Fundamental Mathematics

Math 347 F1, Fall 2010

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Instructor: Prof. Zhong-Jin Ruan (ruan@math.uiuc.edu)

Classroom: 142 Henry; MWF 2:00-2:50pm

Office Hour: MW 1-1:50pm, or by appointment.

Office: 353 Altgeld Hall

Email: ruan@math.uiuc.edu

Web page: http://www.math.uiuc.edu/~ruan/347F1.html

Textbook: Mathematical Thinking: Problem-Solving and Proofs byD'Angelo and West. 2nd edition.

Homework: A homework assignment will be due in class on the following days:
Aug. 30, Sept 8, 15, 22;
Oct. 6, 13, 20, 27;
Nov. 10, 17, Dec. 1 and 8.
No late homework will be accepted for any reason. If you have a reasonable excuse for missing an
assignment, I will score it by the average of the other assignments.


Exams:There will be two 50-minute exams and a 3-hour final exam.

Exam 1 Monday September 27 (estimated time)
Exam 2 Monday November 1 (estimated time)
Final Exam 1:30-4:30 pm, Friday December 17, 2010.

Grading policy:There will be total of 500 points computed as follows.
Homework10 x 10 pts100 pts
Exams2 x 100 pts200 pts
Final Exam 200 pts
Total 500 pts

Your final grade will be based on the total scores.


HOMEWORK ASSIGNMENTS
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HW #1. Practice Homework: Page 20, 1.7, 1.15, 1.20, 1.30, 1.41, .149a, 1.49c.
Hand-in Homework: Page 20, 1.13, 1.14, 1.32, 1.44a, 1.47a, 1.50. (Due Monday, Aug. 30)
HW #2. Practice Homework: Page 44, 2.4, 2.9, 2.22, 2.25, 2.27.
Hand-in Homework: Page 2.23, 2.26, 2.35, 2.38a, 2.47, 2.49 and the following problem:
Use the contradiction method to show that ``there are no non-zero rational numbers r and s such that (sqrt 3) = r + s (sqrt 2)'', where `sqrt 3' denotes `the square root of 3'. (Due Wednesday, Sept. 8.)
HW#3. Hand-in Homework: Page 71, 3.15, 3.26, 3.49b), 3.56 a) and b), 3.57. (Due Wednesday, Sept 15.)
HW#4. Hand-in Homework: Page 95, 4.20 a), b), 4.24, 4.25 a), b), c), 4.33 a), 4.34 a), c), 4.36, 4.47. (Due Wednesday, Sept 22).

1st Exam: Monday September 27, 2010, 2:00-2:50pm at classroom.

HW#5, Book, 13.22a), b); Handout material: Page 38, #4, #6, #10, Page 43, #2, #7, #13, #18. (Due Wednesday October 6).
HW#6, Handout material: Page 43, #5, Page 50, #3, #7, #13a, and the following problem:
Carefully prove that sup(S) = 2 if S ={x rational such that 0< x <2}. (Due Wednesday October 13).
HW#7, Handout material: Page P59, #5d, #6d, #10, P67, #7, and the following problem:
Use definition to show that x_n = (-2)^n + 1/n does not converge to 1. (Due Wednesday, October 20).
HW#8. (1) Use definition to show that lim(3n+7)/(6n-5) = 1/2;
(2) Let (a_n) be a sequence of real numbers. Show that lim a_n = 0 if and only if lim |a_n| = 0;
Handout material: Page 74, #1, #2 (with 1< x_1 < 2); Page 80, #4, #9. (Due Wednesday, October 27)..

2nd Exam: Monday November 1, 2010, 2:00-2:50pm at classroom.

HW#9. Book, Page 134: #6.8 a) & b), #6.9 b) & c), #6.17, #6.28, #6.46, #6.47, and Extra Credit: #6.37* (Due Wednesday, November 10).
HW#10 Book, Page 134: 6.18, Page 151, 7.5, 7.6, 7.9, and the following problems:
(1) Prove that 2^{1/3} is an irrational number,
(2) Let n be a positive integer greater than or equal to 2. If any prime p <= \sqrt n does not devide n, show that n must be a prime number.
HW#11 Book, Page 151, 7.1, 7.32, 7.34, 7.41, and the following problems
(1) Let a = 3, 5, 8, and 10. Find the inverse of a mod 13, i.e. find x such that ax = 1 in Z_13.
(2) Consider the linear congruence equations 3x =2 in Z_9, and 3x=6 in Z_9. Determine whether we have any solution. If yes, find all solutions.
(Due Monday December 6, 2010)

Final Review: Wednesday, December 15, noon-1pm, 142 Henry.

Final Exam: Friday, December 17, 1:30-4:30pm, 142 Henry