OFF ON A TANGENT A Fortnightly Electronic Newsletter from the Hope College Department of Mathematics
 March 14, 2007 Vol. 5, No. 12 http://www.math.hope.edu/newsletter.html

Happy Pi Day!
Pi Day Celebrations

Math enthusiasts are celebrating Pi Day today, March 14.  The symbol π was first used by William Jones in 1707, although it didn't become common until 1737 when the Swiss mathematician Euler adopted it.   March 14 also happens to be the birthday of Albert Einstein, among many other famous people.  There are several excellent resources available on the web to help you plan your Pi Day celebrations:
Happy Pi Day!

Lower Michigan Mathematics Competition
• Registration deadline: Noon on Thursday, March 15
• Competition date: Saturday, March 31
• Location: Calvin College, DeVries Hall
The 31st annual Lower Michigan Mathematics Competition (LMMC) will be held Saturday, March 31, 2007 at Calvin College.  The competition consists of a 10-problem exam taken by teams of 2 or 3 undergraduates and will last about three hours.  Lunch will be provided for participants.  The Hope Mathematics Department will provide transportation to and from the competition.  The schedule of the day's activities for the 2007 LMMC is:

 8:45am - 9:15am Check-in and refreshments in the Atrium of DeVries Hall 9:30am -12:30pm Competition 12:30pm -1:30pm Lunch 1:30pm - 2:30pm Solution session

The highest scoring team earns the right to take home the coveted Klein bottle trophy, shown above in the first floor VWF display case, where we hope it will reside next year!

To register for the 2007 LMMC, sign up on one of the registration sheets that will be circulated in classes over the next few weeks, or add your name to the registration sheet on Dr. Cinzori's door by noon on Thursday, March 15.

Colloquium: Thursday After Spring Break
• Professor Randy Pruim, Calvin College
• Calculus -- the way it might have been
• Thursday, March 29
• 4:00 p.m. in VZN 240
You have probably heard of number systems like the integers, the rationals, the reals and the complex numbers. But have you ever seen the hyperreal numbers? The hyperreal numbers are an extension of the reals (much as the complexes extend the reals, or the reals extend the rationals) and can be used as a basis for calculus that it much closer to the spirit of Leibniz, Newton and other early developers of calculus than our modern epsilon-delta approach. The result is infinitesimal calculus (also called non-standard analysis).

I'll give an introduction to the hyperreal number system, some of its properties, and what freshman calculus might have looked like had the hyperreal system been developed before epsilon-delta limits.

Join us for Tea Time on Thursdays before colloquia

As part of our colloquium series this year, the mathematics department will host a "tea time" in the Reading Room (VWF 222) at 3:45 pm.  If tea isn't really your cup of tea, have no fear -- we'll provide some other beverages and snacks, too.  So please join us for a little food and fellowship before you go to the colloquia.  It'll be a great time to chat with the speaker, your professors and other students.

This Day in Mathematics History . . .

On March 14, 1664, Isaac Barrow delivered a two-hour lecture as the first Lucasian professor at Cambridge University.  Generally given a modicum of credit for his contributions to calculus, Barrow was one of the pioneers in calculating tangents to curves and is credited with computing the tangent to the kappa curve.  Barrow has two major mathematics works to his credit -- the first on geometry and the second on optics -- although he is probably most widely recognized in connection with his student, Isaac Newton, who was the second Lucasian professor at Cambridge.  The Lucasian chair at Cambridge has been held by some of the best known and most influential mathematicians, including Charles Babbage, George Stokes, Paul Dirac, and its current occupant, Stephen Hawking.

To read more about Barrow or other Lucasian professors, please visit http://en.wikipedia.org/wiki/Isaac_Barrow.

Problem Solvers of the Fortnight

Picking two points at random on a circle C of radius 1 (using a uniform distribution on the circle so that each point on the circle has an equal probability of being chosen), the expected value of the length of the chord connecting the two points is 4/π.  Using a rotation of the circle, we may let point A be (1,0) without loss of generality.   Let B = (cos θ, sin θ) be the other point, where θ is the angle formed by the radii OA and OB.  The Law of Cosines then gives that the length L of the chord AB is L = √ (2 - 2 cos θ ).  The expected value of the length of the chord is the average value of this function over the circle -- i.e. as θ goes from 0 to 2π. Thus, the expected value of the length of the chord AB is

1/(2π) $\int$0  √ (2 - 2 cos θ ) dθ = 4/π

Congratulations to Greg Huizen, Sam Baker and Dirk Van Bruggen for correctly computing the expected length of a chord formed by picking two points at random on a unit circle with a uniform probability distribution.

Problem of the Fortnight

Let {an} be a (possibly infinite) sequence of positive integers.  A creature like

is called a continued fraction and is sometimes denoted by [a0, a1, a2, a3, . . . ].  A fact that is well known by those who know it well is that π can be represented as the continued fraction [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, . . . ]  The first few convergents are 3, 22/7, 333/106, and 355/113.  The very large term 292 means that the convergent [3, 7, 15, 1] = 355/113 is a very good approximation to π (accurate to 6 decimal places), a fact first discovered by astronomer Tsu Ch'ung-Chih in the fifth century A.D.

While you're on spring break, ponder our problem of the fortnight:  Find the exact value (no decimal approximations allowed) of the continued fraction [1, 1, 1, 1, . . . ].

Write your answer on the back of your favorite picture of Einstein and drop it by the Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by 3:00 p.m. on Friday, March 30.  As always, be sure to include your name, you math class(es) and the name(s) of your professor(s) -- e.g. Isaac Newton, Professor Barrow, Calculus 1 -- on your solution.

The most beautiful thing we can experience is the mysterious. It is the source of all true art and science
~ Albert Einstein in What I Believe