**Fundamental Mathematics**

_______________________________________

**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.

**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 |

Your final grade will be based on the total scores.

**HOMEWORK ASSIGNMENTS **
** ********************************************************* **
** PART I**

******
** HW #1 ** Practice Homework: Page 20, 1.15, 1.20, 1.41, 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**

******

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.

** ***************************************
PART III
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Additional Problem: Show that 5^{1/3} is an irrationall number.

Problem 1: Let a, b, and n be positive integers. Show that the congruence linear equation \bar{a} \bar{x} = \bar{b} has an solution if and only if gcd(a, n) | b.

Problem 2: Find the digraph of the map f_3: Z_{13} -> Z_{13} and find the inverse of \bar{3} in Z_{13}.