- 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) 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 {a
n} be a (possibly
infinite) sequence of positive integers. A creature like
is called a
continued fraction
and is sometimes denoted by [a
0, a
1,
a
2, a
3, . . .
]. 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.
Happy
Pi Day! 
