# MAT 150: Honors Mathematics. Introduction

## Stony Brook University - Spring 2016

### Key to math

... Of that kingdom this is the key.
Nursery Rhyme

Mathematicians like the mathematics that most people are not aware about.

The course answers to the question What is the mathematics that mathematicians like?
Students who will also find this mathematics attractive will be helped to enter it.

Each lecture is aimed to surprise and reshape the students' perception of mathematics. The course is organized as a collection of short, important, and self-contained subjects, which are mostly independent from each other. They are picked up from various parts of mathematics: number theory, geometry, set theory, combinatorics, topology, optimization theory, etc. and complement the content of advanced mathematical courses.

No preliminary knowledge of advanced mathematics is required.

The course is targeted towards students of the following three types:

• freshmen who have not yet decided what major to choose
• beginning math majors
• junior and senior math majors
A freshman would be helped to decide whether mathematics deserves her/his attention and efforts, whether to become a mathematics major, or to choose another major subject.

A beginner math major would learn plenty of nice mathematics and would be helped to decide if she/he is ready to join the honors program in mathematics immediately.

The course will provide students with a rich collection of classical mathematics of prime importance, which otherwise may not reach students, because is not included in the main math courses.

### Textbook

The Enjoyment of Mathematics by Hans Rademacher, Otto Toeplitz.

The book is available from Amazon for about $13. Certainly, the course will not follow the book section by section, but the style would be similar. From the introduction to this book: We select a sequence of subjects, each one complete in itself, none requiring more than an hour to read and understand. The subjects are independent, so that one need not remember what has gone before when reading any chapter.'' 1. The Sequence of Prime Numbers 2. Traversing Nets of Curves 3. Some Maximum Problems 4. Incommensurable Segments and Irrational Numbers 5. A Minimum Property of the Pedal Triangle 6. A Second Proof of the Same Minimum Property 7. The Theory of Sets 8. Some Combinatorial Problems 9. On Waring's Problem 10. On Closed Self-Intersecting Curves 11. Is the Factorization of a Number into Prime Factors Unique? 12. The Four-Color Problem 13. The Regular Polyhedrons 14. Pythagorean Numbers and Fermat's Theorem 15. The Theorem of the Arithmetic and Geometric Means 16. The Spanning Circle of a Finite Set of Points 17. Approximating Irrational Numbers by Means of Rational Numbers 18. Producing Rectilinear Motion by Means of Linkages 19. Perfect Numbers 20. Euler's Proof of the Infinitude of the Prime Numbers 21. Fundamental Principles of Maximum Problems 22. The Figure of Greatest Area with a Given Perimeter 23. Periodic Decimal Fractions 24. A Characteristic Property of the Circle 25. Curves of Constant Breadth 26. The Indispensability of the Compass for the Constructions of Elementary Geometry 27. A Property of the Number 30 28. An Improved Inequality Notes and Remarks This is the only textbook which is recommended. The whole book will not be covered, but this is an excellent source for reading. ### Prerequisites • Score of level 4 or better on the math placement exam, • or MAT 123 passed with a grade C or better, • or a permission by the instructor. Priority given to students in the University's honors programs. Notice that the prerequisites do not include any Calculus. High-school algebra suffices. Though the course is not easy: as any mathematics, it relies on proofs. This is why MAT 200: Logic, Language and Proof is an advisory co-requisite. ### Lecturer Oleg Viro Professor, Ph.D. 1974, Doctor Phys-Mat.Sci. 1983, both from Leningrad State University Arrived at Stony Brook in 2007. Office: Math Tower 5-110 Phone: (631) 632-8286 Email: oleg.viro AT math.stonybrook.edu Web page: www.math.stonybrook.edu/~oleg Research fields: Topology and Geometry, especially low-dimensional topology and real algebraic geometry. ### Lectures time and location Mondays and Wednesdays 5:30pm - 6:50pm in Library W4530 ### Grading policy The final grade will be based on • Homeworks 20%, • Midterm 40%, • Final exam 40%. Homeworks will be assigned once a week. A typical homework will consist of two - three problems. In MAT 150, the main purpose of homeworks is to keep students aware about the material of lectures: definitions, examples, statements of theorems, some proofs. More creative challenging problems (e.g., compose a problem or generalize a theory given in a lecture) will be proposed, but rarely and only as non-compulsory bonus problems. The exams, both midterm and final will serve the same purpose: to verify whether the material is familiar. ### Disabilities If you have a physical, psychological, medical, or learning disability that may impact your course work, please contact Disability Support Services or call (631) 632-6748. They will determine with you what accommodations are necessary and appropriate. All information and documentation is confidential. Students who require assistance during emergency evacuation are encouraged to discuss their needs with their professors and Disability Support Services. For procedures and information go to the Evacuation Guide for People with Physical Disabilities. ### Academic Integrity Each student must pursue his or her academic goals honestly and be personally accountable for all submitted work. Representing another person's work as your own is always wrong. Faculty are required to report any suspected instances of academic dishonesty to the Academic Judiciary. For more comprehensive information on academic integrity, including categories of academic dishonesty, please refer to the academic judiciary website. ### Critical Incident Management Stony Brook University expects students to respect the rights, privileges, and property of other people. Faculty are required to report to the Office of Judicial Affairs any disruptive behavior that interrupts their ability to teach, compromises the safety of the learning environment, or inhibits students' ability to learn. ## Lectures ### Lecture 1 #### The sequence of prime numbers The first lecture followed closely the first chapter of the textbook. After the lecture its content was presented in a more formal style. In the Homework 1, a comparison of the two texts was requested. More specifically, the proofs of theorems 1 and 3 had to be done. The statement and proof of theorem which was called Theorem 3 in Lecture 1 were not formally distinguished in the textbook. The proof was not quite clear. The homework required to find them in the textbook, and to answer to the following two questions: What is not said in the proof? What is number$p$in the proof of Theorem 3? and list the differences between the proofs in the lecture and textbook. The number$p$is not defined in the textbook. One can guess what$p$is. Moreover, the numbers in the formula that defines$M$are not specified either. They cannot be the same as in the proof of Theorem 3 in Lecture 1. The textbook is a beautiful classical book. A good mathematical book may, and even should, be somewhere unclear. This gives to a student an opportunity to think, fix the vague piece and improve self-esteem. The third problem (bonus problem) in Homework 1: What sets of primes can you prove to be infinite? • the sets of primes in arithmetic series$4,8,12,16, \dots,4n,\dots$;$1,5,9,\dots,4n+1,\dots$;$6,10,14,18, \dots,4n+2\dots$;$3,8,11,\dots,4n-1,\dots\$?
• the sets of primes with the last digit 3 or 7 (i.e., having the last digit 3 or 7 in the decimal notation)?

### Lecture 2

#### Traversing nets of curves

Lecture notes (incomplete)

### Lecture 3

#### Pythagorean triples

The lecture used a new technique: a pdf computer presentation was used. This is a screen presentation file. It consists of 80 pages. Pages next to each other differ usually just by one sentence. The pdf file was used as a background for xournal. This allowed to write over the screen. Almost all the proofs were inserted during the lecture.