Professor:  Lauren Rose, Albee 305, rose@bard.edu

Class schedule:  T/Th 11:50 - 1:10, Hegeman 204

Textbook: Proofs and Fundamentals, by Professor Ethan Bloch, 2nd Edition.

Course Requirements: There will be a midterm and a final exam. Your grade will be determined roughly by homework(30%), midterm(25%), final exam(25%), and a final project, attendance and class participation(20%).

Homework: Homework will be discussed in class, and several proofs will be handed in each week.  You will also be expected to do homework problems at the board.

Attendance:  Attendance is required.  More than 2 absences will affect your grade.

Most Important: READ THE BOOK!  If you’ve never read a math book before, this is the time to start.  I will expect you to have read the relevant sections BEFORE I discuss them in class.

This course provides a bridge between the calculus sequence and the more abstract mathematics in upper level courses. You will learn about the language of mathematics, the structure of mathematical proof, and how to communicate mathematics to others.  We will cover topics that arise in most branches of mathematics, such as sets, functions, relations, and cardinality.

TENTATIVE SCHEDULE:  subject to change

Assignments: bold = hand in, (purple) = challenge problems,  grey = tentative.)

Project Topics for Math 261

Project Guidelines: 3 people per group, 1 group can have 2 people.

1) Class presentation (20-25 min)  introduction, background, motivation, etc.

2) Proof write up (1 per group): brief introduction, cite sources including websites.

1. R:  Which natural numbers are sums of two or more consecutive natural numbers? For example, 7 = 3+4, 10 = 1+2+3+4.
2. R:  How many regions are determined by n lines in a plane?  For example, two non-parallel lines determine 4 regions.  Assuming no two lines are parallel and no point lies on more than two lines, find a formula for the number of regions and prove it works.  If you have time, consider the general case.
3. R:  Let a, b be relatively prime (no common factors) natural numbers.  What is the largest n that can’t be written as ax+by for any x and y?  
4. RB: Which triples of natural numbers (a, b, c) are Pythagorean triples? Find the formula and prove that it works.
5. RB: Find and prove the relationship between V, E, and R for a planar graph, where V =  the number of vertices,  E = the number of edges, and R = the number of regions determined by the graph.
6. RB: Which graphs can you draw in a continuous manner, without lifting your pencil or going over any edge twice? This comes from the Konigsberg Bridge Problem
7. B: Prove Fermat's Last Theorem for n = 4.
8. B:  Prove Fermat's Little Theorem, about powers modulo a prime p.
9. B: Prove that ISBN-13 numbers detect single digit errors and transposition errors.
10. B: Prove the Euler/Euclid Perfect Number Thm about perfect numbers and Mersenne primes.

Note: Other topics are possible with permission from the instructor.

Mathematical Theorems:  A theorem is a statement that has been proven to be true.

1.          The Pythagorean Theorem

2.          The Fundamental Theorem of Algebra

3.          The Fundamental Theorem of Arithmetic

4.          The Fundamental Theorem of Calculus

5.           The Quadratic Formula

6.           Fermat’s Last Theorem

7.           The Four Color Theorem

8.           The Product Rule from Calculus

9.          The Intermediate Value Theorem

Mathematical Statements:  A statement can be either true or false, although it is not always easy to tell its truth value.  What are the truth values of the following statements?    

1.          sin2x + cos2x = 1, for all real numbers x.

2.          The sum of two even numbers is even.

3.          The sum of two odd numbers is odd.

4.          The square root of 2 is irrational.

5.          The Euler constant e is rational.

6.          There are finitely many prime numbers.

7.          A number is divisible by 8 if and only if the sum of its digits is also divisible by 8.

8.          A number is divisible by 9 if and only if the sum of its digits is also divisible by 9.

9.          If x and y are rational, so is x+y.

10.  If x and y are irrational, so is x+y.

11.  Is p is prime, so is 2p – 1.

12.  x2+ y2 = z2  has infinitely many integer solutions.

13.  x3+ y3 = z3  has infinitely many integer solutions.

14.  Every polynomial with real coefficients has a root.

15.  If n is divisible by 3, so is n2.

16.  If n2 is divisible by 3, so is n.

17.  x2 > x, for all real numbers x.

18.   Every prime number can be written as the sum of two squares.

19.   Every even number can be written as the sum of two prime numbers.

20.  There are infinitely many pairs of consecutive odd prime numbers.

Extras

Self-Referential Multiple Choice Quiz