Riemann, Mayan MathDate: 5/20/96 at 19:34:44 From: Anonymous Subject: Help please Dear Dr. Math, All of my Algebra II classmates and I are doing a huge math research project. Some of them have seen the info that I have previously gotten from this service and asked me to see if could get them some info as well. Some of the topics are very vague while others are too specific and many of my friends are having a lot of trouble getting information. Any help would very much appreciated. The topics are... A. Riemann - a German mathmatician B. The Mayan number system and calendar C. Probability Thanks in advance, Rob Date: 5/20/96 at 22:59:32 From: Doctor Sarah Subject: Re: Help please Hi - I don't think we can look up topics for your whole Algebra II class! I'll have a go at the first two of the topics you mention below, but I think your classmate who is working on probability will need to get on the Web if he or she can, to look at the resources the Math Forum lists at http://mathforum.org/probstat/probstat.html Asking for information on probability is being a bit too general for us to answer. One of the best spots on the Internet to investigate probability and statistics is the CHANCE database at http://www.dartmouth.edu/~chance/ "The CHANCE data base contains materials designed to help teach a CHANCE course or a more standard introductory probability or statistics course. The CHANCE course is a case study quantitative literacy course intended to make students more informed, and critical, readers of current news that uses probability and statistics as reported in daily newspapers such as "The New York Times" and the "The Washington Post" and current journals and magazines such as "Chance," "Science," "Nature," and the "New England Journal of Medicine." ______________________________________________ RIEMANN http://userwww.sfsu.edu/~rsauzier/Riemann.html Riemann, (Georg Friedrich) Bernhard (1826-66) Mathematician, born in Breselenz, Germany. He studied at Göttingen and Berlin universities, and became professor of mathematics at Göttingen (1859). His early work was on the theory of functions, but he is best remembered for his development of non-Euclidean geometry, important in modern physics and relativity theory. His profound conjecture (the Riemann hypothesis) about the behavior of the zeta (or Riemann) function, which he showed determines the distribution of the prime numbers, has resisted proof since its publication in 1857. An extensive quote from Riemann can be found at http://attila.stevens-tech.edu/math_history/authors/Riemann/hypoth.1.html by RIEMANN ON THE HYPOTHESES WHICH LIE AT THE FOUNDATIONS OF GEOMETRY Plan of the Investigation As is well known, geometry presupposes the concept of space, as well as assuming the basic principles for constructions in space. It gives only nominal definitions of these things, while their essential specifications appear in the form of axioms. The relationship between these presuppositions [the concept of space, and the basic properties of space] is left in the dark; we do not see whether, or to what extent, any connection between them is necessary, or a priori whether any connection between them is even possible. From Euclid to the most famous of the modern reformers of geometry, Legendre, this darkness has been dispelled neither by the mathematicians nor by the philosophers who have concerned themselves with it. This is undoubtedly because the general concept of multiply extended quantities, which includes spatial quantities, remains completely unexplored. I have therefore first set myself the task of constructing the concept of a multiply extended quantity from general notions of quantity. It will be shown that a multiply extended quantity is susceptible of various metric relations, so that Space constitutes only a special case of a triply extended quantity. From this however it is a necessary consequence that the theorems of geometry cannot be deduced from general notions of quantity, but that those properties which distinguish Space from other conceivable triply extended quantities can only be deduced from experience. Thus arises the problem of seeking cut the simplest data from which the metric relations of Space can be determined, a problem which oy its very nature is not completely determined, for there may be several systems of simple data which suffice to determine the metric relations of Space; for the present purposes, the most important system is that laid down as a foundation of geometry by Euclid. These data are - like all data - not logically necessary, but only of empirical certainty, they are hypotheses; one can therefore investigate their likelihood, which is certainly very great within the bounds of observation, and afterwards decide upon the legitimacy of extending them beyond the bounds of observation, both in the direction of the immeasurably large, and in the direction of the immeasurably small. [This goes on for many pages.] http://mathworld.wolfram.com/RiemannCurveTheorem.html Riemann's Theorem If two algebraic plane curves with only ordinary singular points and Cusps are related such that the coordinates of a point on either are rational functions of a corresponding point on the other, then the curves have the same Genus (Curve). This can be stated equivalently as: the Genus of a curve is unaltered by a Birational Transformation. Reference Coolidge, J.L. A Treatise on Algebraic Plane Curves. New York: Dover Publications, p. 120, 1959. http://hkein.ie.cuhk.hk/Education/Math/mactutor/Mathematicians/Riemann.html Georg Friedrich Bernhard Riemann Sept 17 1826 - July 20 1866 Born Hannover, Germany. Died in Italy. Riemann's ideas concerning geometry of space had a profound effect on the development of modern theoretical physics. He clarified the notion of integral by defining what we now call the Riemann integral. Riemann moved from Göttingen to Berlin in 1846 to study under Jacobi, Dirichlet and Eisenstein. In 1849 he returned to Göttingen and his Ph.D. thesis, supervised by Gauss, was submitted in 1851. In his report on the thesis Gauss described Riemann as having a gloriously fertile originality. On Gauss's recommendation Riemann was appointed to a post in Göttingen. Riemann's paper Uber die Hypothesen welche der Geometrie zu Grunde liegen, written in 1854, became a classic of mathematics, and its results were incorporated into Albert Einstein's relativistic theory of gravitation. Gauss's chair at Göttingen was filled by Dirichlet in 1855 and, after his death, by Riemann. Even at this time he was suffering from tuberculosis and he spent his last years in Italy in an attempt to improve his health. Riemann's ideas concerning geometry of space had a profound effect on the development of modern theoretical physics and provided the concepts and methods used later in relativity theory. He was an original thinker and a host of methods, theorems and concepts are named after him. The Cauchy-Riemann equations (known before his time) and the concept of a Riemann surface appear in his doctoral thesis. He clarified the notion of integral by defining what we now call the Riemann integral. He is also famed for the still unsolved Riemann hypothesis. (There's lots more on the Web if your friend has access to it.) _____________________________________________________________ Here's some information on Mayan numerals, although you really need to look up the Web pages to see the diagrams: http://www.vpds.wsu.edu/fair_95/gym/UM001.html THE MAYAN NUMBERS "The Mayan number system is in some respects very similar to ours. They used a symbol for 0 and had symbols for 0 - 19. These symbols are shown below [see the Web page for a chart of the Mayan numerals]. You can see that these are logical and easy to use. The bar symbol represents 5, and the dots are 1's. The numbers can be written with the dots on top of horizontal lines or to the left of vertical lines. They may also be combined with the head symbols (discussed and shown later). [Briefly, the chart shows the following - 0 looks like part of a shell (or a baseball cap without the brim) 1 . 2 .. 3 ... 4 .... 5 _____ 6 . _____ (the dots should be centered over the bars) 7 .. 8 ... _____ _____ 9 .... 10 ____ (bars close together) _____ ____ 11 . 12 .. _____ _____ _____ _____ etc. ] A very interesting twist on the place values after 360 is that they are multiples of 18. The first place value is 1; the second, 20, the third 18*20, the fourth is 18*202; the fourth 18*203, and so forth. Thus the Mayan numerical system would translate into ours as follows: Numeral read Meaning for each row of Equivalent number vertically symbols in our system . . 2*720 (or 2*18*20 squared) 1440 . . . _____ 13*360 (or 13*18*20) 4680 _____ . 1*20 (or 1*20) 20 _____ 5*1 (or 5*1) 5 _____ 6145 Mathematical Reasoning for Elementary Teachers by Calvin T. Long and Duane W. DeTemple. HarperCollins, 1995, pp. 165-167. Mayan head numerals Here are the head numerals for 0 through 19. [Chart of Mayan head hieroglyphs.] Closs, Michael P. "Mathematical Notation of the Maya" Native American Mathematics, edited by Michael P. Closs, University of Texas Press, Austin, 1986. p. 335, figure 11.16 They also used a head symbol representing the moon for 20. Here is a representation of the 31st day in their calendar. You will notice that the head symbol is combined with the "normal" numerical system. [another image] Closs, Michael P. "Mathematical Notation of the Maya" Native American Mathematics, edited by Michael P. Closs, University of Texas Press, Austin, 1986 p. 345, figure 11.20a CONCLUSION Overall, I found the Mayan numerical and calendrical system to be quite complicated. I saw an ancient civilization that had a very accurate, sophisticated, and complex numerical and calendrical system. Their system was a base 20 with a sub-base of 5. They combined this with a set of beautiful head numerals." http://hanksville.phast.umass.edu/yucatan/mayamath.html (You really need to look at these pages to see the images.) Mayan Math The Mayans devised a counting system that was able to represent very large numbers by using only 3 symbols, a dot, a bar, and a symbol for zero, or completion, usually a shell. The chart above shows the first complete cycle of numbers. Like our numbering system, they used place values to expand this system to allow the expression of very large values. Their system has two significant differences from the system we use: 1) the place values are arranged vertically, and 2) they use a base 20, or vigesimal, system. This means that, instead of the number in the second position having a value 10 times that of the numeral (as in 11-1 x 10+1 x 1), in the Mayan system, the number in the second place has a value 20 times the value of the numeral. The number in the third place has a value of (20)^2, or 400, times the value of the numeral. This principle is illustrated in the chart below. [a big number, explained] Sometimes this number will be expressed in the shorthand 3.10.6.13.17 in writings on the Mayan numeration system, especially when discussing dates that are recorded in stelae or monuments. Using this system for expressing numbers has 2 advantages: 1) large numbers can be easily expressed, so long time periods can be recorded; and 2) simple arithmetic can be easily accomplished, even without the need for literacy among the population. In the marketplace, sticks and pebbles, small bones and cacao beans, or other items readily at hand can be used to express the numbers in the same way that they are expressed on the monuments or in the books of the upper classes. Simple additions can be performed by simply combining 2 or more sets of symbols (within their same set). This is shown below. . ... .... _____ + _____ = ____ (the bars should ____ be closer together) For more complicated arithmetic, you must simply remember that you borrow or carry only when you reach 20, not 10, as shown below. [borrowing/carrying image] It is important to note that this number system was in use in Mesoamerica while the people of Europe were still struggling with the Roman numeral system. That system suffered from serious defects: there was no zero (0) in the system, and, as opposed to the Mayan system, the numbers were entirely symbolic, without direct connection to the number of items represented. It is not known whether a system was developed for multiplication and division. -Doctor Sarah, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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