Stony Brook Mathematics Department

Index Announcement Syllabus & Homework 
We will cover material in High School Mathematics: An Advanced Perspective. by Usiskin, Peressini, Marchisotto and Stanley.
Week 1 (Sept 4) Chapter 1. What Is Meant by `An Advanced Perspective'?
Students will take part of the NYS Math III Regents Exam in class on Wednesday Sept 11.
Homework 1 (due Sept 9) p 16 nos 1, 7, 8
Week 2 (Sept 9 & 11) Chapter 2. Real Numbers
omit sections 2.1.3 and 2.1.4 (except for proof of irrationality of e)
Homework 2 (due Sept 17) p 30 nos 5, 6, 12 p 41 nos 1, 4, 7
Week 3 (Sept 17 & 18) Note that Tues Sept 17
follows a Monday schedule. Chapter 2 (end) Complex Numbers; Chapter 3 Functions
Homework 3 (due Sept 23) p 80 nos 1bd, 4, 11 p 98 nos 3, 5 p 103 nos 4, 67ab p 113 nos 1, 2, 7, 8Week 4 (Sept 23 & 25) Chapter 3 Functions (cont.)
Homework 4 (due Oct 2) p 124 nos 1, 4, 7 p 132 nos 1, 5, 8 p 139 nos 2, 6 p 149 nos 1bdfh, 4, 6, 7
Homework 5 (due Oct 9) p 194 nos 12, 15 p 199 no 4 p 204 nos 2, 3 p 217 nos 1, 3, 5
Week 7 (Oct 13 and 15) Monday: Midterm.
Wednesday: Chapter 5. Integers and polynomials, beginning.
Homework 7 (due Oct 22) 1. Prove, using the definition C(n,k) = n!/(k!(nk)!), that C(n+1,k+1) = C(n,k) + C(n,k+1). 2. Use this fact to write a complete proof of the Binomial Theorem: (a+b)^{n} = C(n,0)a^{n} + C(n,1)a^{n1}b + ... + C(n,n)b^{n}. 3. Also use this fact to show how the binomial coefficients C(n,k) can be calculated in a triangle ("Pascal's Triangle") C(0,0) C(1,0) C(1,1) C(2,0) C(2,1) C(2,2) C(3,0) C(3,1) C(3,2) C(3,3) etc. where the entries along the edges are all ones, and where each interior number is the sum of the two directly above it. 4. Prove that the sum of the elements in the nth row of Pascal's Triangle is exactly 2^{n} and that their alternating sum is 0. 5. The Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, ...; each one is the sum of the two preceding ones. Prove that the nth DIAGONAL sum of the entries in Pascal's Triangle is the nth Fibonacci number: E.g. C(0,0) = 1 C(1,1) = 1 C(1,0)+C(2,2)=2 C(2,1)+C(3,3)=3 C(2,0)+C(3,2)+C(4,4)=5 C(3,1)+C(4,3)+C(5,5)=8 C(3,0)+C(4,2)+C(5,4)+C(6,6)=13 Hint: use 1. and induction.
Homework 8 (Due October 29) 1. Explain in your own words the "row reduction" method for finding the g.c.d. (a,b) of integers a and b, and why it works. Specifically, show how going by integral row operations from 1 0 a ( ) 0 1 b to x y d ( ) z w 0 gives d = (a,b) = x a + y b. p q r Hint Show that the entries ( ) in each of the matrices in the process s t u satisfy p a + q b = r and s a + t b = u, by induction.Week 9 (Oct 28 and 30) Primes and Prime Factorization. Base representations.
Homework 8 (due Nov 6) 1. Write out the proof of the statement: if a prime divides the product of two integers, it must divide at least one of the factors. 2. Work out the binary long division 1010101/11111 (341/31) to five binary digits past the "decimal" point, and convert your answer into decimal notation.
No homework assigned for this week.
Week 11 (Nov 11) Chapter 7: Isometries (cont.)
Calculate the coordinates of the image of (x,y) after rotation by 45^{o} about the point (1,3).
Calculate the coordinates of the image of (x,y) after reflection in the line 2x+3y=5.
Homework 10 (due Nov 20) 1. Prove "Jason's Formula:" T_{P} R_{P,theta} = R_{0,theta}T_{P}. Translate this formula into words. 2. Let F: R^{2} > R^{2} be defined by F = R_{(1,3),pi/6} (i.e. rotation by pi/6 about the point (1,3)). Write F(x,y) in terms of x and y. 3. Let F: R^{2} > R^{2} be defined by F = r_{{y=2x5}} (i.e. reflection in the line y=2x5). Write F(x,y) in terms of x and y.