Math 629 - News

Math 629 - Spring 2003

News

May 3, 2003: For those of you that read German and want to view the essential source of the original work on Dedekind cuts as first described by Richard Dedekind, please follow this link. http://www.math.kun.nl/werkgroepen/gmfw/bronnen/dedekind2.html You get the title page but the remainder is an e-book, not scanned pages.

April 28, 2003: If you prefer to take a final exam rather than the final essay-type assignment, you may. Here is a link. You may prefer this option because you can use previously learned facts. BTW: This is a variation of an exam I've given a couple of times as a qualifier for mathematics education phd students.

April 28, 2003: For those of you needing some extra time on the assignment, you can take until Monday, May 5 to complete the work. ALSO:::: I have graded all electronic submission of the position paper that I have received. If you haven't received a grade, you better resubmit.

Also, for those of you wanting more information on the Poincare conjecture by way of more detail, here is a link to a short paper written by the Fields medalist (and master expositor) John Milnor. (http://www.math.sunysb.edu/~jack/PREPR/poincare03.pdf)

April 27, 2003: (1) You all know that graphing calculators are now ubiquitous. Most students are fully able to solve equations using them. But did you know that universal equation solvers have been around for more than two centuries. That's even before electricity. Such a device was reported in the Encyclopedie of Diderot, published in 1751. For information on this subject, please visit the URL: http://www.math.unifi.it/archimede/archimede_inglese/curve/visita/macchina.htm
The method, not surprisingly, is truly geometric, relying on the similarity of specific right triangles. Following the links you can see an original drawing of the device.

(2) An interesting infinity problem: A numerical palindrome is a sequence of numbers which is the same when read left to right or right to left. For example '2332' is a numerical palindrome of length four. Naturally, there are palindromes of any length. Now consider all real numbers in the interval [0,1]. Each has a decimal representation, of course. Given a specific length n, let P_n be the set of all numbers in [0,1] that have a palindrome of length n in their decimal representation. Define Q_n = [0,1 ] - P_n. Question: for which n is Q_n finite? Countable? Uncountable? Hint. This problem is not terribly easy. I have solutions for n=2, 3, and 4. Beyond that it gets tricky.

April 23, 2003: Solutions to problem set 12 are posted.

April 22, 2003: Solutions to problem set 11 are posted.

For local students only: On the last class day, April 28, I will ask each of you to give a prepared approximately 10 minute talk on the following aspects of this course: (i) What surprised me most about the history of mathematics, (ii) What lessons I have learned that I can apply in my own, job, and (iii) How do I now perceive the relevance and importance of having an understanding of this history of a subject.

April 18, 2003: Solutions to problem set 10 are posted.

April 18, 2003: Some of you have had a little difficulty with the Euler formula problem. So, here is a hint. This link will also give you some further dicussion of modular arithmetic. Among other things modern data encryption could not take place without modular arithmetic. It is the foundation of much of number theory. For modular arithemetic there are also online references. You might check out

http://www.unf.edu/~mdedeo/BasicNumbThy.html
http://www-sal.cs.uiuc.edu/~steng/cs497_01/lecture3.ppt
http://klein.math.okstate.edu/~wrightd/crypt/lecnotes/node12.html
http://www.ecu.edu/si/cd/interactivate/discussions/cipher.html

April 17, 2003: Update, Dunwoody, the mathematician alluded to in this article, has retracted his proof. However, recently a Russian mathematician, Grigori (Grisha) Perelman has claimed to have solved this problem in a paper posted to the ArXiv.

April 16, 2003: The Poincare conjecture, one of the most difficult problems of the twentieth century, has apparently been solved. You may read about the conjecture and the person who solved it at http://abcnews.go.com/sections/scitech/WhosCounting/whoscounting020505.html. If the proof is not quite ready for prime time, you may resume work on it for the $1m prize available. Here is a description of the problem, taken from the Clay Mathematical Institute.

If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is "simply connected," but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This question turned out to be extraordinarily difficult, and mathematicians have been struggling with it ever since.

Here's a picture of Henri Poincare (1854-1912).

April 8, 2003: As you have noted from the homework assignment page, there will be no final exam. While this has not served as a disappointment to most of you, some concern has been expressed about how grades will be assigned. Below is the revised grading scheme.

Activity	Points	Final Grade	Points
Homework	200	A	540-600
Term paper	200	B	480-539
Position paper	100	C	420-479
Book Report	100
Total	600

However, if you wish to have a final exam, please let me know. I will send one, and then I will assign a grade according to the previous schedule.

April 6, 2003: New homework assignments have been posted. Information about the final is given. See the assignments page.

April 2, 2003: End of course news. The last day of classes here on campus is April 29. Hence no work will be assigned after that. What about the final exam? The final exam will be a comprehensive set of problems of both the "work out" and "essay" variety. You will be given about a week to complete the exam.

Homework is in the process of being graded. I've gotten behind. Sorry.

April 2, 2003: Christian Huygens (1629-1695), in his day the leading scientist in Europe, produced the first pendulum clock, though it did not solve open longitude problem --- that being to accurately measure the longitude of a ship at sea. Just eight years later he noticed that when two of his clocks were placed side-by-side the pendula would in very short time synchonize with each other exactly 180^o out of phase. When disturbed from this state they would quickly resynchronize. Huygens attributed it to imperceptible motions of the clock cases though he could not precisely solve the problem giving a mathematical answer. This problem has recently been resolved by four researchers at Georgia Tech. Their (M. Bennett, M. F. Schatz, H. Rockwood, and K. Wiesenfeld) article can be found at http://www.physics.gatech.edu/schatz/pubs/royclocks2.pdf.
Sympathetic vibrations such as Huygen's clocks have inspired many generations of studies of similar phenomena.

March 31, 2003: In the transitions chapter the diagram for the proof of the Steven proof that the center of gravity of a triangle is on the median is not labeled. That has been corrected.

March 31, 2003: The Vieta method for computing pi is the first known closed formula for its computation. Here is a link to a Maple worksheet where the first fourty terms are computed. Effectively, this means we are considering polygons of 2⁴⁰sides. From the gee-whiz department, we can safely say that if the construction were actually made for a circle of radius 1, the length of each side would be about width of 4 electrons. Here is the Maple worksheet itself if you want to compute more digits.

March 31, 2003: There is an excellent link (furnished by one of the students in the class) on the decline of Islamic influence and interest in mathematics and science. Here it is: http://www.al-kawn.com/Ihsan/decline.htm The article begins with the following few sentences.

The decline of Islamic Science and Astronomy in the middle ages

The decline of Muslim scientific and astronomical activity after a brilliant and successful start has attracted the attention of a number of Muslim writers and historians since the 17th century. One of the most proposed explanations was directed to explain the decline with political causes such as the Crusades in the twelve-century and the destruction (especially of libraries and men of knowledge) by the Moguls invasions in the thirteen-century. Another explanation attribute the decline to economic causes such as the decline ... All these explanations have some truth in them however these explanations do not explain the loss of motivation in science in Muslim societies as a whole. ...

Basically, the gist of it is that mathematics, science and astronomy fell victim to the politics and political fashion of Islam.

March 25, 2003: More than a trillion!!! Between September and December 2002 (over about 600 hours on a HITACHI SR8000/MP machine with 1TB storage) Yasumasa Kanada, and a team of ten, computed 1.03 ×10¹² hexadecimal digits, using both See:

48 arctan( 1/49 ) +128 arctan

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-20 arctan

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239

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+48 arctan

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110443

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and

176 arctan

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+28 arctan

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239

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-48 arctan

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682

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+ 96 arctan

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12943

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This was converted to 1.24 ×10¹² decimal digits. Kanada estimates the method is about twice the speed of the AGM methods he was previously using - and has other advantages to make such a huge computation feasible. For more information see: http://www.super-computing.org/ and http://www.cecm.sfu.ca/personal/jborwein/Pi_Talk.html and http://www.hints.org/~kanada/

March 24, 2003: Solutions to homework 9 are posted.

March 21, 2003: Solutions to homework 8 have been posted. I have just about completed reading the book reports, and I want to say that many of these reports are among the very best I've received in all the years I've been teaching this course. I've seen a maturity of understanding and great skill at writing. Well done.

In the lecture on Wednesday, I added some material on the medieval notions of infinity, which I've since added toward the end of the chapter on Medieval mathematics. Please read it over.

March 15, 2003: New homework assignments have been posted.

March 6, 2003: Here is a collection of axioms equivalent to the Euclid parallel lines axiom. It sweeps across history and across all of geometry. Thanks to you all for your work on this.

***I have added a few more choice paragraphs from the "I used my teacher's math book" essay.

****Please, when you turn in papers identify yourself on the paper and the assignment on the paper. Also identify the assignment on the subject line if you send it by e-mail. If you fax papers, be sure your name and assignment number are both on the paper.

March 4, 2003: I have posted The solutions to homework 6 and to homework 7 have been posted.

Also, I've clarified the hint on the sum of odd numbers problem by specifying the nature of what the proposition actually is.

Also next week is spring break here on campus. There is no assigned homework for that week. Please continue the reading portion of the course.

February 26, 2003: Hint on Problem 11. To apply induction you need the proposition

P(n) = n^3

to prove.
It is obvious that the left side, e.g. 7+9+11, furnishes P(n). That is

P(1) := 1 =1³
P(2) := 3 + 5=2³
P(3) := 7 + 9 + 11=3³
and so on.

The proposition is that the certain sum is the number cubed. So, given arbitrary n you need to express the left side. That is, you need to find the general term. Then you are in business to carry out the induction. The induction is actually very easy once you find P(n).

February 26, 2003: Homework scores are being sent by e-mail. Also, when you submit homework, please label it correctly. The homework assignment number is the same number as appears in the left column of the assignment.

IMPORTANT: It is important to use the very latest problem assignments for the homework. If you printed more than a week ago, you should check for the latest version. I am writing this because some of you are doing unassigned problems.

Finally many of you provided excellent responses to the question about using textbooks for several generations. You make quite good points. I agree there certainly is value to using a book for a long time. I wish I could teach from the calculus book I learned from so many years ago. I would really be able to judge students' understanding a whole lot better than now. Here is a link to select (somewhat random) responses.

By the way, you know that pi is known to 206,000,000,000 places. Here's a bit of triviality for your students. Typing this value at 10 digits per inch, the length of the typescript would be 206,000,000,000/10/12/5280= 325126. 26 miles, more than the distance to the moon.

February 25, 2003: The next homework assignment is posted.

February 21, 2003: Please... When you submit your homework by e-mail., include at the top your name and the homework assignment number. Often, during a typical day, I have twenty or thirty files and program open, and I can easily loose track of your work, if it isn't clearly identified. Also, be sure the subject line of the e-mail. identifies which homework it is. Thanks.

February 21, 2003: There has been some confusion about the order of the numbering in the Ionian system. It is true that by the nature of the numbering, the order should not be important. However, ordering is more or less natural in writing. Thus it comes to this: In problem 4, if you assume that ordering is important you get one value; if you assume otherwise you can get another value. Can you see what they are?

February 14, 2003: The material for the position paper is now posted. My experience has been that a sizable number of folks that take this course are not only very good writers but like to write as well. Here's a good opportunity to really explore what the history of mathematics is all about.

The book report due date is changed to March 3.

February 9, 2003: Here is a hint for problem 11, which was not assigned. But it provides all the tools needed for problem 13. You need to figure out how to get a segment of length sort(p). Hint. Use the Pythagorean theorem. I hope this helps.

February 8, 2003: We have re-posted the problems for the Greek mathematics chapters. The changes are significant. However, no problems have have been already assigned have changed numbers. So the assignment due on Monday is in tact. The next two assignments have numbers corresponding to the new posting. (FYI, the Greek problems sections consists of about sixty separate files. I encourage everyone to take Math 696 this summer to find out how this is done. That is, how we convert mathematics to web-ready formats.)

Also: The topic of the position paper is yet to be announced.

Finally: Our courses this summer are Math 451 - first summer term and Math 696 - second summer term. However, the entire summer budget is at risk because of State budget problems.

February 6, 2003: As you have been studying the work of Pythagoras, you've noticed that when discussing perfect numbers, suddenly we moved from the 5th century BCE up to last year. This is what I call the "fast-forward" feature of the notes. The reason this current information is embedded in the chapter on Pythagoras is to emphasize that the study of certain topics has ranged over such a broad span of time. The study of primes in modern mathematics has profound applications to many areas of mathematics. Indeed, one of the great unsolved problems in mathematics, the Riemann Hypothesis, can be expressed in terms of prime numbers. But the real purpose of this news item is to put you in touch with the "Primes Pages" , an authoritative source of information about prime numbers. Read the pages and join the GIMPS.

What is the current largest prime? Well, if you printed the notes a week ago you would have read that it was found a couple of years ago, where in fact a new prime was found just over a year ago. It was found on 14 November 2001 by the team of Michael Cameron, George Woltman, Scott Kurowski et. Al. Its value is: .

2^13466917-1

How big is this number? It has over four million digits, and is therefore one of two known primes called "megaprimes" because they have more than a million decimal digits. Naturally, prime numbers this large have no practical application at this time. However, the quest itself shows the absolutely amazing tenacity of humankind.

Four million digits is plenty big, but there are numbers that are bigger. As you no doubt know the number

is not rational, that is to say, it has no repeating decimal representation. As you know 3.14 is a rough and ready approximation to it. It's accurate enough for the classroom, but for serious work you may want to use the best known approximation. How many digits? about 206 billion. :) Now that's big. It was found just last December by Professor Yasumasa Kanada. Here is a list of computations of

February 5, 2003: More mathematician/mathematics name dropping in political/sociological/religious works. Perusing the papers and works of the founding fathers of the United States, we find that the general population had some familiarity with the ancient mathematicians. For example, we find in Thomas Paine's, The Age of Reason, the following passage

When we consider the lapse of more than three hundred years intervening between the time that Christ is said to have lived and the time the New Testament was formed into a book, we must see, even without the assistance of historical evidence, the exceeding uncertainty there is of its authenticity. The authenticity of the book of Homer, so far as regards the authorship, is much better established than that of the New Testament, though Homer is a thousand years the most ancient. It is only an exceedingly good poet that could have written the book of Homer, and therefore few men only could have attempted it; and a man capable of doing it would not have thrown away his own fame by giving it to another. In like manner, there were but few that could have composed Euclid's Elements, because none but an exceedingly good geometrician could have been the author of that work.

In this part of the book Paine is attempting to establish the authority of the New Testament, and is using the well known and established works of Homer and Euclid to reason his case.

Also, in Paine's The Rights of Man (1791-1792), the very first sentence of the introduction reads, "What Archimedes said of the mechanical powers, may be applied to Reason and Liberty. 'Had we,'said he, 'a place to stand upon, we might raise the world.' "

From the writings of Thomas Jefferson, the letter to Thomas Lomax Monticello on Mar. 12, 1799 contains the following passage:

DEAR SIR,

-- I have to acknolege the receipt of your favor of May 14 in which you mention that you have finished the 6. first books of Euclid, plane trigonometry, surveying & algebra and ask whether I think a further pursuit of that branch of science would be useful to you. There are some propositions in the latter books of Euclid, & some of Archimedes, which are useful, & I have no doubt you have been made acquainted with them. Trigonometry, so far as this, is most valuable ...

As is apparent, Jefferson is writing to Mr. Monticello about what materials he should read and understand in his personal advancement in scholarship. Jefferson also advises Mr. Monticello about other scientific subjects worthy of study. In another letter*, dated July 5, 1814, Jefferson talked about the value of formal education in the classics.

But why am I dosing you with these Ante-diluvian topics? Because I am glad to have some one to whom they are familiar, and who will not receive them as if dropped from the moon. Our post-revolutionary youth are born under happier stars than you and I were. They acquire all learning in their mothers' womb, and bring it into the world ready-made. The information of books is no longer necessary; and all knolege which is not innate, is in contempt, or neglect at least. Every folly must run it's round; and so, I suppose, must that of self-learning, and self sufficiency; of rejecting the knolege acquired in past ages, and starting on the new ground of intuition. When sobered by experience I hope our successors will turn their attention to the advantages of education. I mean of education on the broad scale, and not that of the petty academies, as they call themselves, which are starting up in every neighborhood, and where one or two men, possessing Latin, and sometimes Greek, a knolege of the globes, and the first six books of Euclid, imagine and communicate this as the sum of science. They commit their pupils to the theatre of the world with just taste enough of learning to be alienated from industrious pursuits, and not enough to do service in the ranks of science.

*The unusual spellings above are from the period.

You will note the use of these references is to use well established name recognition and respect for the names to establish a point or make a case. This is still done today, but rather rarely is the cited name a mathematician.

All this brings us to an interesting idea: For a term paper, one could trace such name dropping or classical mathematical/mathematician referencing, as a tool for making a political case. Any takers? It would require some considerable research, but it may well be publishable.

February 4, 2003: As you've been reading about Thales, you've noted that he is attributed to proving theorems that eventually became axioms in Euclid. This is the slow process of how rigor evolved during the three hundred years that range from Thales to Euclid. One of the questions I want you to consider is how mathematics has been "rigorized" over the many centuries since. Let me start you out by indicating that the question of whether we have currently achieved an ultimate level of rigor is one of the questions that philosophers and historians of mathematics ponder even today. This link is to a current thread of a discussion board of Historia Mathematica where a number of historians, professional and amateur have ventured their opinions. It makes interesting reading. If you wish to join Historia Mathematica, please do so. The entries in the link are ordered chronologically as they have been posted

February 3, 2003: Grades: I have begun to e-mail grades for homework papers. In most of these early assignment the grades are rather high. However, some of you lost points because you interpreted some of the question as short answer. When the question is open-ended, you should spend some time looking things up. There is such a wealth of material on the web, there is no reason why good explanation cannot be given for each essay type question.

Your answers the the teach by example question ala the Ahmes papyrus. Here is a link to some of the answers I received. A number of you cited that students learning in this system may not be flexible problem solvers in their future lives. However, let me point out that the Egyptian civilization lasted longer than the entire CE (Common Era) by nearly one thousand years. Don't misconstrue my position on this point, but it makes you think that the system was very, very functional.

Your answers on how to make a right angle were interesting as well, with many new ideas (to me) put forward.

As to why primitive cultures may need multiplication and division, someone ventured the idea that the division of stakes and computing payouts in gambling may have come quite early. Also, a couple mentioned that the first application of multiplication and division may have occurred in tribal government, with the computation of tribute owed and/or taxes. The people would have little need to use such advanced tools.

January 31, 2003: The Wall Street Journal of January 27, 2003: In a recent announcement by Bill Gates (i.e. Microsoft) wherein he states that "The West's best scientists should turn their attention to the developing world's diseases," he begins his editorial with a reference to the great 20^{th mathematician David Hilbert. The first paragraph of his essay
reads:}

In 1900, the German mathematician David Hilbert presented the International Congress of Mathematicians with 23 unsolved problems. These were problems he believed could guide mathematics research over the next 100 years, sparking breakthroughs and opening up fields of study. Indeed, research into these problems guided 20^th-century mathematics and led to significant breakthroughs in technology and medicine.

I urge you to read this essay and note how interesting it is that such a very rich man would announce his philanthropic venture via the metaphor of solving important mathematical problems. It is not clear that this sort of introduction would have appeared not too many years ago. A Biblical or certainly more classical introduction would seem more within the standard norms. Note: One or two of you are writing on the 23 problems of David Hilbert.

January 30, 2003: If you have tried and failed to access the paper by Panchenko, please try again. The link has changed to one that will work. Also, the lectures site has a different, simpler look. The same material is present. I removed the frames for better compliance with the ADA.

January 29, 2003: The problems for the Babylonian chapter have been revised for clarity. Please review the changes.

Policy change: In an effort to save trees... From now on, I will be grading your homework online, if you submit it electronically. I will e-mail back to you the paper with the grade if points are taken off. Otherwise, I will simply e-mail your grade. No homework has been returned to date. The first batch will come very soon.

Also, an interesting plausibility argument on how the quadratic equation and its solution may have originated has been prepared. Check it out on the readings page for Egyptian/Babylonian mathematics or just click here.

January 26, 2003: I have added more details on the process of converting decimal numbers to the sexagesimal form. You can find the link to these examples on the Readings page of the Egypt-Babylon chapter.

January 21, 2003: I will be out of town Jan 22-23; so e-mail responses may be delayed. I've looked over homework 1. There are many excellent papers. Thank you for your fine work.

Distance students: Do I have everyone's address?

January 19, 2003: We have included in the origins section a link to the famous engraving by Durer, Melancholia, which shows a 4 x 4 magic square. The engraving itself depicts and alchemist. Go to the reading page of Chapter 1.

You will note in the etching a number of mathematical objects including a sphere, polyhedron, a semicircle, and parallel lines. It is thought by some that the title Melancholia is symbolic of the dejection and frustration that certainly would be a part of the life of anyone searching after the ephemeral Stone, the unending quest of the Alchemist. You will note that modern mathematics was well along before any coherent theory of atomic structure and chemistry (Antoine Laurent Lavosier, 1743-1794) gained a foothold in European scientific circles. Indeed, the great Issac Newton probably spent more time on his alchemical studies that he did with calculus or physics. To my mind, this proves that while chemistry can be learned and studied at many levels, it is conceptually a true milestone for humanity.

January 17, 2003: Because Monday, Jan 20 is a no-class day here on campus, the Monday homework will be due Tuesday, Jan 21.

Remember: all homework assignments pertain to the "Problems" link in each chapter, not to the problems that may be at the end of the readings.

January 15, 2003: The question has been posed by several of you as to the depth and/or length expected as a response to an essay-type or evaluative question. The best answer I can give is that you should provide enough discussion to address the issues at hand. Normally these questions may require a typed page or more. Rarely, can just a sentence or two cover the subject in the depth and range expected. If you find yourself running on for several pages, that's OK. But please keep to the point of the question.

Feel free to consult external sources. There is a wealth of material on the web, by the way. Some of it is a bit polemic; some of it is wrong.

January 13, 2003: Both chapters 1.1 and 1.3 have been reposted with some additional material. In particular read more about magic squares in the chapter on the sources of ancient mathematics.

January 10, 2003: I have reposted the chapter on the origins of mathematics, Chapter 1.1. Included is information on the Blombos stone recently found in South Africa. Also, numerous misspelling were corrected.

NEW: The book report page includes a number of books you might consider.

Math 629 - Spring 2003

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