mirrored file at http://SaturnianCosmology.Org/ For complete access to all the files of this collection see http://SaturnianCosmology.org/search.php ========================================================== see intro.htm.txt The Numerals " . . . Un punto indica un año; dos puntos, dos años; tres puntos, tres años; cuatro puntos, cuatro años. Una barra significa cinco años; dos barras, 10; un punto sobre una barra, seis . . ." (Chilam Balam de Maní.) _________________________________________________________________ _The characteristic principal of the Maya _system of numeration consists of that the used symbols have an intrinsical value. In other words, the symbols themselves contain the multiplicity described. The idea is so elementary, from the semantic point of view, that is within reach to the children's mind: if one dot represents the unity, two dots represent the number 2, three dots, the number 3, and four dots, the number 4. _If we observe a traffic controller_ count the number of trucks that arrive at a construction area: he makes in his notepad one vertical line for each arrived truck until he reaches four vertical lines. Then, the fifth truck is counted with another line but this time he draws a line crossing the previous four vertical lines diagonally. Thus, he makes 1 set of 5 unities. After that, he counts all sets of 5 lines, multiplies by 5 and adds what is left. Well, the maya would do exactly the same thing only annotating dots and after four dots, he would cross the dots or link the dots to draw a bar. At the end, he would count the number of bars and dots left . The difference is that for the maya the dots and bars are already the numbers and not marks or signals only. _Many cultures on Earth _seem to have used a similar ancient system of numeration. But, it happened that they encountered two serious problems: (1) a huge amount of space to represent a large number, e.g. 40, 100 or 2,000, and (2) the improbability of recognizing a total number, with a simple glance, without recurring to laborious additions in each one of the marks. This problem was resolved by all, with the exception of the Mayas and Indostanos, with conventional symbols for the sets of 10, 20, 100, 1000, etc. At that moment, the advantage that every numeral had an operative or intrinsical value was lost. _The Aztecs,_ as you may know, used a vigesimal system learned probably from the Toltecs whose culture was profoundly influenced by the Maya culture. From 1 to 19, points were used or small circles, and ocassionaly, images of fingers (as Mayans did.) For the number 20, however, they used an absolute conventional and arbitrary symbol, a flag. To reach to the count of 400 where they introduced another hieroglyph similar to the image of a tree to symbolize something that in their experience illustrated a big number (with branches and leaves.) Finally, the number 8,000 was represented with a handbag that would contain something very valuable equivalent to the high number of items. _The Babylonians,_ on the other hand, used cuneiform characters. Their system of numeration was hexagesimal (counting from 60 to 60). There are two theories of the origin this system grouped quantities. The first one forwards the hypothesis that it emerged as the fusion of two archaic systems: the double-metric and the metric adopted by the Egyptians and the majority of Mediterranean cultures. The second emerged as a need for a calendar, so it resembles a vague babylonic year of 360 days, identical to the "tun" maya. In every case, the advantage was obvious. A system divisible by 2, 3, 4, 5, 6, 10, 12, 15, 30 and 60 which reduce the difficulties related to division, fractions and the use of irrational numbers. The babylonic system did not work very well in the development of algorithms, and for this reason, there was a vast activity of the mathematicians of that time to prepare tables, "magical" squares and progressions that would allow businesspeople and architects deduct quickly the results of the most common operations. Hogben (1) states that this abundant compilation only has a parallel with our actual technological era and to achieve this it was necessary for the Babylonians to resolve binomials and quadratic equations. _I would like to add that the metric_ and double-metric system have been confirmed in Europe, Asia and Egypt historically. However, scholars have not explained yet the survival of the vigesimal system, of Maya origin, and still recognized in the French system ("quattre vingt"=80), the Basque and the Georgian. Isn't about time to investigate the Pre-hispanic interculturization between America and Europe? _Another example of numerical writing_ that started using the principle of accumulation and modified to represent quantities graphically is the Chinese numeration, and adopted by the Japanese in its modern style before the introduction of arabic numerals to that country. The Egyptians, with their admirable perfection of chronology and geometry, were not be able to originate another system to confront large quantities as they follow the same metric system with different signs to count 10, 100 and 1000. The Greeks and the Romans introduced more confusion by representing each numeral with a letter taken from the alphabet that probably contributed to the scientific decadence of the Medieval Age. _The two trascendental inventions of the Mayan system_ of numerations was first, the bar with a value of 5. Instead of placing in line ten small dots, it was enough to draw two bars achieving a considerable saving of space. The second invention consisted of the ordering of numerals by unities, twenties, twenties of twenties, twenties of twenties of twenties and so on. Each dot or bar is assigned a multiple value when taken in consideration the first, second, third, or fourth position in a column. Obviously, it resulted that the highest numerical ranges are associated with the top positions. * One dot in the 6th position = 3,200,00 * One dot in the 5th position = 160,000 * One dot in the 4th position = 8,000 * One dot in the 3rd position = 400 * One dot in the 2nd position = 20 * One dot in the 1st position = 1 _Simultaneously to this invention_, it was discovered the need of creating a symbol that would fill the spaces in a column where no numeral was written. The Mayan symbol for the "number" zero is one of the most ancient concepts of abstract thinking. If the reader is asked to represent the nothingness you will understand how difficult is to imagine a figure without conventionalism but carries the essence of the idea implicitly. The arabic symbol for 0 will not probably resist a criticism or judgement as a self-explanatory sign. The most intelligent man on Earth will not figure out what it represents without a previous explanation! _Palacios (2) affirms_ that the Mayan concept of zero implies the absence of everything - the empty void is a physical improbability at least in the galaxy we inhabited - and, in reality, the Maya mathematician did not pretend to indicate absence or negation but a sense of completion. Say, to write 20, the zero indicates that twenty was completed and that nothing was lacking. This is an opposite assertion to the concept of absence or lacking. In support of that thesis, we could mention the opinion of Morley (3) who says that the symbol of zero found in the codexes represented a closed fist seen frontally. The closed fist would symbolize that the fingers (and, of course, the numerals because man started counting in this way) are retained within a closed space, integrated and completed. González y Obregón (4) also clarified the Aztecs counted with the fingers bent to complete a fist representing a complete count. _On the other hand, there is a defended point of view_ that the symbol of zero is a snail shell or sea shell for which there are solid rationale and documentary evidence. The shell is a frequent element in the Mesoamerican epigraphy and its vinculation with death, in my opinion, have been firmly established. It is easy to notice that the shell is the remant of a dead molusk. The Mayans must have understood that these empty carcasses were the fingerprint of vanished species. Nevertheless, both concepts are reconciliated in one thought. The termination of Life is also the closure of a cycle, a measure that is completed . . . the final integration. _To summarize_, one dot in the first position is 1, in the second position is 20, in the third position 400, in the fourth position 8,000 and 160,000 in the fifth position. The value of the bar is 5 in the first position, 100 in the second, 2,000 in the third, 40,000 in the fourth and 800,000 in the fifth. [1][iconback.gif] [2][iconup.gif] [3][iconnext.gif] References 1. Hogben, Lancelot. Mathematics in Antiquity. Londres. 1932. 2. Palacios, Enrique Juan. El Calendario y Los Jeroglificos Cronograficos Mayas. Ed. Cultura, México, 1933. p.36. 3. Morley, S.G. An Introduction to the Study of the Maya Hieroglyphs. Bureau of American Ethnology. Smithsonian Institution. Bull. 28, Washington, D.C. 1915. 4. González Obregón, Luis. Historia Patria. J. Ballescá y Cía, México, Tomo II, pp. 119-137. References 1. file://localhost/www/sat/files/intro.htm 2. file://localhost/www/sat/files/control.htm 3. file://localhost/www/sat/files/chapter2.htm