Second Term
Consider whether or not two-dimensional numbers can be divided.
Look for equilibrium values using the recursion rule zn+1 = zn2 + c
Continue exploring the cubic recursion model from the exam. Come see me to learn the calculus that will allow you to find the height of the hump.
Learn about four-dimensional numbers (quaternions) and the dynamical systems made with them.
Make a detailed map of the Mandelbrot set, showing where various types of Julia sets are located. (For example, the five-cycle Julia sets are in this region, seven-cycles are here, etc.)
Design an interactive web site to teach people about two-dimensional numbers.
Read Thomas Kuhn's The Structure of Scientific Revolutions to learn more about the concept of paradigm.
Use the Randomania program to look at the difference between randomness and chaos.
When the computer generates a random number, it's actually a pseudo-random number. Find out how computers generate these numbers. The most common method is based on modular arithmetic!
Analyze the probabilities involved in Yahtzee or poker or some other game that involves chance. Is the scoring for the game fair?
Analyze the card game called "War," which is played purely on the basis of chance. How long should the average game last?
Record a sporting event and analyze the commentary of the sportscasters. Do you feel they demonstrate an understanding of probability? You may need to make several recordings before you get some good examples.
A current political debate concerns the United States census. Some people are advocating that the government should use sampling to count the population, while others insist that we should stick with the current method of attempting to find every single person and count heads one by one. Find out what they're arguing about and why it has political significance.
Part of the debate about evolution involves probability. Creation theorists argue that there hasn't been enough time for development of complicated organisms to have occurred by random mutations. Check into the debate. One possible source is Climbing Mount Improbable by Richard Dawkins.
Look into the application of probability theory to the stock market. The work of Nassim Taleb is especially interesting. There is a link on our chance page to an article by Malcolm Gladwell about Taleb and Taleb has his own web site. There is also a famous book by Malkiel, A Random Walk Down Wall Street, which has been out since I was in college.
Follow the news coverage of a major accident. Get some verbatim quotations. Analyze the view of risk that seems to be underlying what people say. Use the concepts presented by Malcolm Gladwell in "Blow Up." Which people seem to believe in "normal accidents"?
In choosing a good sample size to conduct an opinion poll, does it make a difference what the distribution is in the general population? In the lab on polling, all of the experiments were run with a population that was split 45%-55% between the two choices. Try changing that to a 50%-50% distribution and see if it changes your other conclusions. Then try 70%-30%. Generalize about what difference distribution makes, if any.
Do polling experiments where there are three choices instead of two. See if it influences how you choose sample size.
In our glossary, check the definition of an algebraic number. Many algebraic numbers are irrational. As we saw in class, the set of rational numbers is countable, but the set of real numbers is not. What about algebraic numbers? They are a subset of the real numbers. Are they countable or not? You might look at other subsets of the irrationals to see if they are countable or not.
Look at all the various operations involving transfinite numbers. We have already seen some of the properties, such as aleph null plus aleph still equals aleph null. But what about division and subtraction? What can we say about something like aleph null minus aleph null?
Look into surreal numbers, invented by John Conway. There are a few links on our cardinality page.
To reach a new level of inifinity, we looked at the set of all subsets of counting numbers. This includes some finite subsets, such as {1, 3, 17}, as well as some infinite subsets, such as all of the positive multiples of 3. If we removed the infinite subsets and left only all of the finite subsets, do you think that the result would still be uncountable?
Find out more about Cantor's Continuum Hypothesis. Explain why it cannot be proven to be true or false.
Analyze Mastermind strategies. What situations make Bjorn's strategy fail to win within eight guesses? Is there some number of guesses (like nine, ten, etc.) where Bjorn's strategy is guaranteed to win? What about Nate's and Ms Koch's? And your own, perhaps? Can you come up with something that will have better average performance than the others? Can you guarantee a win within eight guesses? You don't have to answer all of these questions, but this gives you an idea of what to look at.