Corrections for Text, Section 4.5 The handout The Real Numbers is now complete (except for my final touchup and proofreading). You are responsible for the material in that handout (just like earlier handouts). The last homework set fairly well covered the material we talked about. However, here are just a few more practice problems related to sequences. The final exam will be held at the time scheduled in the Course Listings Book: Wednesday. May 10, 10:30 - 12:30. Some information about the final exam From the past: Exam 1 information Exam 2 information May 1-5 will have my regular office hours, and I will be available at some other times too. I'll will be "around" my office and the Math Department 9:30-2 on Monday (maybe till 3:30) Optional Q/A Session on the material after Exam II on Monday, May 8, 5-7 p.m. in the Seminar Room at Cornerstone. After a little reflection, I think a second Q/A session (on the material up through Exam II) probably wouldn't be too helpful--too much material for "random" questions to benefit a whole group of people. But I'll certainly be around to talk about any material that you like.10:30-3 on Tuesday (maybe till 4) Slight modifications in computing the total score, T, for the course: Neither change will make a big difference in T, but each one might help "at the margin.I will discard the lowest score among the homeworks that I graded (leaving 6/7 of them to count). Also, Course Evaluations on the web for this semester are open online at http://evals.wustl.edu . Please take a few minutes to fill out a course evaluation. I would appreciate the feedback. |
| Exam 1 | Exam 2 |
Final
Examination |
Each week I will post here a list of assigned homework to be handed in. These assignments will normally be due in class on Fridays. Late homeworks usually will not be accepted unless there is a special legitimate reason such as illness. When the problem list is first posted here it may not be complete. However, the list of problems due the following Friday will always be complete by noon on the preceding Tuesday. Be sure to check after that that you have the complete list.
The hand-in homeworks should be written up on 8.5 x 11 paper with "clean edges" (not torn out from a spiral bound notebook). Since part of the goal of the course is clear writing (in particular, of proofs), the solution of each problem should be written up quite clearly and legibly, and should follow the "mathematical writing" guidelines that will be handed out in class. (In particular, check yourself by reading aloud the words and symbols you wrote down, exactly as written. The result should be smooth sounding English.)
Talking with other students about homework problems is a good way to learn, but each student must write up his or her own homework. Therefore, no solutions from two students should look too much alike. After all, everybody says things in their own way, makes up their own notation as needed, etc. A good way to avoid "copying" even inadvertently from another student is to talk about problems together without taking any notes away from the conversation. This lets you share understanding and ideas, but forces you to reconstruct your own understanding on paper.
All homeworks will be posted at the link below.
Grading of homeworks: For about half the homework sets, I will select one problem to grade myself, and keep a separate record of those scores. After discarding the lowest one of these scores, the remaining total on these problems (=H1) will carry the weight of one exam in determining your grade for the course.
A grader (Haley Abel) will handle the other homework problems and assign you a score (H2) for them.
In addition to the hand-in homework, I will post a longer list of suggested problems that are not to be handed in. You should be sure you can do them all and (even better) try some others from the text book as well.
What is Mathematics Discussion (handed out in class)
Useful Tautologies (handed out in class)
Practice with Connectives (handed out in class)
Practice with Quantifiers (handed out in class)
Some definitions used in our basic proof practice (handed out in class)
Principia Mathematica
In the period 1910-1913, Bertrand Russell and Alfred North Whitehead published 3 volumes of their work Principia Mathematics . It was an attempt to show that what we would call "symbolic logic" (rather than the set theory we use today) could be used as a foundation for mathematics--that all mathematical objects (including sets) could be defined in terms of logical symbols and all theorems could be proved within symbolic logic. Although their point of view ("logicism") has few if any adherents today, the work did a good job of showing the power of symbolic logic and in clarifying many ideas. The link shows a few pages from Volume 1. I displayed it in class to show what some really, really formal proofs would look like. Notice that their work is so careful that, in setting up mathematics, they get to "1" on p. 245. (It takes another 30 pages or so to get to "2.") We (and research mathematicians) do not write proofs with that much formality. Instead, we use an informal mix of symbols, logic and good English to present a clear argument (and we become more formal when there's confusion of a particularly subtle point. Here is a little more information about the history and significance of the Russell/Whitehead project.A very good History of Mathematics site.
The 4 Color Theorem
Here is some additional background information. I mentioned this theorem in class because it's an example of a "theorem" where the proof ultimately relied on a computer doing a lot of the work. Is something a proof if all the (1476) cases cannot be verified by hand? This really raises the question of redefining "proof" from its classical meaning. Mathematicians disagree on this issue. Why?
Additional Exercises on Simple Proofs (handed out in class)
Pythagoras We proved (2 ways!) in class that the square root of 2 is irrational. This result is ascribed to the "Pythagorean School" --although this group was so secretive that it's not clear whether the result is due to Pythagoras or a later student. This is a result that went against a fundamental tenet of the Pythagorean philosophy which implies that everything in the universe could be measured with a ratio of natural numbers. As discussed in class, if we have a 1 : 1 : sqrt(2) right triangle, then a ruler that is subdivided into equal parts of length 1/q (no matter how big q is) cannot exactly measure the length of the hypotenuse -- because sqrt(2) is not equal to "p 1/q's."
Here is a third (geometric) proof that sqrt(2) is irrational. It too uses the method of contradiction, roughly as follows: "If sqrt(2) is irrrational, then must be a smallest triangle with a certain property. But given this triangle, it's easy to show that there must be a smaller one still (contradiction!). Take a look.
The Basics of Set Theory (handed out in class; this copyrighted material material be freely copied and used as long as the author's name, institution and the copyright are included on all copies))
We mentioned in class that people disagree about whether to call "0" a natural number, because, after all, it historically developed later than the other natural numbers. Here's a link called "How was zero discovered?" that is simply light reading -- no complicated history. See also A History of Zero
Sums of Powers of Natural Numbers (handed out in class)
In class we proved (by induction) a formula for the sum of the first n natural numbers: 1 + 2 + ... + n = n(n+1)/2. The handout looks at finding (recursive) formulas for
S2 = 1^2 + 2^2 + ... + n^2, S3 = 1^3 + 2^3 + ...+ n^3, etc.
Proof of the Equivalence of PMI, PCI and WOP (handed out in class)
The Whole Numbers
This is the complete document including all the separate handouts from class. Here and there, typos have been corrected or the wording improved (I think) is one of the earlier handouts. But there are no significant changes from the material handed out.
Math Gets Its Due... from the St. Louis Post-Dispatch, Sunday February 19, 2006.
Relations (handed out in class--notes and supplementary material)
Andrew Wiles and "Fermat's Last (or Great) Theorem" (NOVA)
Constructing the Integers (handed out in class)
The Rational Numbers (handed out in class))
Functions (handed out in class)
From the WU Record: Is the nature of "proof" changing in mathematics?
Countable Sets (handed out in class)
The Real Numbers (handed out in class)
Corrections for Text, Section 4.5
If your final score is <= the average of your previous exam scores, your total T will be determined by the formula
T = (H1 + H2 + E1 + E2 + F)/5 .
If your final score is > the average of your previous exam scores, your total T will be determined by the formula
T = max { (H1 + H2 + E1 + E2 + F + F )/6 , (H1 + H2 + E1 + E2 + F)/5 }
I will base grades on the number T, rounded to the nearest integer. I will not make up a grading scale until the end of the course, but it is guaranteed that the grading will be no more severe than:
90-100 A (possibly +/-)
80-89 B (possibly +/-)
65-79 C (possibly +/-)
50-64 D
< 50 F
Apart from my office hours, I am willing to schedule a special "help session" meeting once a week (probably at some late afternoon time) if there is sufficient interest. We can talk about this in class in a week or so. Feel free to remind me. However, if only a few people express interest or actually attend, it would be easier for everybody that I just add an extra office hour instead. If we decide on a time, then I will post the time/location on this web page.