Fundamental Mathematics

Math 347B1 and 347D2, Spring 2018

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Instructor: Prof. Zhong-Jin Ruan

Classroom/Time: Section B1: 143 Henry Bld, MWF 9:00 - 9:50am

Section D2: 141 Altgeld Hall MWF 11:00-11:50am

Office Hour: MW 10:00-11:00am at 353 AH or by appointment.

Email: z-ruan@illinois.edu

Course Web Page: https://faculty.math.illinois.edu/~ruan/347Sp18.html

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

Homework: Homework will be assigned each week and will be due in class on the following days:

Part I: Jan 24, 31, Feb 7 and 14;
Part II: Feb 28, Mar 7, 14 and 28;
Part III: Apr 11, 18, 25 and May 2.

No late homework will be accepted. If you have a reasonable excuse for missing an
assignment, I will score it by taking the average of the other assignments.

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

Exam 1 Monday, February 19
Exam 2 Monday, April 2

Final Exam
Math347B1 Class: 1:30 - 4:30pm on Thursday May 10, 2018
Math347D2 Class: 1:30 - 4:30pm on Friday May 4, 2018

Grading policy:There will be total of 500 points computed as follows.
 Homework 10 x 10 pts 100 pts Exams 2 x 100 pts 200 pts Final Exam 200 pts Total 500 pts

HOMEWORK ASSIGNMENTS
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PART I
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HW #1 Practice Homework: Page 20, 1.15, 1.20, 1.r1, 1.43, 1.49.
Hand-in Homework: Page 20, 1.13, 1.32, 1.36, 1.44a, b, 1.47a, 1.50 (Due Wednesday, January 24, 2018)

HW #2 Practice Homework: Page 44, 2.4, 2.9, 2.21, 2.25, 2.27.
Hand-in Homework: Page 44, 2.22, 2.23, 2.35, 2.38, 2.47 and the following problem:
Let n be an positive integer. Show that q = \sqrt n + \sqrt 2 is an irrarional number. (Due Wednesday, January 31, 2018).

HW #3 Hand-in Homework: Page 71, 3.7, 3.17, 3.26, 3.49b, 3.56b, 3.57. (Due Wednesday, February 7, 2018).

HW #4 Hand-in Homework: Page 95, 4.20a, b, 4.24, 4.25a, b, e, 4.33a, b, 4.34a, c, 4.46, and 4.47. (Due Wednesday, February 14, 2018).

Exam1 will be given on Monday, February 19,2018 at classroom during class time. See solution [pdf]

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PART II Real Analysis
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We would like to recommend the following book and chapters:
1. Textbook: chapter 13 and chapter 14.
2. Introduction to Real Analysis by R. Bartle and D. Sherbert, 4th edition: chapter 2 and chapter 3.

HW #5 [pdf], Solution [pdf],
Note: There is a minor typo in HW5.3. We should read: ..., but the number u + 1/n is an upper bound of S. (Not: n + 1/n).

HW #6 [pdf], Solution [pdf],

HW #7 [pdf], Solution [pdf],

HW #8 [pdf], Solution [pdf],

Exam2 Review Practice [pdf],

Exam2 will be given on Monday, April 2, 2018 at classroom during class time.

Exam 2 Solution: Section B1 Solution [pdf], Section D2 Solution [pdf],

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PART III
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HW#9, Textbook Page 134: 6.8a, 6.8b, 6.9b, 6.9c, 6.17, (6.28 delete this problem), 6.46, 6.47. (Due Wednesday, April 11, 2018). Solution: [pdf],

HW#10, Textbook Page 134: 6.18 (Explain your answer), 6.28, 6.37 a), Page 151: 7.1, and the following problem:
Additional Problem: Show that 5^{1/3} is an irrationall number. (Due Wednesday, April 18, 2018) Solution: [pdf],

HW#11 Textbook Page 151: 7.5, 7.6, 7.9, and the following three additional problems.

HW11 Problem 1: Let f_a be the map on Z_{13} defined by f_a (\bar{x}) = \bar {ax} for all \bar{x} in Z_{13}.
(1) Find the digraph of the map f_a : Z_{13} -> Z_{13} for \bar{a} = \bar{3}, \bar{4}, and \bar {7}, respectively.
(2) Find the order and the inverse of \bar{3}, \bar{4}, and \bar{7} in Z_{13}.

HW11 Problem 2: Let a, b, and n be positive integers. Show that the congruence equation \bar{a} \bar{x} = \bar{b} has an solution if and only if gcd(a, n) | b.

HW11 Problem 3: Consider \bar{8} in Z_{12}.
(1) Find all \bar{b} in Z_{12} such that the congruence equation \bar{8} \bar{x} = \bar{b} has a solution.
(2) Find all solutions for each \bar{b} found in part (1).
(Due Wednesday, April 25, 2018) Solution: [pdf],

HW #12: [pdf] (Due Wednesday May 2, 2018). Solution: [pdf]

Part III Review and Practice Problems: [pdf].

Extra Office Hour for Final Exam:
I will hold an extra Office Hour'' at 141 Altgeld Hall on Thursday (May 3) at 3:00-3:50pm. I will try to return your HW12 during this time, and you can check your scores for this class at that time.

Final Exam:
Math347 Section D2 will take Final Exam at classroom 141 Altgeld Hall on May 4 Friday at 1:30-4:30pm.
Math347 Section B1 will take Final Exam at classroom 143 Henry Bldg on May 10 Thursday, at 1:30-4:30pm.
Notice that:
1.Final Exam is a close book exam. So no book, no lecture notes, and no cell phone during the exam.
2. Bring your student ID with you in case we need it.
3. Please come to exam on time. No make up exam unless you have a strong reason, getting my approval before the exam.