Math 1165 Summary of Lectures
  1. Place Value, why is .9999... =.9bar the same as 1.
  2. Integer arithmetic, rational and repeating decimals and one-pile nim. Also, expressing numbers in other bases. Two methods, repeated subtraction and repeated division.
  3. Bouton's  Nim, balanced configurations, fractions in other bases.
    See problems about Bouton's Nim.
  4. Cantor and Fibonacci  representations and even more exotic representations.
    See representations of integers for the problem.
  5. Decanting Problem and The Euclidean Algorithm The dinner bill spliting problem, which asks the following. Suppose that a dollars and b cents differs by one cent from b dollars and a cents, what are a and b?
  6. Primes and the Fundamental Theorem of Arithmetic and the irrationality of the square root of two.
  7. Induction and inductive sets. An inductive set is a set of real numbers that has zero as an element, and is closed under the operation of adding 1. In other words for any x in the set, x+1 is also in the set. We talked about addition problems, divisibility problems, inequalities, all horses are the same color, and one tiling problem.
  8. Divisibility properties, counting divisors, LCM, and GCD; finding representation in negative bases.
  9. More on Fibonacci numbers. Theorem. Suppose a,b,c,d, and e are consecutive Fibonacci numbers. Then (ad-bc)+(be-cd)=0. So this, coupled with the fact that 1x3-1x2=1 shows that each such expression ad-bc is either 1 or -1. What is the limit of the ratios of the Fibonacci numbers, 1/1, 2/1, 3/2, 5/3, 8/5, etc. Yes, its the golden mean.
  10. Finally, can a 2^n by 2^n punctured checker board always be tiles by L-shaped triominoes?
  11. First Test. September 25.
  12. Solving first and second order recurions
  13. Cardinality. Bijection of one set A onto another set B. Making zero disappear, ie finding a bijection from the closed interval [0,1] to the half open interval (0,1]. Also, the dual experiments with the balls numbered 1,2,3,4, ... . Why we should not expect to be able to use our intuition about infinite cardinals.
  14. The open interval (0,1) is not equivalent to the natural numbers 1,2,3,4... Cantor's diagonalization procedure.
  15. Modular arithmetic
  16. Using characteristic functions to prove set identities.
  17. Second test about November 6
  18. Counting procedures, Inclusion/Exclusion
  19. Permutations vs. Combinations
  20. Binary Relations
  21. Equivalence relations
  22. Posets, counting binary relations