For Friday, February 7, 2020

For Friday:

  • Watch Video of an informal explanation of why the square root of 2 is irrational (Numberphile) (8 or 9 minutes long). He rambles interestingly for a while about paper and peeing toward the sun, but he does eventually give a proof.
  • Now, write up a nicely written mathematical style proof that the square root of 2 is irrational.  You can and should follow the logic in the video (back up and replay as much as you like), but you need to write it formally, not just copy down his informal ramblings.  The script of the video is not a nicely written mathematical proof.  There are some details that need to be dealt with, so be careful and fill in any little holes as carefully as you can.  (Please *don’t* look up write-ups of this proof online or in your text; use only the video and have the writing part be your own work.)
  • What you did above is a proof by contradiction.  Answer these two:
    • What did you assume for contradiction?
    • What contradiction did you reach?
  • Did you notice the other tiny proof by contradiction contained in the bigger one (about even squares)?  Do the same here:
    • What was the statement of this little fact?
    • What did you assume for contradiction to prove it?
    • What contradiction did you reach?
  • Please bring your work to class, as always, in case I check homework.
  • There will be a badges quiz on Friday.  We will have Sets I, II, III and IV, as well as Proofs I.
    • We will do examples of Proofs I in class before the quiz, but basically the idea with that is to be able to figure out what you have to assume to set up proof by contradiction (the first sentence of the proof, typically).  This is really practice in negation, more than anything else, and in formal writing.
    • If you have mastered your other badges, and you wish to read ahead, Sets II is very easy to do by self-study and is covered in Hammack, 1.5-1.7.  Otherwise, focus on the badges we have covered in class and we’ll get to this in due time.
    • Remember, you can attempt as many or few of the badges as you like on the quiz each time.  Your score for each badge can only go up, not down, so you are welcome to attempt badges you’ve already earned, to get feedback and practice, without any penalty.  But you may wish to focus on badges you have not yet earned.
  • With remaining time, study for quizzes, catch up on reading the text and do practice problems, as always.

Wednesday, February 5th, 2020

For Wednesday:

  • Please make an attempt to fill out the worksheet on negation handed out in class.  We haven’t studied this yet, so some of it may be tricky, but just dive in and try to give it a go.  We’ll take them up in class.
  • Budget 10 minutes to watch these two videos I made on the topic of today’s lecture:
  • There will be a proof quiz on Wednesday (we are catching up on quizzes).  It will be another direct proof (Chapter 4, Hammack), but it will be more challenging than the first proof quiz, in one or several of the following ways:
    • It may require you to read and understand a novel definition and apply it.
    • It may require more steps, or breaking into cases.
    • It may require more creativity.
  • Note: I will shortly (but not up yet) post solutions to a few recent homework problems, so you have more things to compare to.