Math 170: Ideas in Mathematics (Spring 2007)

Mondays, Wednesdays and Fridays 10 am - 11 am in DRL A8

Faculty: Nate Ackerman TA: Daniel Pomerleano TA: Chintan Mehta
E-mail: nate AT math.upenn.edu E-mail: dpomerle AT math.upenn.edu E-mail: chintanm AT math.upenn.edu

This is the approximate schedule of what should be covered in class with the associated reading. Check back to this page for updates throughout the course. With regards to the reading column, Math Jazz refers to the textbook by Burger and Starbird while Magic Numbers.  Refers to the textbook by Gross and Harris. 
Day of Week Date Topics Reading
Monday January 8
Introduction, Explanation of Grading, Math Anxiety.
Begin Coincidences, Birthday Problem
Math Jazz: Chapter 1
Magic Numbers: Chapter 5
Wednesday January 10
Related and Unrelated Events
Monte Hall Game
How to Calculate Probabilities
Wikipedia on Monte Hall
Friday January 12
More Coincidences
Finishing up with Probabilities
Calculate Probabilities
Wikipedia on Probability
Monday January 15 Martin Luther King Jr. Day (no class)
Wednesday January 17
Chaos, Iterations and Small Variations.
Butterfly Effect
Rounding Error
Population Formula
Math Jazz: Chapter 2
Tutorial on Dynamical Systems
Wednesday January 24
Complex numbers, Complex plane
Mandelbrot and Julia sets
Coastlines of infinite length
Complex Numbers, Julia and Mandlebrot Sets
Wednesday January 31
Fractals
Sierpinski Gasket
Cantor Line
Definition of Dimension
Math Jazz: Chapter 8
Box Counting Dimension
Scaling Dimension
Sierpinski Triangle
Koch Curve
List of Fractals by Dimension
Monday February 5
Definition by recursion
Factorial
Proof by Induction
Magic Numbers: Chapter 7
Mathematical Induction
Recursion
Wednesday February 7
Fibonacci numbers
Breeding Rabbits
Math Jazz: Chapter 6
Fibonacci Sequence
Friday February 9
Golden Ration
Relationship between Squares and Rectangles
Beauty and the Ancient Greeks
Math Jazz: Chapter 7
Golden Ratio
Monday February 12
Golden Ration and Fibonacci numbers
Continued Fractions
Limits, Cauchy Sequences, Real numbers
Rational and Irrational numbers
Equivalence classes
Continued Fraction
Cauchy Sequence
Rational Number
Irrational Number
Equivalence Class
Friday February 17
Pascal's Triangle
Some Nice Properties
Sierpenski's Gasket and Pascal's Triangle
Magic Numbers: Chapter 6
Pascal's Triangle (Wikipedia)
Properties of Pascal's Triangle
Pascal's Triangle Applet
Monday February 20
Binomial Formula
Expanding out (x + y)^n
Prove by induction that this corresponds to Pascal's Triangle
Magic Numbers: Chapter 6
Binomial Theorem
Wednesday February 22
Prime numbers
Relative Prime Numbers
Divisibility
Magic Numbers: Chapter 10
Prime Number
Composite Number
Friday February 24
Rational numbers
Square root 2 is not rational
Irrational Numbers
Magic Numbers: Chapter 12.6
Magic Numbers: Chapter 14
Rational Number
Irrational Number
Friday February 24
Modular Arithmetic
Examples of Clocks and Computers
Operations in modular arithmetic
Magic of Numbers: Chapter 15
Magic of Numbers: Chapter 16
Modular Arithmetic
Friday February 24
Show that Z/(p) is a field when p is prime (but don't define field)   
Explain difference between Z/(p) and Z/(m) where m is not prime
Magic of Numbers: Chapter 17
Monday February 26
Basic Codes
(Possibly) Talk about Enigma
Magic of Numbers: Chapter 21
Wednesday February 28
Fermat's Little Theorem
Taking kth Roots
Magic of Numbers: Chapter 19
Fermat's Little Theorem
Friday March 2
Public Key Cryptography
RSA
Math Jazz: Chapter 4
Magic of Numbers: Chapter 22
Public Key Cryptography
RSA
Monday March 12
Limits on Public Key Cryptography
Limits of Computer Power
 
Wednesday March 14
Primitive Recursive Functions
Encoding Primitive Recursive Functions
Primitive Recursion (Wikipedia)
Primitive Recursion (PlanetMath)
Primitive Recursion (Wolfram)
Friday March 16
Halting Problem
Non-Primitive Recursive Functions
Cantor's Diagonal Argument
Halting Problem (Wikipedia)
Halting Problem 1
Halting Problem 2
Halting Problem (PlanetMath)
Cantor's Diagonal Argument (Wikipedia)
Cantor's Diagonal Argument (PlanetMath)
Monday March 19
Recursive/Computable Functions
mu Operator
Church-Turing Thesis
Recursive Function (Wikipedia)
Recursive Functions (Stanford)
Recursive Functions (Wolfram)
mu Operator (Wolfram)
Church-Turing Thesis (Wikipedia)
Church-Turing Thesis (Stanford)
Church-Turing Thesis (Wolfram)
Computable Functions (Wikipedia)
Wednesday March 21
Recursive/Computable Sets
Recursively/Computably Enumerable Sets
Recursive Set (Wikipedia)
Recursive Set (Wolfram)
Recursive Set
Recursively Enumerable Set (Wikipedia)
Recursively Enumerable Set (Wolfram)
Recursively Enumerable Set (PlanetMath)
Monday March 26
Turing Degrees/Reducibility
Recursive Relative to a Function
Turing Jump
Turing Degree (Wikipedia)
Turing Reduction (Wikipedia)
Turing Jump (Wikipedia)
Wednesday March 28
Turing Machine
Turing Machine (Wikipedia)
Turing Machine (Stanford)
Turing Machine (Wolfram)
Turing Machine Simulator 1
Turing Machine Simulator 2
Wednesday February 28
Connections to Modern Computers
Churches Thesis
 
Friday March 2
Encoding Computer Programs as Numbers
Computer Programs Which Run on Programs
 
Monday March 12
Halting Problem
Why it isn't a solution to Churches thesis
Difference between Computable and Computably Enumerable
 
Wednesday March 14
Definition of Size of a set
Monte Hall Game
Calculate Probabilities
 
Friday March 16
Related and Unrelated Events
Examples of finite and infinite sets
 
Monday March 19
Operations on Infinite Sets
+, *, Exponent
 
Wednesday March 21
There are more reals than natural numbers
 
Friday March 23
4th Dimension
 
Monday March 26
4th Dimension
 
Wednesday March 28
Topology
Equivalence with Clay
 
Friday March 30
Bridges of Konigsberg
 
Monday April 2
Knots
 
Wednesday April 4
Extra Time
 
Friday April 6
Extra Time
 
Monday April 9
Extra Time
 
Wednesday April 11
Extra Time
 
Friday April 13
Extra Time
 
Monday April 16
Extra Time
 
Wednesday April 18
Extra Time
 
Friday April 20
Extra Time
 

This page was created by Nate Ackerman, and last revised on January 5, 2007.