CATALOG DESCRIPTION: Foundations of mathematics including logic, set theory, combinatorics, and number theory. Prerequisite: MATH172.
Instructor: Dr. Eric Rowell. email: rowell@math.tamu.edu, webpage: www.math.tamu.edu/~rowell. Office Milner 210.
Place and Time: This course meets TR 9:35am-10:50am in
BLTN 018.
Course Objectives: Understand and communicate in the language of higher mathematics. This course is designed to provide a foundation for further study of mathematics beyond calculus. A major part of this foundation is learning to write proofs, and will be the main objective of the course.
Text: Foundations of Higher Mathematics (3rd Ed.) by Fletcher and Patty.
Material Covered: Chapters 1-6, and parts of chapters 7 and 8, time permitting. Additional topics may be added.
Grading: Your grade will be based upon 2 Midterms (20pts each), Homework (20pts), a Writing Project (15pts), a Journal (10pts), and a Final Exam (25pts). Notice these add up to 110 points. The grading scale is the usual one: 90-100% A, 80-89 % B, 70-79% C, 60-69 % D, 0-59% F. The final exam is Friday, May 4 12:30pm-2:30pm. Midterm I will be Thursday, February 15, 2007. Midterm II March 27, 2007.
Writing Project: The writing project will be a research paper of 5-10 pages on a topic in mathematics, for example a famous theorem or conjecture, a branch of mathematics etc. This will be turned in towards the end of the term. The topic must be approved by Dr. Rowell. More information will be given later.
Journals: Occasionally there will be questions that come up in class that we do not have time to pursue. These will be assigned as journal entries. You will be asked to think about these problems and articulate your ideas in your journal. In addition you will be asked to write a description of your chosen topic for the writing project at some point. The journal will be turned in twice during the term for grading. The emphasis will be on effort and clarity rather than correctness of answers. The journal is not to be used for anything but the specific topics assigned as journal entries, and should be written legibly, using complete sentences and following the rules of grammar, punctuation etc.
Homework: Weekly homework sets will be turned in to be graded. As writing proofs takes considerable practice, the homework sets will likely be quite time-consuming. Your lowest homework score will be dropped. Late homeworks count as a 0, and may or may not be graded.
Course Policies: Late homework and make-ups for missed exams will only be allowed for a university approved excuse in writing. Wherever possible, students should inform the instructor before an exam is missed. Consistent with University Student Rules, students are required to notify an instructor by the end of the next working day after missing an exam. Otherwise, they forfeit the right to a make-up.
An Aggie does not lie, cheat, or steal or tolerate those who do. Copying work that was done by others is an act of scholastic dishonesty and any instance of it will be prosecuted according to University Student Rules.
Students with disabilities can get assistance from the Office of Services for Students with Disabilities, Tel. 845-1637.
Copyright Policy: All printed materials disseminated in class or on the web are protected by Copyright laws. One photocopy (or printout from the web) is allowed for personal use. Multiple copies or sale of any of these materials is strictly prohibited.
Important Announcements:
Exam I will be Thursday, February 15, 2007
Journal I due Tuesday, March 6, 2007
E. Rowell's office hours cancelled the week of Feb. 25. S. Witherspoon will hold her regular office hours.
Midterm II will be Tuesday March 27, 2007.
Writing project due: Tuesday April 24th
Final exam: Friday May 4th, 12:30pm
Three (3) announcements:
1) 5 points extra credit will be added
to your final exam score (which is out
of 150) if you attend and write 1/2 page
summary of Rob Benedetto's lecture
TUESDAY NIGHT. Here are the details:
Math Awareness Month Lecture
Speaker: Rob Benedetto
Amherst College
Title: The abc Conjecture: An Introduction
Date: Tuesday, April 24
Time: 7:30-8:30 PM
Location: Blocker 158
Rob Benedetto is a number theorist who works in arithmetic dynamics. The
abc conjecture is a famous open problem in elementary number theory with
many notable consequences, including Fermat's Last Theorem.
2) I will have extra office hours
Tuesday May 1st 1-3:30pm.
3) Officially, your journal II is due
Thursday April 26th. However, there will be no penalty if you turn them in
Tuesday May 1st by 3:30pm (during my finals week office hours).
Homework Assignments:
Homework Assignment 1: due Thursday, 1/25 11am.
1.1: 3(a)(b)(c)(d)(g)(h), 7
1.2: 15(a), 25(b), 27, 28(a), 30, 31, 36, 37, 39
Homework Assignment 2: due Thursday, 2/1 11am.
1.3: 41(a)(b)(d)(g), 42, 45, 47, 49
1.4: 57, 59, 64, 71
1.5: 77, 79, 86, 89, 90
Homework Assignment 3: due Thursday 2/8 11am
1.6: 92, 95
2.1: 3,5,7,10,13
2.2: 22, 25, 29(b),(c),(e), 39, 41, 47.
Homework Assignment 4: due Thursday 2/15 11:00am
2.2: 35, 40, 49.
2.3: 54, 56, 57, 59, 62, 71.
Homework Assignment 5: due Thursday, 3/1 11:00am
3.1: 1(a),(g),(n),(q), 2, 6, 15
3.2: 17, 18, 21, 22, 35, 37(a)
Homework Assignment 6: due Thursday, 3/8 11:00am
3.3: 42, 45, 48, 54, 55, 56, 58, 60, 61, 65(a), 67
3.4: 82, 83, 92, 93.
Homework Assignment 7: due Thursday, 3/22 11:00am
3.5: 102, 103, 104, 106, 108, 109, 110, 116, 117, 118(c),(d)
Homework Assignment 8: due Thursday, 3/29 11:00am
4.1: 4(a)(d)(f), 5(a)(d)(f), 9, 13
4.2: 19, 20, 21, 25.
Homework Assignment 9: due Thursday, 4/5 11:00am
4.3: 26,31,34,36,39,43
4.4: 50,51,54(a),58,59
Homework Assignment 10: due Thursday 4/19 11:00am
4.5: 66(a,b,c),70,73,75,77a,80,84
Chapter 5: 1,6,7(a,d),32-34,38
Homework Assignment 11: due Thursday 4/26 11:00am
6.1: 3, 6, 7, 9, 11, 12, 14, 19, 20
6.2: 29, 33
6.3: 35, 36, 37, 41, 46, 48, 52, 54
Journal Topics:
#1. Finish the proof that a^3+b^3=c^3 has no solutions for primes a,b and c. Start from the assumption b=2, so that 8=c^3-a^3=(c-a)(c^2+ac+a^2).
#2. Read and respond to the controversy over Perlman's proof of the Poincare Conjecture and Dr. Yau's involvement.
Some references: The New Yorker Article
An NPR interview (audio)
Feel free to use other references, but be sure to cite them if you do.
Use as much space as you need to explain the controversy and express your opinion.
#3. Explain Russell’s Paradox in plain English so that anyone can understand it.
#4. Outline or describe your writing project
paper. This should be at least 1
page.
#6. Suppose you live on a the surface of a donut. You have 3 houses needing gas, water and electricity. There is one source for each utility. Can you connect up each utility to each house without having the lines cross? It is known that you cannot do this on the sphere (or plane).
Journal II Topics
#1 Let F be the operation "add one" and G be the operation "negative reciprocate" in Conway's rope arithmetic game. Show that (FGFGFG)=identity in two ways:
a) usual arithmetic: let t be a fraction, F(t)=t+1, and G(t)=-1/t.
b) with diagrams: let t be any tangle, and show that (FGFGFG) does not change the tangle (you might try it with some jump-ropes to see how it works, and you may have to give the ropes a tug to get them to see that they are the same.)
#2 Show that any 4 digit integer a=a1a2a3a4 is divisible by 11 if and only if the alternating sum of its digits is divisible by 11: by showing a=(a1-a2+a3-a4)(mod 11).
#3 With Conway's rope arithmetic game, suppose you are only allowed to make a total of 4 moves. Write down all of the numbers that you can get in 4 moves or fewer starting at 0 excluding infinity. (For example FFGF(0)=FFG(1)=FF(-1)=F(0)=1 is one possibility). Now using your list, which fractions can you get in 5 or fewer moves? Can you make a conjecture about how many different numbers you can get after at most n moves? (Hints: with 0 moves there is one number: {0}, with 1 move there are 2: {0,1}, with 2 moves there are 4: {-1,0,1,2}. Be careful, for example -4/23 can be obtained after just 12 moves! It might help to write a computer program to compute a few more. The best you can hope for is a recursive formula, like the Fibonacci sequence.)