- Math Net: Mathematical Software
There was no response. The server could be down or is not responding.
- Cyber-Prof
'Document contains no data'
- Communications in Visual Mathematics
'This Page Has Moved' (to http://www.geom.umn.edu/locate/CVM)
Maybe you should check your links?
---------------------------------------------------------
If, three centuries ago, Machin would have chosen the input vector: "liste:=[Pi,arctan(1/3),arctan(1/5),arctan(1/8),arctan(1/239)];" he would have obtained: "vector_found=[0,1,-1,1,0]" thus missing his historic formula.
You are certainly aware of the problem because it occurs whenever we are looking for a relation between Pi and other constants among which an integer relation (with smaller coefficients) already exists.
Is it possible to modify the Maple program (or the algorithm) so as to introduce the constraint that the integer coefficient of Pi be different from zero?
Thank You!
---------------------------------------------------------
Thank you for your time.
Daniel Lewis
---------------------------------------------------------
> new formulas for $ 2 and 2(2)$, formulae that permit digits to be extracted from their expansions.
new formulas for what?!
Keith
---------------------------------------------------------
If one tabulates the values of t, mao and div for increasing n one observers that t often varies widely, even for successive values of n e.g. for n = 26 t = 10 whereas for n= 27 t = 111. What is surprising is that there are sequences of n for which the value of t the same for all numbers in the sequence, and also are the corresponding mao and div values. e.g. for n = 98, 99,100,101 and 102 , t = 25, mao = 7 and div = 18. It would also seem that whenever t is the same for a sequence of n the mao and div are also the same for that sequence.
I wondered if for a given mao what div values will be generated, and conversely for a given div what mao values will be generated, I found the following. If I select a value for div then there will be a maximum value for mao, maomax say, and this is generated for a minimum value of n. Repeat of maomax or lower values of mao occur as n increases. All values of mao from maomax to zero are generated (and often repeated) but always strictly in sequence. e.g. if I select div = 9 then maomax = 3 generated at n = 17. Then for n = 48, 52 and 53 mao = 2; for n = 160, 168 and 170 mao = 1; and for n = 512 mao = 0.
If I select a value for mao there is a minimum value for div, divmin say, generated at a minimum value of n. Repeat of divmin or higher values of div occur as n increases. All values of div from divmin to infinity are generated (and often repeated) but always strictly in sequence. e.g. if I select mao = 2 then divmin = 5 generated at n = 3. Then for n = 6 div = 6; for n = 12 and 13 div = 7; for n = 24 and 26 div = 8 and so on. Another interesting parameter is the maximum number generated during the process of reducing n to unity, I call this xmax. In some cases the xmax equals n itself e.g. if n = 24, xmax = 24. In most cases however xmax is greater than n. e.g if n = 26 ,xmax = 40. That is at some step in the process of reducing 26 to unity the value 40 is generated. Now the interesting thing is that many values of n share the same xmax e.g. n = 19, 29, 38, 50 and many other n values have an xmax of 88. More surprising is that in some cases xmax is the same for a sequence of values of n. Their is a unique example of this, an xmax of 9232. This is the xmax for n =107 thru to 111, and for n = 124 thru to 126 and numerous other sequences of n. In some of, or part of these sequences t,mao and div are also the same.The value of xmax = 9232 seems to be of special significance, over the range n = 2 to 5000 xmax = 9232 for 1225 of the n values.
A common xmax is one sub-group of n, there are other sub-groups where xmax is a linear function of n i.e. xmax = kn + c. The sub-group for k = 1 corresponds to the condition when xmax = n. k can however have values such as 1.125, 1.5, 1.6875, 2.25, 3.0, 3.375, 4.5, and 6.75 and others, all have associated values for c. e.g. when k = 2.25 c = 2.5. All k values seem to be the ratio of integers like 3/2, 9/8 etc.
That is it for now, any interest or comments ? Regards Ivan.
PS I believe that I failed to post previously,please excuse a beginner.
---------------------------------------------------------
EDITORS: Arthur B. Powell and Marilyn Frankenstein
ABSTRACT:
This 440-page collection brings together classic,
previously-published articles and new research to
present the emerging field of ethnomathematics from a
critical perspective, challenging particular ways in
which Eurocentrism permeates mathematics education.
The editors identify several of the field's broad themes:
reconsidering what counts as mathematical knowledge,
considering interactions between culture and
mathematical knowledge, and uncovering hidden and
distorted histories o f mathematical knowledge. The
book offers a diversity of perspectives on
ethnomathematics. It develops both theoretical and
practical issues from various disciplines including
mathematics, mathematics education, history,
anthropology, cognitive psychology, feminist studies, and
African studies written by authors from Brazil, England,
Australia, Mozambique, Palestine, Belgium, and the
United States.
=================================================
TABLE OF CONTENTS:
Acknowledgments
Foreword/ U. D'Ambrosio
Introduction/ A. B. Powell and M. Frankenstein
SECTION 1. Ethnomathematical knowledge/ A. B. Powell
and M. Frankenstein
Chapter 1. Ethnomathematics and its Place in the History
and Pedagogy of Mathematics/ U. D'Ambrosio
Chapter 2. Ethnomathematics/ M. Ascher and R. Ascher
SECTION 2. Uncovering Distorted and Hidden History of
Mathematical Knowledge/ A. B. Powell and M.
Frankenstein
Chapter 3. Foundations of Eurocentrism in Mathematics/
G. G. Joseph
Chapter 4. Animadversions on the Origins of
Western Science/ M. Bernal
Chapter 5. Africa in the Mainstream of Mathematics
History/ B. Lumpkin
SECTION 3. Considering interactions Between/ A. B.
Powell and M. Frankenstein
Culture AND mathematical knowledge
Chapter 6. The Myth of the Deprived Child: New Thoughts
on Poor Children/ H. P. Ginsburg
Chapter 7. Mathematics and Social interests/ B. Martin
Chapter 8. Marx and Mathematics/ D. J. Struik
SECTION 4. Reconsidering What Counts as Mathematical
Knowledge/ A. B. Powell and M. Frankenstein
Chapter 9. Difference, Cognition, and Mathematics
Education/ V. Walkerdine
Chapter 10. An Example of Traditional Women's Work as a
Mathematics Resource/ M. Harris
Chapter 11. On Culture, Geometrical Thinking and
Mathematics Education/ P. Gerdes
SECTION 5. Ethnomathematical Praxis in the Curriculum/
A. B. Powell and M. Frankenstein
Chapter 12. Ethnomathematics and Education/ M. Borba
Chapter 13. Mathematics, Culture, and Authority/ M.
Fasheh
Chapter 14. Worldmath Curriculum: Fighting Eurocentrism
in Mathematics/ S. E. Anderson
Chapter 15. World Cultures in the Mathematics Class/ C.
Zaslavsky
SECTION 6. Ethnomathematical Research/ A. B. Powell
and M. Frankenstein
Chapter 16. Survey of Current Work in
Ethnomathematics/ P. Gerdes
Chapter 17. Applications in the Teaching of
Mathematics and the Sciences/ R. Pinxten
Chapter 18. An Ethnomathematical Approach in
Mathematical Education: A Matter of Political Power/ G.
Knijnik
Afterword/ G. Gilmer
Contributors
Index
A volume in the SUNY series, Reform in Mathematics
Education, Judith Sowder, editor
=================================================
REVIEWS:
"This volume brings focus to the issues of access and
equity within mathematics and identifies ways to assist
teachers in providing quality mathematics to
traditionally underserved and underrepresented students.
Culturally responsive pedagogy is an area that is sorely
lacking given the fact that our nation's classrooms are
becoming increasingly diverse. We cannot have enough
work in this area. Such material should be required for
teacher preparation as well as professional
development." Sharon Nelson-Barber, Far West
Laboratory for Education Research and Development
"This is a collection of some of the most important papers in ethnomathematics. The authors provide insightful and historical analyses of the development and use of mathematical concepts. Traditionally, this perspective is absent from discussions in mathematics education, yet this book makes a unique contribution to the literature." William F. Tate, University of Wisconsin-Madison
ABOUT THE EDITORS: Arthur B. Powell is Associate Professor in the Academic Foundations Department at Rutgers University-Newark. He has co-authored Math: A Rich Heritage; translated Sona Geometry: Reflections on the Tradition of Sand Drawings in Africa South of the Equator, and co- translated Sipatsi: Technology, Art and Geometry in lnhambane.
MariIyn Frankenstein is Professor at the Center for Applied Language and Mathematics, College of Public and Community Service at the University of Massachusetts, Boston. She has also written Basic Algebra and Relearning Mathematics: A Different Third R: Radical Maths.
Together, they cofounded the Criticalmathematics Educators Group and are members of the Radical Teacher Editorial Collective.
HOW TO ORDER:
Ethnomathematics/ Powell and Frankenstein, 440 pages
$22.95 paperback ISBN 0-7914-3352-8
$68.50 hardcover ISBN 0-7914-3351-X
State University of New York Press
c/o CUP Services
P.O. Box 6525, Ithaca, NY 14851
800-666-2211 (Orders, USA only)
607-277-2211 (Orders)
800-688-2877 (Fax orders)
By email, contact
Janice Heidrich
heidrija@sunypress.edu
------------------------------------------------------------
If one tabulates the values of t, mao and div for increasing n one observers that t often varies widely, even for successive values of n e.g. for n = 26 t = 10 whereas for n= 27 t = 111. What is surprising is that there are sequences of n for which the value of t the same for all numbers in the sequence, and also are the corresponding mao and div values. e.g. for n = 98, 99,100,101 and 102 , t = 25, mao = 7 and div = 18. It would also seem that whenever t is the same for a sequence of n the mao and div are also the same for that sequence.
I wondered if for a given mao what div values will be generated, and conversely for a given div what mao values will be generated, I found the following. If I select a value for div then there will be a maximum value for mao, maomax say, and this is generated for a minimum value of n. Repeat of maomax or lower values of mao occur as n increases. All values of mao from maomax to zero are generated (and often repeated) but always strictly in sequence. e.g. if I select div = 9 then maomax = 3 generated at n = 17. Then for n = 48, 52 and 53 mao = 2; for n = 160, 168 and 170 mao = 1; and for n = 512 mao = 0. If I select a value for mao there is a minimum value for div, divmin say, generated at a minimum value of n. Repeat of divmin or higher values of div occur as n increases. All values of div from divmin to infinity are generated (and often repeated) but always strictly in sequence. e.g. if I select mao = 2 then divmin = 5 generated at n = 3. Then for n = 6 div = 6; for n = 12 and 13 div = 7; for n = 24 and 26 div = 8 and so on.
Another interesting parameter is the maximum number generated during the process of reducing n to unity, I call this xmax. In some cases the xmax equals n itself e.g. if n = 24, xmax = 24. In most cases however xmax is greater than n. e.g if n = 26 ,xmax = 40. That is at some step in the process of reducing 26 to unity the value 40 is generated. Now the interesting thing is that many values of n share the same xmax e.g. n = 19, 29, 38, 50 and many other n values have an xmax of 88. More surprising is that in some cases xmax is the same for a sequence of values of n. Their is a unique example of this, an xmax of 9232. This is the xmax for n =107 thru to 111, and for n = 124 thru to 126 and numerous other sequences of n. In some of, or part of these sequences t,mao and div are also the same.The value of xmax = 9232 seems to be of special significance, over the range n = 2 to 5000 xmax = 9232 for 1225 of the n values. A common xmax is one sub-group of n, there are other sub-groups where xmax is a linear function of n i.e. xmax = kn + c. The sub-group for k = 1 corresponds to the condition when xmax = n. k can however have values such as 1.125, 1.5, 1.6875, 2.25, 3.0, 3.375, 4.5, and 6.75 and others, all have associated values for c. e.g. when k = 2.25 c = 2.5. All k values seem to be the ratio of integers like 3/2, 9/8 etc.
That is it for now, any interest or comments ? Regards Ivan. ------------------------------------------------------------