table No 1.Mayan numerals
                     Credit:  dCodeur(https://www.dcode.fr/mayan-numbers)