Zvezdelina Entcheva Stankova

Visiting Professor


713 Evans Hall

Department of Mathematics

University of California at Berkeley

Berkeley, CA 94720-3840

Tel: 510-642-3768

Fax: 510-642-8204

Email: stankova@math.berkeley.edu

Office Hours TTh 9:30-10:50am in Evans 713.

Webpage: http://math.berkeley.edu/~stankova

Click below for the syllabus:

Course Syllabus for MATH 55, Fall 2014.

GSIs finalized office hours were revised in the syllabus on 9/6/2014. Check them out and keep them with at all times. You are welcome to visit any GSI's office hours, along with the professor's office hours.

For enrollment questions:

Do NOT contact the instructor or the GSIs. We have no control over enrollment. Contact instead Thomas Brown, 965 Evans.

Email is ONLY for emergencies (e. g., medical and family emergencies).

Email is NOT for resolving enrollment questions, or asking for letters, or for discussion of midterm results, or for any discussion about how the student is doing in the class or how to improve. I have received lately a number of such emails, to all of which the response is to come see me in person in office hours (bringing all necessary documentation with you). You are also welcome to visit any GSI's office hours to discuss math questions or how to improve in the course; the GSIs are very qualified to discuss any math question. This is clearly written in the syllabus and was discussed in detail during the first lecture.

Videos on Game and Geometry Puzzles from the Numberphile Channel, containing many of the proof and problem-solving ideas discussed in class:

1) Three Squares: http://youtu.be/m5evLoL0xwg

2) Free the Clones: https://www.youtube.com/watch?v=lFQGSGsXbXE

Midterm 1 Results:

Solutions to Midterm 1.

Here is some Statistics from Midterm 1. For example, an exam with 79-81 points is in the top 25% of the class. Note that the median is 61/100 points, which indicates that the exam was hard. Undoubtedly, the hardest problem was the True/False question #1, on which there were only two perfect scores. Overall, there was one perfect exam 100/100 and a bunch of 96/100 and 95/100. The range was [8,100].

The GSIs will return the exams on Wed, Oct. 1, in sections. Exams not returned to the GSIs after the 10 minute viewing period will not be considered for regrading. When asking for a problem to be regraded, please, keep in mind that such requests rarely result in a change of the score. The GSIs will review your whole problem and your score may go up or down. Thus, ask for a regrade only if you have a solid reason and only after you read carefully the official solutions. The GSIs will turn down any requests that are not substantiated. The same grading rubrik is applied to all students equally in the class, and it will not be changed for any particular student. Students who miss section on Wed need to pick their exams at a conveninent time for the GSI or in office hours. Any requests for regrading after Monday, Oct. 6, will not be considered.

Homework Assignments and Notes:

HW5 Solutions; HW4 Solutions.

HW Solutions are posted about a day before the quiz and will be taken off the web in a week. Do NOT ask for solutions to be posted earlier: you must attempt to do your homework without help from posted solutions. If you are late copying them, or you lose them, or some other thing happens: do NOT ask us for the files of the previous solution since we do NOT distribute electronic files of the HW solutions. Instead, ask your classmates for the HW solution files. Make sure you download and save the solutions as soon as they are posted, to avoid having to ask your classmates later on for them.

If not specified odd or even exercises, it is assumed only even exercises, e.g., #2-8 means 2,4,6,8.

Homework 5B, due Mon, Sept 30:

Read 4.4 (may skip Computer Arithmetic with Large Integers and Pseudoprimes). Write: #2,4,6(a)(c),8(try an example first),10,12(b),16* (do (a) with brute force if necessary),20,22,32,34,38(a),40,54,55.

Homework 5A, due Mon, Sept 29:

Read 4.3. Write #4,10,12,13,16(b)(d),18(a)(b)*,20(a)(b),24(a)(b),26(a)(b),28,30,32(c),40(f). Extra challenge: #11*,36*-37*.

Hints: In #10: argue by contradiction and assume that there is an odd divisor of m; this odd divisor is denoted by k in the hints; the hint is asking you to show the given factorization and use it to prove that 2^m + 1 will turn out to be composite in this situation; try some small cases for m to get a feeling for what is going on. In #11*: argue by contradiction, get rid of all inconvenient functions to turn the statement only about integers, and then use the prime factorization of these integers. In #12: use the hint; if it is hard at first, try it in an example with a small n, e.g., n=3, 4. In #13: the asterisk * may be “overrated”; try out an example? In #18(b)*: write out ALL divisors of the given number and add them up using the formula for the geometric progression. In #20: memorizing all powers of 2 up to 2^{10} is good.

Read 4.2 (up to p. 249, inclusive). Write #2,4,29,31,32.

Hints: In #31, show first that 10^n is congruent to 1 (mod 3) for any natural number n; then use the decimal expansion of a positive integer a and modular arithmetic mod 3 to show that a is congruent to the sum of its digits (mod 3). Try exactly the same approach for divisibility by 9. In #32: similarly, start by noting that 10 is congruent to -1 (mod 11), then show that 10^n is congruent to 1 (mod 11) when n is even, and to -1 when n is odd; finish by using the decimal expansion of your number a and replacing all powers of 10^n by +1 or -1, respectively. Try all these divisibility criteria on 3- or 4-digit numbers to see how they work out in practice.

Homework 4B, due Wed, Sept 24:

Read 2.4. Write #6(a)(b)(d)(e), #8,10(d), #12(d), #14(c)(f)(g), #16(c)(g), #22, #26(e)(f), #32(a)(d), #34(b)(c), #40.

Read 4.1. Write #6, #8, #10(d)(e), #12(c), #14(d), #18, #28(a)(d), #32(c), #36, #38, #40.

Homework 4A, due Wed, Sept 24:

Read 2.3. Write #2,6(a)(b)(d),12,14(a)(b)(e),20,22(b)(c),26,30(d),40,50,54,64,74(b). Extra challenge: 76.

Read 2.5. Write #2,4,6,8,10,16,18,20(use bijections),24.

Homework 3B, due Wed, Sept 17:

Read 2.1. Write #10,12,16,18,20,22,24,26,32(b)(d),38.

Read 2.2. Write #2,4,12,14,16(d),18(c),24,26(b),30,44.

Homework 3A, due Wed, Sept 17:

Read 1.8. Write #4,6,10,14,18,22,30,32,34,36,42,44.

This HW will be tough for those who are not experienced with problem solving and proofs, which is natural at this level. Hence some hints are listed here. Do your best.

Hints: #4 (arrange a,b,c in increasing order in your cases); #6 (x being odd and y being even is symmetric to another case); #8 (definitely try some small cases first before you find your example); #10 (Could they BOTH be perfect squares? Could (n+1)^2 - n^2=1? Why is this relevant in the problem?) #18 (place r on the real line between two consecutive integers, exactly one of which will be the required unique n in the problem; why?; what could go wrong if we allowed for r to be rational?) #22 (Do it as the hint suggests. And then apply the inequality from Example 14 to x^2 and 1/x^2 (instead of x and y) for a direct proof.) #30 (If they were such integers, how large can possibly be |y|?) #32 (The hint is giving you specific formulas for the integers x, y, and x. Substitute these formulas into the Pythagorean equation to show that they satisfy the equation. Do these formulas give infinitely many triplets of solutions? Why?) #34 (Adapt the proof that square root of 2 is irrational.) #36 (What kind of a number lies exactly in the middle between a rational and an irrational number? Use the average to get a formula for this middle number and show why it must be irrational by contradiction.) #42 (Direct tiling should do it. Try it.) #44 (What happens if you color the 5 x 5 board as a regular chessboard? Try finding an invariant.)

Homework 2B, due Wed, Sept 10:

Read 1.6. Write #2,4,8,12,16,18,24,28. Challenge: #35*.

Read 1.7. Write #6,8*,10,12,18,20,24,26,30,34.

Homework 2A, due Wed, Sept 10:

Read 1.3. Write #6,8,10(b)(c),14,24,30,32,60,62(a).

Read 1.4. Write #6,10,16,20,32,50,60.

Read 1.5. Write #6,14,20,24,30,40,46.

Note 1: The first quiz is on Wed, Sept. 3 and will be on the material from HW1. For the quizzes, you are allowed to have a cheat sheet containing material related to course. The cheat sheet can be only 1 page (one-sided!) of a regular 8”x11” sheet. It has to be hand-written by you (no zeroxing, copying, pasting, etc.)!

Homework 1, due Wed, Sept 3:

Read 1.1. Write #2,4,8,12,14,18,28,30,34,38. Extra Challenge: #40, 49.

Read 1.2. Write #2,12,16,24,28,40,42. Extra Challenge: #36, 38.

Note 1: The implication "p --> q" can be read in a number of ways. One of the most confusing ways to read it is "p only if q", which means "if p is true then q must be true", i.e., "if p then q". I would suggest avoiding, whenever possible, the expression "only if" as it is doubly dangerous. Indeed, in everyday life it is used to mean "if and only if", while in math it is used only in one direction and it is often counterintuitive. For example, Problem #3 from 1.1 can be translated mathematically as: "If p then [q1 AND (not q2) AND (not q3)].", where p: "You graduate.", q1: "You fulfilled all requirements for your major.", q2: "You owe money to the university.", q3: "You owe a library book."

Note 2: The first quiz is on Wed, Sept. 3.