# MATH : HONORS CALCULUS-3 HOMEWORK 6: SOLUTIONS

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 MATH : HONORS CALCULUS-3 HOMEWORK 6: SOLUTIONS 25-1 Find the absolute value and argument(s) of each of the following. (ii) (3 + 4i) 1 (iv) i (ii) Put z = 3 + 4i. From z 1 z = 1, we have z 1 z = 1 and arg(z 1 ) + arg(z) = 0: z 1 = 1 z = = 1 5. arg(z 1 ) = arg(z) = arctan 4 3 (iv) Let ω be the real seventh root of 3 + 4i. From ω 7 = 3 + 4i, we have ω 7 = 5 and 7 arg(ω) = tan : ω = 7 5, arg(ω) = 1 7 arctan 4 3. There are seven 7th roots of 3 + 4i, each separated by θ = 2π/7, so for = 0, 1,..., 6. arg(z) = 1 7 arctan π Describe the set of all complex numbers z such that (ii) z = z 1 (iv) z a + z b = c (ii) Note that z 0, so z = z 1 if and only if zz = z 1 z if and only if z = 1. Hence the set is the complex unit circle. (iv) If c / R 0, there is no solution. If b a > c, then c < b a b z + z a, so there is no solution. If c = b a, then the set is the line segment between a and b. If c = 0, there is a solution if and only if a = b, in which case, the only solution is z = a. Now assume b a < c. The curve formed by the set is such that the sum of the distances to a and to b is constant for every point on the curve, so by definition the curve is an ellipse with focal points a and b Prove that if a 0,..., a n 1 are real and a + bi (for a and b real) satisfies the equation z n + a n 1 z n a 0 = 0, then a bi also satisfies this equation. (Thus the nonreal roots of such an equation always occur in pairs, and the number of such roots is even). Conclude that z n + a n 1 z n a 0 is divisible by z 2 2az + (a 2 + b 2 ). 1

2 Let σ : C C be defined by x + iy x iy. Put α = a + ib and β = x + iy, and note that σ(α + β) = σ((a + x) + i(b + y)) = (a + x) i(b + y) = (a ib) + (x iy) = σ(α) + σ(β), and σ(αβ) = σ((ax by) + i(bx + ay)) = (ax by) i(bx + ay) = (a ib)(x iy) = σ(α)σ(β). It follows that if z = a + bi satisfies z n + a n 1 z n a 0 = 0, then 0 = σ(0) = σ(z n + a n 1 z n a 0 ) = σ(z n ) + σ(a n 1 z n 1 ) + + σ(a 0 ) = σ(z) n + a n 1 σ(z) n a 0, as desired. The polynomial has roots a + bi and a bi and hence has factor (z (a + bi))(z + (a bi)) = z 2 2az + (a 2 + b 2 ) Prove that if ω is an nth root of 1, then so is ω. A number ω is called a primitive nth root of 1 if {1, ω, ω 2,..., ω n 1 } is the set of all nth roots of 1. How many primitive nth roots of 1 are there for n = 3, 4, 5, 9? Let ω be an nth root of 1, with ω 1. Prove that n 1 =0 ω = 0. If ω n = 1, then (ω ) n = (ω n ) = 1 = 1. Let ζ n = cos 2π n + i sin 2π n ; the n roots of unity are {1, ζ n,..., ζn n 1 }. We prove that ζn is a primitive nth root of 1 if and only if gcd(, n) = 1. Note that a root of unity appears in {1, ζ n, ζ 2 n, ζ 3 n,..., ζ (n 1) } twice if and only if m 1 m 2 (mod n) for some 0 m 1 < m 2 n 1. That is, if and only if (m 1 m 2 ) n for some 0 m 1 < m 2 n 1. This is equivalent to m n for some 1 < m < n 1. Since m < 1, this is only possible if and only if and n share some factor. This completes the proof. Let ϕ(n) be the number of positive integers less than n that are co-prime to n. Then ϕ(3) = 2, ϕ(4) = 3, ϕ(5) = 4, and ϕ(9) = 6. Z Z 2

3 Using the geometric sequence formula (since ω 1), n 1 ω = ωn 1 ω 1 = 1 1 ω 1 = 0. = Prove that if z 1,..., z lie on one side of some straight line through 0, then z z 0. Show further that z 1 1,..., z 1 + z 1 0. all lie on one side of a straight line through 0, so that z If the line is the real axis, then either the imaginary parts of the z i are all positive or all negative, so that the sum of the z i also has the property. In particular, z z 0. Now suppose the line is not the real axis, and let θ be the angle between the positive real line and the ray extending into the imaginary part of the plane. Let ω be the complex number with modulus 1 and argument θ. Then {z 1 ω 1,..., z n ω 1 } is a subset of either the upper half or the lower half of the plane. By the previous argument we are done. If the line is the real axis, then I(z i ) and I(z 1 i ) have opposite signs, so in particular, the imaginary part of the sum is not zero. If the line is not the real line, then one can proceed as in part by noting that the inverses all lie on one side of the reflection of the line across the real line Prove that if a 0,..., a n 1 are any complex numbers, then there are complex numbers z 1,..., z n (not necessarily distinct) such that z n + a n 1 z n a 0 = n (z z i ). Prove that if a 0,..., a n 1 are real, then z n + a n 1 z n a 0 can be written as the product of linear factors z + a and quadratic factors z 2 + az + b all of whose coefficients are real. We induct on n. For n = 1 the statement is clear. Suppose it were true for some 1, and let a 0,..., a be any complex numbers. By the Fundamental Theorem of Algebra, the polynomial z +1 + a z + + a 0 has a root, say z +1. But then z +1 + a z + + a 0 z z +1 is a monic polynomial of degree. By the inductive hypothesis, for some z 1,..., z. It follows that as desired. z +1 + a z + + a 0 z z +1 = (z z i ) +1 z +1 + a z + + a 0 = (z z i ), 3

4 By part, the polynomial can be written as n (z z i). If any of the z i have non-zero imaginary part, then Problem 25-7 implies z i {z 1,..., z n }; consequently, the corresponding linear factors can be multiplied to produce a quadratic factor with real coefficients, as desired Let A be a set of complex numbers. A number z is called, as in the real case, a limit point of the set A if for every (real) ε > 0, there is a point a in A with z a < ε but z a. Prove the two-dimensional version of the Bolzano-Weierstrass Theorem: If A is an infinite subset of [a, b] [c, d], then A has a limit point in [a, b] [c, d]. Prove that a continuous (complex-valued) function on [a, b] [c, d] is bounded on [a, b] [c, d]. Prove that if f is a real-valued continuous function on [a, b] [c, d], then f taes on a maximum and minimum value on [a, b] [c, d] Let A 1 = [a, b] [c, d]. For n 1, let A 2n denote either the closed left or closed right half of A 2n 1 that contains infinitely many points, and let A 2n+1 denote either the closed upper half or closed lower half of A 2n that contains infinitely many points. For each positive integer n, let z n be some point in A n A. Then {x n } is Cauchy: for any ε > 0 there is a large N such that the points {z n } n>n = {(x n, y n )} n>n are contained in a rectangle with length and width less than ε. Hence z n z for some z [a, b] [c, d]; by construction, z is a limit point of A. If f is not bounded above, then for each n N there exists a point x n such that f(x n ) > n; the x n may be chosen so that { f(x n ) } n=1 is non-decreasing. By Bolzano-Weierstrass, there is a subsequence {x n } =1 converging to some x [a, b] [c, d], and by continuity, f(x n ) f(x). But { f(x n ) } n=1 has no convergent subsequence by construction, a contradiction. By part, the range of f is non-empty and bounded above; in particular, sup f exists. Call it M. Now for each n N, there exists a point z n [a, b] [c, d] such that M 1/n f(z n ) M. By the Squeeze Theorem, f(z n ) M. The z n are bounded, and by part, contain a convergent subsequence z n z; by continuity, f(z) = f( lim z n ) = lim f(z n ) = M. Hence the maximum is attained. Now let g = f; the minimum of f is the maximum of g, which is attained by the previous argument Let f(z) = (z z 1 ) m1 (z z ) m for m 1,..., m > 0. Show that f (z) = (z z 1 ) m1 (z z ) m m α(z z α ) 1. Let g(z) = m α(z z α ) 1. Show that if g(z) = 0, then z 1,..., z cannot all lie on the same side of a straight line through z. A subset K of the plane is convex if K contains the line segment joining any two points in it. For any set A, there is a smallest convex set containing it, which is called the convex hull of A; if a point P is not in the convex hull of A, then all of A is contained on one side of some straight line through P. Using this information, prove that the roots of f (z) = 0 lie within the convex hull of the set {z 1,..., z }. 4

5 By the product rule, f [ (z) = (z z1 ) m1 m α (z z α ) mα 1 (z z ) m ] = (z z 1 ) m1 (z z ) m m α (z z α ) 1 If z 1,..., z lied on the same side of a straight line through z, then z z 1,..., z z lie on the same side of a straight line through 0. By the contrapositive of Problem 25-11, if z 1,..., z lie on one side of some straight line through 0 and z z 1 = 0, then z1 1,..., z 1 all do not lie on the same side of a straight line through 0. By 25-7, (z z α ) 1 0. Since the m α > 0, we have a contradiction. m α (z z α ) 1 0, Let A be the convex hull of {z 1,..., z }, and note that the z i are in A. If a root z of g was not in the convex hull, then all of A is contained on one side of some straight line through z; this contradicts the result in. 5

### 2 Complex Functions and the Cauchy-Riemann Equations

2 Complex Functions and the Cauchy-Riemann Equations 2.1 Complex functions In one-variable calculus, we study functions f(x) of a real variable x. Likewise, in complex analysis, we study functions f(z)

### Class XI Chapter 5 Complex Numbers and Quadratic Equations Maths. Exercise 5.1. Page 1 of 34

Question 1: Exercise 5.1 Express the given complex number in the form a + ib: Question 2: Express the given complex number in the form a + ib: i 9 + i 19 Question 3: Express the given complex number in

### Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis

Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >

### 1 Review of complex numbers

1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely

### COMPLEX NUMBERS. a bi c di a c b d i. a bi c di a c b d i For instance, 1 i 4 7i 1 4 1 7 i 5 6i

COMPLEX NUMBERS _4+i _-i FIGURE Complex numbers as points in the Arg plane i _i +i -i A complex number can be represented by an expression of the form a bi, where a b are real numbers i is a symbol with

### DEFINITION 5.1.1 A complex number is a matrix of the form. x y. , y x

Chapter 5 COMPLEX NUMBERS 5.1 Constructing the complex numbers One way of introducing the field C of complex numbers is via the arithmetic of matrices. DEFINITION 5.1.1 A complex number is a matrix of

### Zeros of Polynomial Functions

Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### Proofs are short works of prose and need to be written in complete sentences, with mathematical symbols used where appropriate.

Advice for homework: Proofs are short works of prose and need to be written in complete sentences, with mathematical symbols used where appropriate. Even if a problem is a simple exercise that doesn t

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

### n th roots of complex numbers

n th roots of complex numbers Nathan Pflueger 1 October 014 This note describes how to solve equations of the form z n = c, where c is a complex number. These problems serve to illustrate the use of polar

### Mathematics. (www.tiwariacademy.com : Focus on free Education) (Chapter 5) (Complex Numbers and Quadratic Equations) (Class XI)

( : Focus on free Education) Miscellaneous Exercise on chapter 5 Question 1: Evaluate: Answer 1: 1 ( : Focus on free Education) Question 2: For any two complex numbers z1 and z2, prove that Re (z1z2) =

### HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

### Math 504, Fall 2013 HW 3

Math 504, Fall 013 HW 3 1. Let F = F (x) be the field of rational functions over the field of order. Show that the extension K = F(x 1/6 ) of F is equal to F( x, x 1/3 ). Show that F(x 1/3 ) is separable

### COMPLEX NUMBERS AND SERIES. Contents

COMPLEX NUMBERS AND SERIES MIKE BOYLE Contents 1. Complex Numbers Definition 1.1. A complex number is a number z of the form z = x + iy, where x and y are real numbers, and i is another number such that

### x if x 0, x if x < 0.

Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

### Polynomials and the Fast Fourier Transform (FFT) Battle Plan

Polynomials and the Fast Fourier Transform (FFT) Algorithm Design and Analysis (Wee 7) 1 Polynomials Battle Plan Algorithms to add, multiply and evaluate polynomials Coefficient and point-value representation

### 11.7 Polar Form of Complex Numbers

11.7 Polar Form of Complex Numbers 989 11.7 Polar Form of Complex Numbers In this section, we return to our study of complex numbers which were first introduced in Section.. Recall that a complex number

### Series Convergence Tests Math 122 Calculus III D Joyce, Fall 2012

Some series converge, some diverge. Series Convergence Tests Math 22 Calculus III D Joyce, Fall 202 Geometric series. We ve already looked at these. We know when a geometric series converges and what it

### CHAPTER THREE LOCATING ZEROS OF POLYNOMIALS 1. APPROXIMATION OF ZEROS

January 5, 2010 CHAPTER THREE LOCATING ZEROS OF POLYNOMIALS 1. APPROXIMATION OF ZEROS Since the determination of the zeros of a polynomial in exact form is impractical for polynomials of degrees 3 and

### 3 Contour integrals and Cauchy s Theorem

3 ontour integrals and auchy s Theorem 3. Line integrals of complex functions Our goal here will be to discuss integration of complex functions = u + iv, with particular regard to analytic functions. Of

### Trigonometric Functions and Equations

Contents Trigonometric Functions and Equations Lesson 1 Reasoning with Trigonometric Functions Investigations 1 Proving Trigonometric Identities... 271 2 Sum and Difference Identities... 276 3 Extending

### Zeros of a Polynomial Function

Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

### Metric Spaces. Chapter 7. 7.1. Metrics

Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

### MOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu

Integer Polynomials June 9, 007 Yufei Zhao yufeiz@mit.edu We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealing

### THE COMPLEX EXPONENTIAL FUNCTION

Math 307 THE COMPLEX EXPONENTIAL FUNCTION (These notes assume you are already familiar with the basic properties of complex numbers.) We make the following definition e iθ = cos θ + i sin θ. (1) This formula

### CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY

January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.

### Functions and Equations

Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

### 6. Define log(z) so that π < I log(z) π. Discuss the identities e log(z) = z and log(e w ) = w.

hapter omplex integration. omplex number quiz. Simplify 3+4i. 2. Simplify 3+4i. 3. Find the cube roots of. 4. Here are some identities for complex conjugate. Which ones need correction? z + w = z + w,

### Zeros of Polynomial Functions

Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of

### Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00

18.781 Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00 Throughout this assignment, f(x) always denotes a polynomial with integer coefficients. 1. (a) Show that e 32 (3) = 8, and write down a list

### Roots and Coefficients of a Quadratic Equation Summary

Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and

### PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

### MATH 2300 review problems for Exam 3 ANSWERS

MATH 300 review problems for Exam 3 ANSWERS. Check whether the following series converge or diverge. In each case, justify your answer by either computing the sum or by by showing which convergence test

### x a x 2 (1 + x 2 ) n.

Limits and continuity Suppose that we have a function f : R R. Let a R. We say that f(x) tends to the limit l as x tends to a; lim f(x) = l ; x a if, given any real number ɛ > 0, there exists a real number

### 3.4 Complex Zeros and the Fundamental Theorem of Algebra

86 Polynomial Functions.4 Complex Zeros and the Fundamental Theorem of Algebra In Section., we were focused on finding the real zeros of a polynomial function. In this section, we expand our horizons and

### The Method of Partial Fractions Math 121 Calculus II Spring 2015

Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method

### March 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions

MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial

### Complex Numbers Basic Concepts of Complex Numbers Complex Solutions of Equations Operations on Complex Numbers

Complex Numbers Basic Concepts of Complex Numbers Complex Solutions of Equations Operations on Complex Numbers Identify the number as real, complex, or pure imaginary. 2i The complex numbers are an extension

### TOPIC 4: DERIVATIVES

TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the

### Lecture 14: Section 3.3

Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

### CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,

### DRAFT. Further mathematics. GCE AS and A level subject content

Further mathematics GCE AS and A level subject content July 2014 s Introduction Purpose Aims and objectives Subject content Structure Background knowledge Overarching themes Use of technology Detailed

### Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below.

Infinite Algebra 1 Kuta Software LLC Common Core Alignment Software version 2.05 Last revised July 2015 Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below. High School

### MATHEMATICS (CLASSES XI XII)

MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)

### 88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a

88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small

### Matrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo

Matrix Norms Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 28, 2009 Matrix Norms We consider matrix norms on (C m,n, C). All results holds for

### 5.3 Improper Integrals Involving Rational and Exponential Functions

Section 5.3 Improper Integrals Involving Rational and Exponential Functions 99.. 3. 4. dθ +a cos θ =, < a

### TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let

### by the matrix A results in a vector which is a reflection of the given

Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that

### Week 13 Trigonometric Form of Complex Numbers

Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working

### Differentiation and Integration

This material is a supplement to Appendix G of Stewart. You should read the appendix, except the last section on complex exponentials, before this material. Differentiation and Integration Suppose we have

### Induction Problems. Tom Davis November 7, 2005

Induction Problems Tom Davis tomrdavis@earthlin.net http://www.geometer.org/mathcircles November 7, 2005 All of the following problems should be proved by mathematical induction. The problems are not necessarily

### minimal polyonomial Example

Minimal Polynomials Definition Let α be an element in GF(p e ). We call the monic polynomial of smallest degree which has coefficients in GF(p) and α as a root, the minimal polyonomial of α. Example: We

### PYTHAGOREAN TRIPLES PETE L. CLARK

PYTHAGOREAN TRIPLES PETE L. CLARK 1. Parameterization of Pythagorean Triples 1.1. Introduction to Pythagorean triples. By a Pythagorean triple we mean an ordered triple (x, y, z) Z 3 such that x + y =

### MATH 361: NUMBER THEORY FIRST LECTURE

MATH 361: NUMBER THEORY FIRST LECTURE 1. Introduction As a provisional definition, view number theory as the study of the properties of the positive integers, Z + = {1, 2, 3, }. Of particular interest,

### GRE Prep: Precalculus

GRE Prep: Precalculus Franklin H.J. Kenter 1 Introduction These are the notes for the Precalculus section for the GRE Prep session held at UCSD in August 2011. These notes are in no way intended to teach

### Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

### 8 Primes and Modular Arithmetic

8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.

### MATH 52: MATLAB HOMEWORK 2

MATH 52: MATLAB HOMEWORK 2. omplex Numbers The prevalence of the complex numbers throughout the scientific world today belies their long and rocky history. Much like the negative numbers, complex numbers

9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

### 3. INNER PRODUCT SPACES

. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

### The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

### DIFFERENTIABILITY OF COMPLEX FUNCTIONS. Contents

DIFFERENTIABILITY OF COMPLEX FUNCTIONS Contents 1. Limit definition of a derivative 1 2. Holomorphic functions, the Cauchy-Riemann equations 3 3. Differentiability of real functions 5 4. A sufficient condition

### z = i ± 9 2 2 so z = 2i or z = i are the solutions. (c) z 4 + 2z 2 + 4 = 0. By the quadratic formula,

91 Homework 8 solutions Exercises.: 18. Show that Z[i] is an integral domain, describe its field of fractions and find the units. There are two ways to show it is an integral domain. The first is to observe:

### The Rational Numbers

Math 3040: Spring 2011 The Rational Numbers Contents 1. The Set Q 1 2. Addition and multiplication of rational numbers 3 2.1. Definitions and properties. 3 2.2. Comments 4 2.3. Connections with Z. 6 2.4.

### Mathematics Review for MS Finance Students

Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,

### Monotone maps of R n are quasiconformal

Monotone maps of R n are quasiconformal K. Astala, T. Iwaniec and G. Martin For Neil Trudinger Abstract We give a new and completely elementary proof showing that a δ monotone mapping of R n, n is K quasiconformal

### THE GOLDEN RATIO AND THE FIBONACCI SEQUENCE

/ 24 THE GOLDEN RATIO AND THE FIBONACCI SEQUENCE Todd Cochrane Everything is Golden 2 / 24 Golden Ratio Golden Proportion Golden Relation Golden Rectangle Golden Spiral Golden Angle Geometric Growth, (Exponential

### 4.3 Lagrange Approximation

206 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION Lagrange Polynomial Approximation 4.3 Lagrange Approximation Interpolation means to estimate a missing function value by taking a weighted average

### MATH 110 Spring 2015 Homework 6 Solutions

MATH 110 Spring 2015 Homework 6 Solutions Section 2.6 2.6.4 Let α denote the standard basis for V = R 3. Let α = {e 1, e 2, e 3 } denote the dual basis of α for V. We would first like to show that β =

### Mathematical Induction. Mary Barnes Sue Gordon

Mathematics Learning Centre Mathematical Induction Mary Barnes Sue Gordon c 1987 University of Sydney Contents 1 Mathematical Induction 1 1.1 Why do we need proof by induction?.... 1 1. What is proof by

### U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

### 10.2 Series and Convergence

10.2 Series and Convergence Write sums using sigma notation Find the partial sums of series and determine convergence or divergence of infinite series Find the N th partial sums of geometric series and

### AREA AND CIRCUMFERENCE OF CIRCLES

AREA AND CIRCUMFERENCE OF CIRCLES Having seen how to compute the area of triangles, parallelograms, and polygons, we are now going to consider curved regions. Areas of polygons can be found by first dissecting

### MATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4.

MATH 55 SOLUTIONS TO PRACTICE FINAL EXAM x 2 4.Compute x 2 x 2 5x + 6. When x 2, So x 2 4 x 2 5x + 6 = (x 2)(x + 2) (x 2)(x 3) = x + 2 x 3. x 2 4 x 2 x 2 5x + 6 = 2 + 2 2 3 = 4. x 2 9 2. Compute x + sin

### Practice with Proofs

Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using

### Algebra 1 Course Title

Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

### Zeros of Polynomial Functions

Zeros of Polynomial Functions Objectives: 1.Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions 2.Find rational zeros of polynomial functions 3.Find conjugate

### Linear Algebra I. Ronald van Luijk, 2012

Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.

### MITES Physics III Summer Introduction 1. 3 Π = Product 2. 4 Proofs by Induction 3. 5 Problems 5

MITES Physics III Summer 010 Sums Products and Proofs Contents 1 Introduction 1 Sum 1 3 Π Product 4 Proofs by Induction 3 5 Problems 5 1 Introduction These notes will introduce two topics: A notation which

### 3 0 + 4 + 3 1 + 1 + 3 9 + 6 + 3 0 + 1 + 3 0 + 1 + 3 2 mod 10 = 4 + 3 + 1 + 27 + 6 + 1 + 1 + 6 mod 10 = 49 mod 10 = 9.

SOLUTIONS TO HOMEWORK 2 - MATH 170, SUMMER SESSION I (2012) (1) (Exercise 11, Page 107) Which of the following is the correct UPC for Progresso minestrone soup? Show why the other numbers are not valid

### 4. An isosceles triangle has two sides of length 10 and one of length 12. What is its area?

1 1 2 + 1 3 + 1 5 = 2 The sum of three numbers is 17 The first is 2 times the second The third is 5 more than the second What is the value of the largest of the three numbers? 3 A chemist has 100 cc of

### Ideal Class Group and Units

Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals

### Factoring Polynomials

Factoring Polynomials Sue Geller June 19, 2006 Factoring polynomials over the rational numbers, real numbers, and complex numbers has long been a standard topic of high school algebra. With the advent

### Homework 5 Solutions

Homework 5 Solutions 4.2: 2: a. 321 = 256 + 64 + 1 = (01000001) 2 b. 1023 = 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = (1111111111) 2. Note that this is 1 less than the next power of 2, 1024, which

### Mathematical Induction

Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,

### POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

### Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013

Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013 D. R. Wilkins Copyright c David R. Wilkins 1997 2013 Contents A Cyclotomic Polynomials 79 A.1 Minimum Polynomials of Roots of

### Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

### Answer Key for California State Standards: Algebra I

Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

### Pre-Calculus Review Problems Solutions

MATH 1110 (Lecture 00) August 0, 01 1 Algebra and Geometry Pre-Calculus Review Problems Solutions Problem 1. Give equations for the following lines in both point-slope and slope-intercept form. (a) The

5.3 SOLVING TRIGONOMETRIC EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Use standard algebraic techniques to solve trigonometric equations. Solve trigonometric equations

### Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

### (x + a) n = x n + a Z n [x]. Proof. If n is prime then the map

22. A quick primality test Prime numbers are one of the most basic objects in mathematics and one of the most basic questions is to decide which numbers are prime (a clearly related problem is to find