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Riemann, Mayan Math

Date: 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,

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   

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   

"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, (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   



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.]   

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. 


Coolidge, J.L. A Treatise on Algebraic Plane Curves. New York:
Dover Publications, p. 120, 1959.   

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 

(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:   


"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 

[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
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


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."   

(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

                 [a big number, explained]            

Sometimes this number will be expressed in the shorthand 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 

-Doctor Sarah,  The Math Forum
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