Week | Sections in text (estimated) | Test | Administrivia |
---|---|---|---|

Week 1: Aug. 27-31 | 1.2-1.5 | ||

Week 2: Sept. 3-7 | 1.6, 2.1, 2.2 | No class on Monday (Labor Day) | |

Week 3: Sept. 10-14 | 2.2, 2.3 | Monday: last day to drop without a W or change grading option to pass/fail | |

Week 4: Sept. 17-21 | 2.5, 3.1, 3.3 | ||

Week 5: Sept. 24-28 | 3.4-3.6 | ||

Week 6: Oct. 1-5 | 7.4 | Midterm #1 | |

Week 7: Oct. 8-12 | 4.1, 4.2 | ||

Week 8: Oct. 15-19 | 4.3, 5.1, 5.2 | ||

Week 9: Oct. 22-26 | real numbers (not in text) | Midterm #2 | |

Week 10: Oct. 29 - Nov. 2 | 6.1-6.3 | Monday: last day to drop or change a pass/fail option to letter grade | |

Week 11: Nov. 5-9 | 6.4-6.6 | ||

Week 12: Nov. 12-16 | 8.1-8.4 | ||

Week 13: Nov. 19-23 | Thanksgiving break! | ||

Week 14: Nov. 26-30 | 8.5-8.8 | ||

Week 15: Dec. 3-7 | amusing topic and review |

I encourage you to read Section 1.5. I talked about the most important proof techniques in it and you had an in-class exercise on one more, but there are a few others that it wouldn't hurt you to think about before you have to use them.

At this point, you might want to try reading the proof again and seeing if it makes more sense now that we've done some examples in class.

If you'd like to know more about the Hilbert problems, you can find his original address here, and if you'd like to see some of the people involved in the solution to Hilbert's Tenth, you can watch this trailer for a movie about Julia Robinson. Davis, Putnam, and Matiyasevich all appear. (One fact that is not obvious from this video is that Constance Reid was not only a mathematical biographer but also Julia Robinson's sister.)

The first midterm I gave in 2710 last spring can be found here. Don't worry about #7 -- we haven't gotten to that point yet.

Here's another induction problem for you: Suppose that you have a string of Hs and Ts generated by putting down a row of coins and looking to see if they're heads or tails: for instance, HTTTHTHTH. Now you can modify this string in a certain way: you're allowed to remove any H and flip the coin(s) on either side of it, but not to close the gap created by removing the H (for instance, HTTTHTHTH could be turned into HTTH.HHTH, where the dot represents the gap, or HTTTHTHH.). You can keep doing this over and over and possibly get rid of all the coins in this way: for instance, HTTHH can become .HTHH, then .HTT., then ..HT., then ...H., and finally ..... at the end. On the other hand, you can't reduce TTT at all! Prove that you can remove all the coins in this way iff the number of heads in the original string is odd.

Here's a carefully-written version of the statement and "proof" I gave you in class today. It will make clear that one possible complaint doesn't lead to a problem.

**Statement:** All math professors are female.

**Proof:** Let P(n) be "In any set of n math professors, all of them have the same sex." Our base case is 1: Clearly, in a set of 1 math professor, only one sex will be represented. Now suppose that P(k) is true: in any set of k math professors, all of them have the same sex. We want to show that in any set of k+1 math professors, all of them have the same sex. Suppose you have a set of k+1 math professors. Take a subset of them of size k, leaving out person A. By our induction hypothesis, we know that everyone in that subset has the same sex. Now we remove one person from that set and put person A in instead. Since we have another set of size k, we know that everyone in this set has the same sex. This means that person A has the same sex as everyone else in the set of size k, so everyone in the group of size k+1 has the same sex and the induction hypothesis is proven. And since you know a female math professor, you can see that all math professors are female!

The homework tips you all provided on your homework last week can be found here.

As a bonus, you can think about the function I defined today that is 1 at every rational number and the identity at every irrational number. Come see me if you'd like to talk about what it means for a function to be continuous.

Second of all, you can test your understanding of injectivity and surjectivity by defining functions from, for instance, the negative integers to the positive integers that have each combination of those properties.

To understand the geometric interpretation of taking nth roots of complex numbers a little bit better, choose a manageable n and find the nth roots of 8 cis (pi/4), 8 cis (2pi/3), and 1 cis (2pi/3). Then plot them. How do the plots differ from each other? Why?

Good luck to all of you as you study for your finals. It's been fun teaching you!