Day 12: mat385

Last time

Next time

Today:

• Announcements:
  • I've finally gotten a key together for the first exam, made up of your work. Have a look (here's the exam itself).

    Feel free to ask questions about your exams if you have any.

  • Homework 3.1: Not graded yet (sorry about that; had to grade an exam this weekend, and I'm a little behind).

  • This is our last new material before the exam, which is coming up next week (on the weekend, as before).

• Today: we'll hit a couple more problems from Section 3.3: Analysis of Algorithms; then start section Section 4.1: Sets
  • The gcd stuff is certainly a little mysterious. I've given you a couple of homework problems along those lines to work. Our objective last time was to get a better bound on total number of divisions required for a starting value of $a$ in $gcd(a,b)$. Our author shows that the number of divisions $E(a)$ is bounded above by \[ E(a) \le 2 \log_2(a) \]

    We got a much tighter bound last time.

    So I start with exercises 39 and 40 (which rely on two that you're to do for homework), and show two other bounds for the number of divisions.

    And the scratch work.

  • The focus of the worst case scenario of the gcd falls on Fibonacci numbers, which nature loves, and which the gcd doesn't. Successive Fibonacci numbers have the most divisions for their given size.

    I share a game with you, just in time for Easter, based on a mysterious property of the Fibonacci numbers (every natural number is either Fibonacci, or it can be expressed as a sum of non-consecutive Fibonacci numbers in a unique way).

    You can play this game to take Easter candy from all your young relatives and friends (just don't tell them who taught you this game!).

  • Finally I get started with section 4.1, hitting the highlights up to Power sets. We'll use these to show that there are many sizes of infinity, which is a bit mind-blowing. Coming soon!
  • The marked up pdf (so far....)

• Links:
  • Nothing new to report.