Number theory bsc notes.pdf Mathematical Proof Number. Introduction to Analytic Number Theory " This book is the first volume of a two-volume textbook for undergraduates and is indeed the crystallization of a course offered by the author at the California Institute of Technology to undergraduates without any previous knowledge of number theory., proofs that there are infinitely many primes, without analysis This proof has the disadvantage that it does not exhibit in nitely many primes, but only shows that it is impossible that there are nitely many..

### 1 ANALYTICAL NUMBER THEORY Maharshi Dayanand University

Analytic Number Theory Department Mathematik. ANALYTIC NUMBER THEORY NOTES 7 we obtain Z 1 0 A (a)2A ( 2a)da counts the number of triples (x,y,z) with x +z = 2y. This includes jA jtrivial solutions, so we want to see this integral is larger., 1 In nitude of Primes: Number Theory via Topology This ingenious proof (and the original motivation to construct this talk, in fact!) is due to Israeli-American mathematician Hillel (Harry) Fursten berg, and was.

Prerequisites: A course on complex analysis is a must as well as competence with proofs and the basic concepts of real analysis. A course in elementary number theory would be helpful, but is certainly not required for the motivated student. Outline1 1 Proof 2 Direct Proof 3 Proof by Contradiction 4 Proof by Induction 5 Proof without Words 6 Proofs вЂњFrom the BookвЂќ 1Most of this discussion is вЂ¦

development of logic and mathematicsвЂќ through the Оµ-notation and provides an analysis of AckermannвЂ™s consistency proofs for primitive recursive arithmetic and for the п¬Ѓrst compre- hensive mathematical system, the latter using the substitution method. Contents Introduction 5 0.1. Cold open 5 0.2. Administrivia 6 0.3. More on problems of number theory 6 0.4. Course plan (subject to revision) 7 Chapter 1.

O. Forster: Analytic Number Theory 0. Notations and Conventions Standard notations for sets Z ring of all integers N 0 set of all integers в‰Ґ 0 N 1 set of all integers в‰Ґ 1 CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers aand bwe say adivides bif there is an integer csuch that b= ac.

Abstract Three proofs of the prime number theorem are presented. The rst is a heavily analytic proof based on early accounts. CauchyвЂ™s residue theorem and various results relating to the Number Theory, Analysis and Geometry, alleged publication date 2012 [4unpub] Katz, N., Appendix: Lefschetz pencils with imposed sub-varieties [5unpub] Katz, N., Hooley parameters for families of exponential sums over п¬Ѓnite п¬Ѓelds Here is a list of my publications Bombieri, E. and Katz, N., A Note on Lower Bounds for Frobenius Traces, Enseignement Math. 56 (2010), 203-227. MR2743570Katz

The next proof is unique among all known proofs of the infinitude of the set of primes. It uses a topological argument rather than an analytic or algebraic argument. This proof was given by Fuerstenberg, whose picture is below, in 1955. Contents Introduction 5 0.1. Cold open 5 0.2. Administrivia 6 0.3. More on problems of number theory 6 0.4. Course plan (subject to revision) 7 Chapter 1.

1 In nitude of Primes: Number Theory via Topology This ingenious proof (and the original motivation to construct this talk, in fact!) is due to Israeli-American mathematician Hillel (Harry) Fursten berg, and was A PRIMER OF ANALYTIC NUMBER THEORY Thisundergraduateintroductiontoanalyticnumbertheorydevelopsanalytic skills in the course of a study of ancient questions on

One branch of mathematics is Analytic Number Theory that is thought to number theory problems using the methods and ideas of mathematical analysis. Outline1 1 Proof 2 Direct Proof 3 Proof by Contradiction 4 Proof by Induction 5 Proof without Words 6 Proofs вЂњFrom the BookвЂќ 1Most of this discussion is вЂ¦

CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers aand bwe say adivides bif there is an integer csuch that b= ac. Algebraic Number Theory: study individual solutions. вЂ“ Solve x 2 +y 2 = p, and x 2 +y 2 =n using prime factorization in the Gaussian integers. вЂ“ Solve x 3 +y 3 =z 3 вЂ¦

(1) Elementary Number Theory(book name) by David M. Burton How I used the book: This book introduced me formally to the notions of congruence, the proof of FermatвЂ™s little theorem, the proof of EulerвЂ™s theorem, and the beautiful proofs of quadratic CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers aand bwe say adivides bif there is an integer csuch that b= ac.

### Math 248 Methods of Proof on Mathematics (pdf)

8 Proofs of the Infinitude of the Set of Primes вЂ“ Mathematics. These lecture notes are the only required reading for the course. Homework questions are included in the notes - please see the assignments page to find out when they were assigned. Introduction to the course ( PDF ), Abstract Three proofs of the prime number theorem are presented. The rst is a heavily analytic proof based on early accounts. CauchyвЂ™s residue theorem and various results relating to the.

Number theory bsc notes.pdf Mathematical Proof Scribd. A PRIMER OF ANALYTIC NUMBER THEORY Thisundergraduateintroductiontoanalyticnumbertheorydevelopsanalytic skills in the course of a study of ancient questions on, no proof; Fermat claimed that the margin of the book was too small for it to п¬Ѓt. For more than 350 years, FermatвЂ™s Last Theorem was considered the hardest open question in mathematics, until it was solved by Andrew Wiles in 1994. This, then, is the most recent major breakthrough in mathematics. Ihave included some historical topics in number theory that Ithink are interesting, and that.

### MATH 100 вЂ“ Introduction to the Profession Proofs

Lectures on Analytic Number Theory www.math.tifr.res.in. One branch of mathematics is Analytic Number Theory that is thought to number theory problems using the methods and ideas of mathematical analysis. development of logic and mathematicsвЂќ through the Оµ-notation and provides an analysis of AckermannвЂ™s consistency proofs for primitive recursive arithmetic and for the п¬Ѓrst compre- hensive mathematical system, the latter using the substitution method..

Algebraic Number Theory: study individual solutions. вЂ“ Solve x 2 +y 2 = p, and x 2 +y 2 =n using prime factorization in the Gaussian integers. вЂ“ Solve x 3 +y 3 =z 3 вЂ¦ proofs that there are infinitely many primes, without analysis This proof has the disadvantage that it does not exhibit in nitely many primes, but only shows that it is impossible that there are nitely many.

Number Theory. Final Exam from Spring 2013. Solutions 1. (a) (5 pts) Let d be a positive integer which is not a perfect square. Prove that PellвЂ™s equation x2 dy2 = 1 has a solution (x;y) with x > 0, y > 0 and y even. (b) (7 pts) Find a solution (x;y) to PellвЂ™s equation x2 28y2 = 1 with x > 0 and y > 0. Hint: (b) can, of course, be solved by the standard method, but you may use the proof of Theme 2: Primes Primes numbers occupy very prominent role in number theory. A prime number p is an integer greater than 1 that is divisible only by and itself.

O. Forster: Analytic Number Theory 0. Notations and Conventions Standard notations for sets Z ring of all integers N 0 set of all integers в‰Ґ 0 N 1 set of all integers в‰Ґ 1 Contents Introduction 5 0.1. Cold open 5 0.2. Administrivia 6 0.3. More on problems of number theory 6 0.4. Course plan (subject to revision) 7 Chapter 1.

Analytic Number Theory Andrew Granville 1 Introduction What is number theory? One might have thought that it was simply the study of numbers, but that is too of analytic objects linked to algebraic number theory: Bernoulli polynomi- als and numbers, the gamma function, and zeta and L-functions of Dirichlet characters, which are the simplest types of L-functions.

of analytic objects linked to algebraic number theory: Bernoulli polynomi- als and numbers, the gamma function, and zeta and L-functions of Dirichlet characters, which are the simplest types of L-functions. Number Theory Naoki Sato

1 In nitude of Primes: Number Theory via Topology This ingenious proof (and the original motivation to construct this talk, in fact!) is due to Israeli-American mathematician Hillel (Harry) Fursten berg, and was 1 In nitude of Primes: Number Theory via Topology This ingenious proof (and the original motivation to construct this talk, in fact!) is due to Israeli-American mathematician Hillel (Harry) Fursten berg, and was

Theme 2: Primes Primes numbers occupy very prominent role in number theory. A prime number p is an integer greater than 1 that is divisible only by and itself. 1 In nitude of Primes: Number Theory via Topology This ingenious proof (and the original motivation to construct this talk, in fact!) is due to Israeli-American mathematician Hillel (Harry) Fursten berg, and was

Methods of proof (direct, contradiction, conditional, contraposition); valid and invalid arguments. Examples from set theory. Quantified statements and their negations. Functions, indexed sets, set functions. Proofs in number theory, algebra, geometry and analysis. Proof by induction. Equivalence and well-defined operations and functions. The axiomatic method. 4 lectures. 2. Required 1 In nitude of Primes: Number Theory via Topology This ingenious proof (and the original motivation to construct this talk, in fact!) is due to Israeli-American mathematician Hillel (Harry) Fursten berg, and was

Theme 2: Primes Primes numbers occupy very prominent role in number theory. A prime number p is an integer greater than 1 that is divisible only by and itself. Number Theory. Final Exam from Spring 2013. Solutions 1. (a) (5 pts) Let d be a positive integer which is not a perfect square. Prove that PellвЂ™s equation x2 dy2 = 1 has a solution (x;y) with x > 0, y > 0 and y even. (b) (7 pts) Find a solution (x;y) to PellвЂ™s equation x2 28y2 = 1 with x > 0 and y > 0. Hint: (b) can, of course, be solved by the standard method, but you may use the proof of

Introduction to Analytic Number Theory " This book is the first volume of a two-volume textbook for undergraduates and is indeed the crystallization of a course offered by the author at the California Institute of Technology to undergraduates without any previous knowledge of number theory. Abstract Three proofs of the prime number theorem are presented. The rst is a heavily analytic proof based on early accounts. CauchyвЂ™s residue theorem and various results relating to the

## Number Theory Florida State University

Mathematical Diversity and Elegance in Proofs Who will be. Proof theory was created early in the 20th century by David Hilbert to prove the consistency of the ordinary methods of reasoning used in mathematics| in arithmetic (number theory), analysis and set theory., B.Sc. Mathematics Analytic Number Theory Analytic Number Theory DIVISIBILITY:Suppose , then we say that divides b if b is a multiple of a. If a divides b then a is also.

### Lectures on Proof Theory

Number Theory Analysis and Geometry Princeton University. In additive number theory we make reference to facts about addition in 1 contradistinction to multiplicative number theory, the foundations of which were laid by Euclid at about 300 B.C., of analytic objects linked to algebraic number theory: Bernoulli polynomi- als and numbers, the gamma function, and zeta and L-functions of Dirichlet characters, which are the simplest types of L-functions..

Theme 2: Primes Primes numbers occupy very prominent role in number theory. A prime number p is an integer greater than 1 that is divisible only by and itself. Contents Introduction 5 0.1. Cold open 5 0.2. Administrivia 6 0.3. More on problems of number theory 6 0.4. Course plan (subject to revision) 7 Chapter 1.

analytic number theory and algebraic number theory. Throughout, we denote by П‡a real primitive character of modulus Dwith Dgreater than a suп¬ѓciently large computable number. Abstract Three proofs of the prime number theorem are presented. The rst is a heavily analytic proof based on early accounts. CauchyвЂ™s residue theorem and various results relating to the

Number Theory. Final Exam from Spring 2013. Solutions 1. (a) (5 pts) Let d be a positive integer which is not a perfect square. Prove that PellвЂ™s equation x2 dy2 = 1 has a solution (x;y) with x > 0, y > 0 and y even. (b) (7 pts) Find a solution (x;y) to PellвЂ™s equation x2 28y2 = 1 with x > 0 and y > 0. Hint: (b) can, of course, be solved by the standard method, but you may use the proof of 1 Introduction 1.Complex analysis is in my opinion one of the most beautiful areas of mathemat-ics. It has one of the highest ratios of theorems to de nitions (i.e., a very low

Math 782: Analytic Number Theory (InstructorвЂ™s Notes)* Analytic Versus Elementary: Terminology (Analytic Number Theory makes use of Complex Analysis and Elemen-tary Number Theory does not; but it isnвЂ™t so simple to distinguish.) Writing an integer as a sum of two squares. This is the rst of a few examples of how Complex Analysis can be used to answer a question seemingly unrelated to it (1) Elementary Number Theory(book name) by David M. Burton How I used the book: This book introduced me formally to the notions of congruence, the proof of FermatвЂ™s little theorem, the proof of EulerвЂ™s theorem, and the beautiful proofs of quadratic

1 Introduction 1.Complex analysis is in my opinion one of the most beautiful areas of mathemat-ics. It has one of the highest ratios of theorems to de nitions (i.e., a very low One branch of mathematics is Analytic Number Theory that is thought to number theory problems using the methods and ideas of mathematical analysis.

development of logic and mathematicsвЂќ through the Оµ-notation and provides an analysis of AckermannвЂ™s consistency proofs for primitive recursive arithmetic and for the п¬Ѓrst compre- hensive mathematical system, the latter using the substitution method. Number Theory Naoki Sato

Prerequisites: A course on complex analysis is a must as well as competence with proofs and the basic concepts of real analysis. A course in elementary number theory would be helpful, but is certainly not required for the motivated student. proofs that there are infinitely many primes, without analysis This proof has the disadvantage that it does not exhibit in nitely many primes, but only shows that it is impossible that there are nitely many.

Analytic Number Theory Andrew Granville 1 Introduction What is number theory? One might have thought that it was simply the study of numbers, but that is too Methods of proof (direct, contradiction, conditional, contraposition); valid and invalid arguments. Examples from set theory. Quantified statements and their negations. Functions, indexed sets, set functions. Proofs in number theory, algebra, geometry and analysis. Proof by induction. Equivalence and well-defined operations and functions. The axiomatic method. 4 lectures. 2. Required

Introduction to Analytic Number Theory " This book is the first volume of a two-volume textbook for undergraduates and is indeed the crystallization of a course offered by the author at the California Institute of Technology to undergraduates without any previous knowledge of number theory. Analytic Number Theory Andrew Granville 1 Introduction What is number theory? One might have thought that it was simply the study of numbers, but that is too

Theme 2: Primes Primes numbers occupy very prominent role in number theory. A prime number p is an integer greater than 1 that is divisible only by and itself. Methods of proof (direct, contradiction, conditional, contraposition); valid and invalid arguments. Examples from set theory. Quantified statements and their negations. Functions, indexed sets, set functions. Proofs in number theory, algebra, geometry and analysis. Proof by induction. Equivalence and well-defined operations and functions. The axiomatic method. 4 lectures. 2. Required

ANALYTIC NUMBER THEORY NOTES 7 we obtain Z 1 0 A (a)2A ( 2a)da counts the number of triples (x,y,z) with x +z = 2y. This includes jA jtrivial solutions, so we want to see this integral is larger. dressed in a course in number theory. Proofs of basic theorems are presented in an interesting and comprehensive way that can be read and understood even by non-majors with the exception in the last three chapters where a background in analysis, measure theory and abstract algebra is required. The exercises are care-fully chosen to broaden the understanding of the concepts. Moreover, these

The references are вЂњIntroduction to Analytic Number TheoryвЂќ by T.M. Apostol, 1991 Analytic Number Theory notes by A. Hildebrand, вЂњAn- alytic Number for undergraduatesвЂќ by H.H. Chan. proofs that there are infinitely many primes, without analysis This proof has the disadvantage that it does not exhibit in nitely many primes, but only shows that it is impossible that there are nitely many.

1 Introduction 1.Complex analysis is in my opinion one of the most beautiful areas of mathemat-ics. It has one of the highest ratios of theorems to de nitions (i.e., a very low Prerequisites: A course on complex analysis is a must as well as competence with proofs and the basic concepts of real analysis. A course in elementary number theory would be helpful, but is certainly not required for the motivated student.

Prerequisites: A course on complex analysis is a must as well as competence with proofs and the basic concepts of real analysis. A course in elementary number theory would be helpful, but is certainly not required for the motivated student. Proof theory was created early in the 20th century by David Hilbert to prove the consistency of the ordinary methods of reasoning used in mathematics| in arithmetic (number theory), analysis and set theory.

In additive number theory we make reference to facts about addition in 1 contradistinction to multiplicative number theory, the foundations of which were laid by Euclid at about 300 B.C. Is there some reason that contradiction proofs should be avoided at all costs? вЂ“ Jim Belk Mar 17 '14 at 5:09 Overuse of proof by contradiction leads students to believe that every proof should be a proof by contradiction, meaning that it becomes the first strategy they attempt eve though most of the time it makes things more confusing.

O. Forster: Analytic Number Theory 0. Notations and Conventions Standard notations for sets Z ring of all integers N 0 set of all integers в‰Ґ 0 N 1 set of all integers в‰Ґ 1 In additive number theory we make reference to facts about addition in 1 contradistinction to multiplicative number theory, the foundations of which were laid by Euclid at about 300 B.C.

### MATH 100 вЂ“ Introduction to the Profession Proofs

Number Theory TU/e. 1 In nitude of Primes: Number Theory via Topology This ingenious proof (and the original motivation to construct this talk, in fact!) is due to Israeli-American mathematician Hillel (Harry) Fursten berg, and was, Q2, Theorem (MertensвЂ™ Second theorem; HW Th. 427). в€‘ pв‰¤x 1/p = loglogx+C1 +O(1/logx) (x в‰Ґ 2), for some constant C1. Proof. We use Abel summation, with.

### Introduction to Analytic Number Theory Mathematical

Number Theory TU/e. Math 782: Analytic Number Theory (InstructorвЂ™s Notes)* Analytic Versus Elementary: Terminology (Analytic Number Theory makes use of Complex Analysis and Elemen-tary Number Theory does not; but it isnвЂ™t so simple to distinguish.) Writing an integer as a sum of two squares. This is the rst of a few examples of how Complex Analysis can be used to answer a question seemingly unrelated to it A PRIMER OF ANALYTIC NUMBER THEORY Thisundergraduateintroductiontoanalyticnumbertheorydevelopsanalytic skills in the course of a study of ancient questions on.

The next proof is unique among all known proofs of the infinitude of the set of primes. It uses a topological argument rather than an analytic or algebraic argument. This proof was given by Fuerstenberg, whose picture is below, in 1955. development of logic and mathematicsвЂќ through the Оµ-notation and provides an analysis of AckermannвЂ™s consistency proofs for primitive recursive arithmetic and for the п¬Ѓrst compre- hensive mathematical system, the latter using the substitution method.

1 In nitude of Primes: Number Theory via Topology This ingenious proof (and the original motivation to construct this talk, in fact!) is due to Israeli-American mathematician Hillel (Harry) Fursten berg, and was (The Prime Number Theorem is the central result of analytic number theory since its proof involves complex function theory. Our concerns, by contrast, lie within algebraic number theory.) There are several alternative proofs of EuclidвЂ™s Theorem. We shall give one below. But п¬Ѓrst we must establish the Fundamental Theorem of Arithmetic (the Unique Factorisation Theorem) which gives prime

Theme 2: Primes Primes numbers occupy very prominent role in number theory. A prime number p is an integer greater than 1 that is divisible only by and itself. Theme 2: Primes Primes numbers occupy very prominent role in number theory. A prime number p is an integer greater than 1 that is divisible only by and itself.

Is there some reason that contradiction proofs should be avoided at all costs? вЂ“ Jim Belk Mar 17 '14 at 5:09 Overuse of proof by contradiction leads students to believe that every proof should be a proof by contradiction, meaning that it becomes the first strategy they attempt eve though most of the time it makes things more confusing. of analytic number theory. This minicourse is an introduction to classical results in analytic number theory, presenting fundamental theorems with detailed proofs and highlighting the вЂ¦

dressed in a course in number theory. Proofs of basic theorems are presented in an interesting and comprehensive way that can be read and understood even by non-majors with the exception in the last three chapters where a background in analysis, measure theory and abstract algebra is required. The exercises are care-fully chosen to broaden the understanding of the concepts. Moreover, these Prerequisites: A course on complex analysis is a must as well as competence with proofs and the basic concepts of real analysis. A course in elementary number theory would be helpful, but is certainly not required for the motivated student.

mathematical surveys number jo an introduction to the analytic theory of numbers by raymond ayoub 1963 american mathematical society providence, rhode island The references are вЂњIntroduction to Analytic Number TheoryвЂќ by T.M. Apostol, 1991 Analytic Number Theory notes by A. Hildebrand, вЂњAn- alytic Number for undergraduatesвЂќ by H.H. Chan.

The next proof is unique among all known proofs of the infinitude of the set of primes. It uses a topological argument rather than an analytic or algebraic argument. This proof was given by Fuerstenberg, whose picture is below, in 1955. 452 Chapter 10. Number Theory and Cryptography Computers today are used for a multitude of sensitive applications. Customers utilize electronic commerce to make purchases and pay their bills.

CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers aand bwe say adivides bif there is an integer csuch that b= ac. Introduction to Analytic Number Theory " This book is the first volume of a two-volume textbook for undergraduates and is indeed the crystallization of a course offered by the author at the California Institute of Technology to undergraduates without any previous knowledge of number theory.

- On a New Method in Elementary Number Theory which leads to an Elementary Proof of the Prime Number Theorem. Proceedings of the National Academy of Sciences. U.S.A. , vol. 35, вЂ¦ A PRIMER OF ANALYTIC NUMBER THEORY Thisundergraduateintroductiontoanalyticnumbertheorydevelopsanalytic skills in the course of a study of ancient questions on

Introduction to Analytic Number Theory " This book is the first volume of a two-volume textbook for undergraduates and is indeed the crystallization of a course offered by the author at the California Institute of Technology to undergraduates without any previous knowledge of number theory. 1 In nitude of Primes: Number Theory via Topology This ingenious proof (and the original motivation to construct this talk, in fact!) is due to Israeli-American mathematician Hillel (Harry) Fursten berg, and was

proofs that there are infinitely many primes, without analysis This proof has the disadvantage that it does not exhibit in nitely many primes, but only shows that it is impossible that there are nitely many. 452 Chapter 10. Number Theory and Cryptography Computers today are used for a multitude of sensitive applications. Customers utilize electronic commerce to make purchases and pay their bills.

CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers aand bwe say adivides bif there is an integer csuch that b= ac. (1) Elementary Number Theory(book name) by David M. Burton How I used the book: This book introduced me formally to the notions of congruence, the proof of FermatвЂ™s little theorem, the proof of EulerвЂ™s theorem, and the beautiful proofs of quadratic

Number Theory Naoki Sato

analytic number theory and algebraic number theory. Throughout, we denote by П‡a real primitive character of modulus Dwith Dgreater than a suп¬ѓciently large computable number. Introduction to Analytic Number Theory " This book is the first volume of a two-volume textbook for undergraduates and is indeed the crystallization of a course offered by the author at the California Institute of Technology to undergraduates without any previous knowledge of number theory.

1 Introduction 1.Complex analysis is in my opinion one of the most beautiful areas of mathemat-ics. It has one of the highest ratios of theorems to de nitions (i.e., a very low Number Theory Naoki Sato

(The Prime Number Theorem is the central result of analytic number theory since its proof involves complex function theory. Our concerns, by contrast, lie within algebraic number theory.) There are several alternative proofs of EuclidвЂ™s Theorem. We shall give one below. But п¬Ѓrst we must establish the Fundamental Theorem of Arithmetic (the Unique Factorisation Theorem) which gives prime O. Forster: Analytic Number Theory 0. Notations and Conventions Standard notations for sets Z ring of all integers N 0 set of all integers в‰Ґ 0 N 1 set of all integers в‰Ґ 1

Algebraic Number Theory: study individual solutions. вЂ“ Solve x 2 +y 2 = p, and x 2 +y 2 =n using prime factorization in the Gaussian integers. вЂ“ Solve x 3 +y 3 =z 3 вЂ¦ O. Forster: Analytic Number Theory 0. Notations and Conventions Standard notations for sets Z ring of all integers N 0 set of all integers в‰Ґ 0 N 1 set of all integers в‰Ґ 1

The next proof is unique among all known proofs of the infinitude of the set of primes. It uses a topological argument rather than an analytic or algebraic argument. This proof was given by Fuerstenberg, whose picture is below, in 1955. Contents Introduction 5 0.1. Cold open 5 0.2. Administrivia 6 0.3. More on problems of number theory 6 0.4. Course plan (subject to revision) 7 Chapter 1.