OFF ON A TANGENT
A Fortnightly Electronic Newsletter from the Hope College Department of Mathematics
October 20, 2004 Vol. 3, No. 4

Tomorrow's colloquium will feature problems, puzzles, and pizza! 

Professor Emeritus John VanIwaarden will present puzzles and problems for students to solve during tomorrow's colloquium. These will be fun problems to work on and should stretch your mind a little.  To stretch your stomach a little, pizza will be served at the end of the problem session.  So come and enjoy working on some interesting problems and eating some great pizza.  This all starts at 3:30 p.m. Thursday, October 21 in VWF 104.


Tea Time!

Since tomorrow's colloquium involves food (and we don't want you to be too full to enjoy it) we will not have tea time before the colloquium. Tea time will return, however, before next week's colloquium at 1:30 p.m. in the reading room (VWF 222).  



Next week's colloquium will feature Michigan Tech. professor

Jianping Dong  from Michigan Technological University will present a colloquium on Friday next week at 2:00 p.m. (Note day and time change.)  She will talk about statistical genetics and will also provide information about the graduate programs in the mathematical sciences at Michigan Tech.

Today is the last day to sign up for the Michigan Undergraduate Mathematics Conference

The Department of Mathematics at Central Michigan University will be hosting the seventh annual Michigan Undergraduate Mathematics Conference (MUMC) on Saturday, October 30, 2004 from 9:00 a.m. to 5:00 p.m.  Hope College will be taking a group of students and faculty.  They will leave early in the day and return in the evening.

There will be student presentations as well as presentations on careers in mathematics, information about mathematics graduate programs and REU programs.  For those interested in attending, the deadline to sing up is today, October 20.  Contact Prof. Darin Stephenson if you are interested in attending (he has a sign-up sheet outside his office door, VWF 210).  Visit the MUMC web page at http://calcnet.cst.cmich.edu/org/mumc/  for more information about the conference.


Tomorrow is the last day to sign  for the MATH Challenge

You have until tomorrow, Thursday, October 21, to sign up to take the Michigan Autumn Take Home Challenge (or MATH Challenge).  The competition will take place on the Saturday morning November 6. Teams of two or three students take a three-hour exam consisting of ten interesting problems dealing with topics and concepts found in the undergraduate mathematics curriculum. 

For more information about this competition visit http://www.mcs.alma.edu/mathchallenge/.  If you are interested in competing, you need to sign up with Prof. John Stoughton.  You can sign up on the sign-up sheet on his door or send him an e-mail at stoughton@hope.edu.


Information session about the GRE is scheduled for tomorrow

Taking the Graduate Record Exam (GRE) is a requirement for entrance into many graduate schools.  The Hope Pew Society and the Office of Career Services are sponsoring an information session on the GRE. Professor Charles Behensky of the Department of Psychology will discuss the mechanics of the GRE, what students might do to prepare for the exam, and answer questions.  The session will be tomorrow, October 21, from 5:30 to 6:30 p.m. in 1000 Science Center.

For more information, go to the Career Service’s GRE web page: http://www.hope.edu/student/career/GRE.html. The site provides more information on the GRE, including subject test dates, and announces the availability of some practice test software.


Problem Solvers of the Fortnight

This fortnight nineteen of you proffered solutions to the somewhat weightier problem of determining which of twelve coins was counterfeit and whether the bogus one was heavier or lighter than the others.  Congratulations to Daniela Banu, Chris Johnson, Alex Larson, and Josh Morse, who correctly determined that three weighings will decidedly determine (without any luck) the counterfeit coin and whether it is heavier or lighter than the others.  For their weighty labors, our four problem solvers may claim their sweet rewards by dropping by Dr. Pearson's office.    


Problem of the Fortnight

No coins this time. Just a neat problem!

The carousel on Windmill Island here in Holland has been around a long time and is starting to show signs of wear. In fact, the deck of the carousel has needed a paint job for a couple years now, but before it could be painted, Herm VanderVeedenVanderMeen (the carousel master) wanted to know exactly how much paint would be needed to do the job. The problem was, though, he couldn't figure out how to determine the area of the carousel deck, which is an annular ring, because the motor and gears in the middle of the merry-go-round prevented him from measuring the radii of the inner and outer circles. If he could have measured those, his job would have been easy! One day, he was talking to a group of Hope College students, who had taken their parents to Windmill Island, and he told them about his dilemma. One of the students, who had taken some really great math courses at Hope, said to him, "I think I can help you out," and she took his tape measure, walked over to the carousel and made a single measurement along a straight line. Her measurement was 30 feet, and after making a few quick calculations in her head, she told the man the area of the carousel deck.

The question is, What measurement did she make (and how did she use it to calculate the area of the carousel deck), and what is the area of the carousel deck?

Write your solution on the back of a pair of World Series tickets (old ticket stubs from Windmill Island will suffice if you have difficulties procuring World Series tickets), and drop them in the Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by 3:00 p.m. on Friday, October 29.



 Got a Math Question?

Ask Elvis ...

... email him at elvis@hope.edu

Note: Student names have been changed for identity protection.

Dear Elvis,
How were/are the values for the known transcendentals such as Pi and e found? Obviously, I can draw a circle and measure circumference and radius to get a decent estimate, but we know these numbers to the millionth decimal. How is this accuracy accomplished and confirmed?
In awe of Mathematics, "Kyle Williams"

Dear "Kyle,"
Thanks for writing -- that's a GREAT question!  I've come across those transcendental numbers many times in my studies of calculus and wondered the same thing.  I didn't know the answer offhand (or offpaw, as the case may be), so I had to do a little digging.  I asked around the department, and Dr. Cinzori transcended the call of duty with his answer.  He writes:

Dear Elvis,
Pi is currently known to over 1,240,000,000,000 decimal digits. Amazingly, the basic formula used to calculate pi to this accuracy was discovered around 1800 by a teen-aged Carl Gauss. Gauss' algorithm is quite straightforward; it uses only addition, subtraction, multiplication, division, and square roots. The algorithm is recursive which means that it begins from an initial guess, applies a formula to produce an output, and then uses that output as the next input to the formula. Doing this repeatedly produces a sequence of numbers that converge to pi. Gauss' algorithm is fast (it converges quadratically) because it more than doubles the number of correct digits with each iteration. Starting with the initial guess 3.14, it only takes 38 iterations to produce over one trillion digits.  You can see Gauss' algorithm at http://pw1.netcom.com/~hjsmith/Pi/Gauss_L.html.

Because the algorithm is so simple, it is easy to code and check for accuracy. Even so, a record is not officially established until it is checked by an independent party using a different algorithm, of which there are many. Many mathematicians have produced fast algorithms for pi. Most famous are Gauss (1777-1855) and Ramanujan (1887-1920), but during the past 20 years great progress (including a quartically convergent algorithm) has been made by Peter and Jonathan Borwein and Simon Plouffe.

If you are interested in learning more, check out the book Pi Unleashed by Jorg Arndt and Christoph Haenel available in VanWylen Library. The book includes a CD with many different routines for calculating pi. The homepage of Yasumasa Kanada, the current record holder for calculating digits of pi, is http://www.super-computing.org/. Prof. Kanada has calculated over one trillion digits of pi. You can also check out Peter Borwein's homepage http://www.cecm.sfu.ca/~pborwein/.

--Dr. Cinzori

Thanks again for the great question, "Kyle"!






Mathography: Srinivasa Aiyangar Ramanujan (1887-1920)

The story of Ramanujan reads almost like a fable.  Born in his grandmother's house in Erode, a small town about 400 km southwest of Madras, India, Ramanujan contracted smallpox at the tender age of two but survived the disease whose death rates have historically been as high as 30%.  (The last case of smallpox in the U.S. occurred in 1949, and smallpox was eradicated from the world only in 1977.) 

In school, Ramanujan proved himself an able scholar in all subjects, but he was particularly entranced by mathematics and in high school picked up a copy of G.S. Carr's book, "Synopsis of elementary results in pure mathematics," a book that contained only theorems, formulas and short proofs and was out-of-date, having been published in 1856.  From this book, Ramanujan taught himself mathematics, providing proofs where none were given and deriving the book's formulas for himself. 

In 1906 Ramanujan went to Pachaiyappa's College to prepare for admission to the University of Madras, but he failed all his examinations, except for mathematics, and thus was denied admission to the university.  But he continued his research in mathematics, with only Carr's outdated book to guide him, and in 1913 sent a letter to G.H. Hardy, professor of mathematics at Oxford, that contained some of the results Ramanujan had obtained.  Impressed by Ramanujan, Hardy invited him to Oxford in 1914.  Apart from his mathematical fellowship with Hardy and his colleague, Littlewood, Ramanujan was miserable: he had no friends outside his mathematical colleagues, finding food that agreed with his diet was extremely difficult, and doctors feared that when he fell seriously ill in 1917 he would die.  By 1918 Ramanujan's health improved, but his homesickness did not.  In 1919 he set sail once again for India, but his health problems returned and he died the following year at the age of 32. 

His mathematical legacy is recorded in the many notebooks he left behind at his death, but because of the idiosyncratic style of writing mathematics that Ramanujan learned from Carr's book, many of the results he recorded in his notebooks seem to come out of thin air.  Nevertheless, Ramanujan made significant contributions to many areas of mathematics, including infinite series (see above on computation of pi).  He was one of India's greatest mathematical geniuses.