Math 261: Proofs and Fundamentals
Course Information: Fall 2009

 

Professor Lauren Rose

Albee 305, ext. 7362

rose@bard.edu

http://math.bard.edu/rose

 

Class schedule: MW 1:30-3:50, Hegeman 106

Office Hours: MW3, T1:30, and by appointment.

Tutoring:  The Math Study Room is open Sunday through Wednesday 7-10pm in RKC 111.  Go there!

Textbook:  Draft of Proofs and Fundamentals, 2nd edition, by Bard’s own Professor Ethan Bloch.  We will cover most of Chapters 1-6, and other topics time permitting.  Please purchase the text from the bookstore.  DO NOT USE LAST YEAR's TEXT.


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.

 

Course Requirements:

  1. Daily or weekly homework assignments (Approximately 40% of your grade)
  2. Midterm and final exams (Approximately 40% of your grade)
  3. A project consisting of a 20 minute class presentation and a 1-3 page write-up. (Approximately 15% of your grade)
  4. Attendance and class participation (Approximately 5% of your grade)

 

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. 

 

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.

__________________________________________


Suggested Project Topics for Math 261


R = research project.  Discover the theorem, come up with a proof, and then present it clearly to the class, giving examples, etc.
B = book project:  I'll give you a book where you can read the proof and then present it to the class, giving examples, motivation, etc.
RB = both, meaning you can discover and prove the theorem on your own, but it is a standard result that can also be found in a book. 

  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:  Consider n lines in the plane.  How many regions are determined by these lines?  For example, two non-parallel lines will determine 4 regions. You should start by assuming that no two lines are parallel, and no point lies on more than two lines, and try to find a formula for the number of regions.  If you have time, you can consider the general case.
  3. R:  Let a and b be natural numbers that are relatively prime.  What is the largest natural number n such that n cannot be written as ax+by for any integers x and y?  (This is the chicken nuggets problem where nuggets come in boxes of size a or b)
  4. RB: Which triples of natural numbers (a, b, c) are Pythagorean triples (meaning a^2 + b^2 = c^2)?
  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 numbers detect single digit errors and transposition errors.
  10. B: Prove the Euler/Euclid Perfect Number Theorem on the correspondence between perfect numbers and Mersenne primes.
  11. RB:  Which prime numbers can be written as the sum of 2 squares, i.e. 5 = 1^2 + 2^2, but 3 is not the sum of 2 squares.



THE SCHEDULE BELOW IS IN PROGRESS. 

Assignments: Homework will be assigned daily and several proofs will be graded weekly. Problems in red* are to be handed in.


Back home

Week

Read

Date   

Monday

Date

Wednesday

1

1.2- 3

8/31

1.2: 1-15 (odd parts)

9/2

 1.3: 1-5, 9-12 (odd parts), 6 (pick one), 7(pick one)

2

1.4- 5

9/7

1.4: 1, 2 (odd parts)

9/9

Assignment #1

1.5: 2odd, 4odd, 6odd, 8*

3

2.1- 4

9/14

2.1.1, 2.2.6, 2.2.7
Be prepared to present at the board Wed.

9/16

Assignment #2

2.3.3*, 2.3.4*, 2.4.4* Due next Mon 5pm.
(2.3.7, 2.3.8 Challenge)

4

2.5- 6

9/21

2.4.2, 2.5.4, 2.5.7

Be prepared to present Wed.

9/23

 Assignment #3
2.4.6*, 2.5.6* Hand in next Tues by 6pm
(2.4.3 Challenge) 

5

3.2, 3.3

9/28

Video: The Proof

For class this Wed:
3.2: 3, 5, 8, 12, 15

9/30

Assignment #4

For class next Monday: 3.3: 2odds, 3odds, 4

Hand in next Wed 4pm:

3.3: 5*, 15*, 16*, paragraph about film:


6

3.4, 4.1

10/5



10/7

Assignment #5
For class next Wed: 4.1: 1 - 5
Hand in next Thurs 4pm: 3.2.14*, 3.3.18*

7

4.2

10/12

 No class:
Fall break

10/14
Assignment #6
For Class Monday: 4.2.1, 4.2.3, 4.2.4
Hand in next Wed 4pm:  4.2.10*, 4.2.12*

8

4.3

10/19

10/21
Midterm Given out, due Wednesday at the beginning of class, NO EXCEPTIONS.
9
Midterm
10/26
Work on Midterm Due Wed at the beginning of class.
10/28
Assignment #7
  • For Monday:  4.3: 3, 4
  • For Wednesday, be prepared to present at the board: 4.4: 1,2,4
  • Hand in next Friday: 4.3.6*, 4.4.7*

10

4.4

11/2



11/4

Assignment #8

For Monday: 5.1: 1-4 (odd parts), 6

For Wednesday, 5.3: 1-3, 9,10 (odd parts)


11

5.1, 2

11/9

11/11

Assignment #9

For Monday 5.2:1, 2, 4, 5

Hand in Wed 5.2.6*, 5.2.7*

(Note: There is no class next Wed)


12

5.3

11/16

11/18

 Advising day, no class
For Monday:
Read Section 6.3 and Try 6.3.1(1)


13

Ch 6.3

11/23

 Assignment #10

Hand in Wed 6.3.1.(6)*, 6.3.8*

11/25

 No Class (Thanksgiving Break)
For Class Monday: 6.3.5, and Read Section 6.5
Challenge: Chicken McNuggets problem

14

Ch 6

11/30

Assignment #10

For Class Wed

6.5 #6

Hand in Wed 6.5.10*, 6.5.11*

12/2

For class Monday: 6.7.1

Challenge: 6.5.7

15

Ch 6, (4.5)

12/7

 Ch 6

12/9

Extra/Presentations

16


12/14

Presentations

12/16

Final Exam Due Thursday

Presentations







 






 


 


 

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.      sin 2x + 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.