This document appears to be a summer project report submitted by a student named Jitendra Thoury to IIT Kanpur in 2015. The report summarizes the book "Euler The Master Of Us All" which covers Euler's contributions to several branches of mathematics. The report provides an overview of each chapter, including chapters on perfect numbers, logarithms, infinite series, analytic number theory, complex variables, algebra, and combinatorics. It highlights some of Euler's key proofs and insights in these areas. The document serves to introduce the reader to Euler's seminal work and provides context for understanding the mathematical concepts and challenges discussed in greater detail in the source book.
A square meets all the properties of a rectangle - it has four sides, four right angles, opposite sides that are parallel and equal in length. Additionally, all four sides of a square are equal in length. In mathematics, categories are defined inclusively so that a square is considered a special case of a rectangle. This makes theorems and proofs simpler by avoiding separate cases for different shapes.
The document discusses the relationship between mathematics and the natural sciences. It notes that mathematics has been remarkably successful at describing natural phenomena, which is unexpected given that mathematics is a product of human thought while nature exists independently. The document raises questions about how the "laws of nature" can be exact copies of the patterns humans discover in their abstract systems of thought. It calls this alignment of the two fields the "unreasonable effectiveness of mathematics."
The document discusses the nature and certainty of mathematics. It describes how mathematics was traditionally viewed through Euclid's formal model of starting from axioms and reaching theorems through deductive reasoning. However, philosophers have debated whether mathematics is discovered or invented, and whether it is empirical, analytic, or synthetic a priori. While mathematics allows for precise calculations and patterns, the exact status of mathematical concepts remains an open and fundamental question.
1. This document discusses different perspectives on mathematics, including its history and nature. It explores whether math is discovered or invented, and how culture and different fields influence mathematical thought.
2. Key aspects of math discussed are its basis in axioms, deductive reasoning, and theorems. Different views are presented on whether mathematical truths are empirical, true by definition, or insights into reality.
3. The document prompts reflection on relationships between math and other domains like logic, religion, intuition, science, language, and beauty. It encourages examining how perception of math may differ depending on factors like one's background or profession.
Problem solving is the process of finding solutions to difficult or complex issues ,w hile reasoning is the action of thinking about something in a logical, sensible way.
The document discusses Venn diagrams, including:
1) A Venn diagram is a diagram used to represent sets and the relationships between them, with circles representing each set and the overlapping areas representing elements in common between sets.
2) Venn diagrams were introduced in 1880 by John Venn and are used in teaching set theory as well as in other subjects like probability, logic, linguistics, and computer science.
3) The document provides an example of how a Venn diagram could be used to compare characteristics of Republicans and Democrats or of Athens and Sparta.
Nature, characteristics and definition of mathsAngel Rathnabai
This document discusses various views of mathematics, including student, parent, and teacher views. It also covers the nature, characteristics, development, and applications of mathematics. Some key points include:
- Mathematics involves finding and studying patterns, and can be seen as a language, way of thinking, and problem solving approach.
- It has developed over time from ancient subjects like geometry to a more modern field incorporating diverse areas.
- Major subfields include algebra, analysis, applied math, with connections to many other domains. Real-world applications span fields like imaging, cryptography, simulation, and bioinformatics.
This is a brief, I mean brief, introduction to mathematics that I used this year. I also introduced the different types of Geometry, and steps to solving a geometry problem.
A square meets all the properties of a rectangle - it has four sides, four right angles, opposite sides that are parallel and equal in length. Additionally, all four sides of a square are equal in length. In mathematics, categories are defined inclusively so that a square is considered a special case of a rectangle. This makes theorems and proofs simpler by avoiding separate cases for different shapes.
The document discusses the relationship between mathematics and the natural sciences. It notes that mathematics has been remarkably successful at describing natural phenomena, which is unexpected given that mathematics is a product of human thought while nature exists independently. The document raises questions about how the "laws of nature" can be exact copies of the patterns humans discover in their abstract systems of thought. It calls this alignment of the two fields the "unreasonable effectiveness of mathematics."
The document discusses the nature and certainty of mathematics. It describes how mathematics was traditionally viewed through Euclid's formal model of starting from axioms and reaching theorems through deductive reasoning. However, philosophers have debated whether mathematics is discovered or invented, and whether it is empirical, analytic, or synthetic a priori. While mathematics allows for precise calculations and patterns, the exact status of mathematical concepts remains an open and fundamental question.
1. This document discusses different perspectives on mathematics, including its history and nature. It explores whether math is discovered or invented, and how culture and different fields influence mathematical thought.
2. Key aspects of math discussed are its basis in axioms, deductive reasoning, and theorems. Different views are presented on whether mathematical truths are empirical, true by definition, or insights into reality.
3. The document prompts reflection on relationships between math and other domains like logic, religion, intuition, science, language, and beauty. It encourages examining how perception of math may differ depending on factors like one's background or profession.
Problem solving is the process of finding solutions to difficult or complex issues ,w hile reasoning is the action of thinking about something in a logical, sensible way.
The document discusses Venn diagrams, including:
1) A Venn diagram is a diagram used to represent sets and the relationships between them, with circles representing each set and the overlapping areas representing elements in common between sets.
2) Venn diagrams were introduced in 1880 by John Venn and are used in teaching set theory as well as in other subjects like probability, logic, linguistics, and computer science.
3) The document provides an example of how a Venn diagram could be used to compare characteristics of Republicans and Democrats or of Athens and Sparta.
Nature, characteristics and definition of mathsAngel Rathnabai
This document discusses various views of mathematics, including student, parent, and teacher views. It also covers the nature, characteristics, development, and applications of mathematics. Some key points include:
- Mathematics involves finding and studying patterns, and can be seen as a language, way of thinking, and problem solving approach.
- It has developed over time from ancient subjects like geometry to a more modern field incorporating diverse areas.
- Major subfields include algebra, analysis, applied math, with connections to many other domains. Real-world applications span fields like imaging, cryptography, simulation, and bioinformatics.
This is a brief, I mean brief, introduction to mathematics that I used this year. I also introduced the different types of Geometry, and steps to solving a geometry problem.
This document provides an overview and history of "Proofs Without Words" (PWWs), which are visual proofs of mathematical theorems without accompanying words or equations. The document begins by discussing the ancient Chinese visual proof of the Pythagorean theorem. It then discusses how PWWs gained recognition in the 1970s when mathematics journals began publishing them regularly. The document presents examples of PWWs from geometry, calculus, and integer sums, providing traditional proofs alongside for comparison. It analyzes how PWWs convey information visually and whether they satisfy the definition of a formal proof. Overall, the document explores the value and philosophical debate around PWWs as a form of visual mathematical reasoning.
Comparative Analysis of Different Numerical Methods of Solving First Order Di...ijtsrd
A mathematical equation which relates various function with its derivatives is known as a differential equation.. It is a well known and popular field of mathematics because of its vast application area to the real world problems such as mechanical vibrations, theory of heat transfer, electric circuits etc. In this paper, we compare some methods of solving differential equations in numerical analysis with classical method and see the accuracy level of the same. Which will helpful to the readers to understand the importance and usefulness of these methods. Chandrajeet Singh Yadav"Comparative Analysis of Different Numerical Methods of Solving First Order Differential Equation" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-4 , June 2018, URL: http://www.ijtsrd.com/papers/ijtsrd13045.pdf http://www.ijtsrd.com/mathemetics/applied-mathematics/13045/comparative-analysis-of-different-numerical-methods-of-solving-first-order-differential-equation/chandrajeet-singh-yadav
Theory of Knowledge - mathematics philosophiesplangdale
This document discusses different philosophical views of mathematics:
- Formalism views mathematics as true or false based only on definitions, but this cannot explain unproven conjectures.
- Platonism sees mathematical truths existing independently and discovered through reason, but new geometries challenged this.
- Empiricism holds mathematics is generalized from experience, but math has greater certainty than science.
Godel's incompleteness theorems showed that for any axiom system, there are true statements not provable within the system, implying mathematics involves more than just applying rules to axioms. Whether mathematics is discovered or created is debated, with some arguing both discovery and creation are involved.
This course introduces students to the nature of mathematics by exploring how it describes patterns found in nature and its applications in daily life. Students will learn that mathematics is more than just formulas, but also a means of understanding aesthetics, logic, and science. The course will survey how mathematics provides tools for managing finances, making choices, appreciating designs, and dividing resources. Students will complete exercises applying mathematical concepts to gain a broader understanding of its dimensions and test their comprehension.
This document discusses the correlation between mathematics and other disciplines. It defines correlation as the relationship between two or more variables where a change in one variable creates a corresponding change in the other. It provides examples of the correlation between different branches of mathematics like algebra and geometry. It also explains how mathematics is correlated with sciences by expressing scientific laws with mathematical equations, with social sciences by using math for maps, dates, and geography, with language by using math terms and definitions, and with fine arts by applying mathematical concepts of ratio, proportion, symmetry and rhythm. It concludes that mathematics provides relationships and applications across many fields of study.
This document provides an introduction to sets and functions for cryptography students. It covers basic concepts like sets and members, set notation, empty and number sets, subsets and power sets, equality of sets, universal sets and Venn diagrams, set operations, and set identities. The document is divided into multiple sections and includes examples and review questions to help explain the topics.
K TO 12 GRADE 7 LEARNING MODULE IN MATHEMATICS (Quarter 3)LiGhT ArOhL
This document provides a lesson plan on solving linear equations and inequalities in one variable algebraically. It begins with reviewing translating between verbal and mathematical phrases and evaluating expressions. The main focus is on introducing and applying the properties of equality, including reflexive, symmetric, transitive, and substitution properties, to solve equations algebraically. Word problems involving equations in one variable are also discussed. The objectives are to identify and apply the properties of equality to find solutions to equations and solve word problems involving one variable equations.
This document provides an introduction to calculus and integration. It states that integration allows us to determine the area under curves and functions. As an example, it notes that if we are given the line f(x) = x, integration can be used to find the area under the line. It then briefly explains that differentiation is the counterpart of integration and allows us to determine the slope or rate of change of a curve or function.
* There are 6 guests besides Alice who will sit in the chairs around the table
* The first chair can be filled by any of the 6 guests.
* Once the first chair is filled, the second chair can be filled by any of the remaining 5 guests.
* Continuing in this way, the number of ways to seat the guests is 6 * 5 * 4 * 3 * 2 * 1 = 720
Since they change seating every half hour and there are 720 possible seatings, it will take 720 / 2 = 360 hours or 15 days for every possible seating to occur.
Benno artmann (auth.) euclid—the creation of mathematics-springer-verlag new ...Vidi Al Imami
This document provides background information on Euclid's Elements, a seminal work in mathematics from ancient Greece. It discusses the contents and organization of the 13 books that make up the Elements. The books cover topics in plane and solid geometry, numbers, proportions, commensurability, constructions, and the Platonic solids. The document also provides context on the origins of mathematics in ancient Greece and key figures like Pythagoras who influenced Euclid's work. It aims to help readers understand and appreciate Euclid's work, which set standards for deductive reasoning in mathematics that are still followed today.
The document summarizes a research paper on the Euler line theorem in geometry. It provides four different proofs of the theorem, which states that the orthocenter, centroid, and circumcenter of any triangle lie on a single line, called the Euler line, with the centroid always lying halfway between the orthocenter and circumcenter. It then gives examples of applying the theorem to solve geometry problems. The paper aims to shed light on the properties and uses of the Euler line, which is an important theorem that is not widely known.
This document provides a preface and table of contents for a book on the theory of polynomials. The preface outlines the book's contents, which include discussions of roots of polynomials, irreducible polynomials, special classes of polynomials, properties of polynomials, Galois theory, Hilbert's theorems, and Hilbert's 17th problem on representing nonnegative polynomials as sums of squares. The table of contents provides further details on the chapters and sections.
Mortad-Mohammed-Hichem-Introductory-topology-exercises-and-solutions-World-Sc...Bui Loi
This document is the preface to the book "Introductory Topology: Exercises and Solutions" by Mohammed Hichem Mortad. The book aims to provide students with a good introduction to basic general topology through solved exercises. It contains over 500 exercises divided into 8 chapters covering topics such as sets, metric spaces, topological spaces, continuity, compactness, connectedness, complete metric spaces, and function spaces. The preface explains that topology can be a difficult subject for students to grasp initially, but that working through exercises is key to learning. It encourages students to attempt exercises on their own before checking the solutions.
The document discusses how mathematicians have historically expanded mathematical systems and concepts to overcome limitations. It provides examples of how the number system was expanded to include complex numbers to allow taking square roots of negative numbers. Geometry was expanded through projective geometry to allow for parallel lines to intersect at a point at infinity. Recently, mathematicians defined new elements for division by zero to fill a gap in algebraic structures. The thesis will present the development of these concepts and discuss properties that fail when moving from fields to newer structures.
Here are the answers to the exercises:
1. The 2007th digit after the period in the decimal expansion of 1/7 is 7, since the expansion repeats with a period of 7 digits (142857...).
2. a) and b) have finite decimal expansions, while c) does not.
3. A = [-1, 2], B = (-∞, -1] ∪ (2, ∞). C = (-∞, 1) ∪ (2, ∞). D = (-∞, 1) ∪ (3, ∞). E = [-1, 2].
The finite sets are A and E.
3. Functions
3
This document provides the preface to a textbook on abstract algebra. It discusses how abstract algebra differs from elementary algebra in its emphasis on concepts and principles over problem solving. The preface aims to make abstract algebraic concepts more meaningful and intuitive for students by tracing their origins and exploring connections to other areas of mathematics. It also outlines changes made for the second edition, including additional exercises and applications.
The Fascinating World of Real Number Sequences.pdfDivyanshu Ranjan
The study of sequences of real numbers is a fascinating and important part of mathematics. It is a central topic in analysis, a branch of mathematics that deals with continuous functions and their properties. A sequence is simply a function whose domain is the set of positive integers, and whose range is a set of real numbers. Sequences of real numbers have many interesting and useful properties, and they are used to model a wide range of mathematical and real-world phenomena.
In this book, we will delve into the properties of sequences of real numbers and explore their connections with other areas of mathematics. We will start with a gentle introduction to sequences, including the definitions and notation used in the study of sequences. We will then move on to more advanced topics, such as convergence and divergence of sequences, Cauchy sequences, and subsequences.
One of the central ideas in the study of sequences is the concept of convergence. A sequence is said to converge if its terms become arbitrarily close to a fixed real number as the index of the sequence increases. We will explore the different types of convergence and their properties, as well as the notion of limit, which is a fundamental concept in analysis.
In addition to convergence, we will also study the properties of divergent sequences, which are sequences that do not converge. We will examine the relationship between convergence and divergence and their connection with real-valued functions.
Throughout the book, we will use a friendly and engaging tone, making the material accessible to a wide range of readers, including students, mathematicians, and anyone with an interest in mathematics. Whether you are a beginner or an expert, you will find something of interest in this book.
Chapter 1: Introduction to Sequences
In this chapter, we will introduce the basic concepts and notation used in the study of sequences of real numbers. A sequence is simply a function whose domain is the set of positive integers, and whose range is a set of real numbers. We will start by defining sequences and their terms, and we will explore some basic properties of sequences, such as their limits and bounds.
We will also introduce the notation used to represent sequences, including the use of the Greek letter n to represent the index of a sequence. This is a crucial piece of notation, as it allows us to express the properties of a sequence in a concise and easily understood way.
Finally, we will look at some examples of sequences, including arithmetic and geometric sequences, and we will explore some of their properties. This will provide a foundation for the more advanced topics we will study later in the book.
Definition of Sequence
A sequence of real numbers is a function whose domain is the set of positive integers, and whose range is a set of real numbers. The elements of the sequence are referred to as terms, and they are denoted by a variable (usually "a_n") with a subscrip
This document is the introduction chapter of a textbook on probability. It introduces fundamental probability concepts like outcome spaces, events, and probability as a function of events. Probability is first discussed in the context of equally likely outcomes, where each outcome has the same probability of occurring. Two common interpretations of probability - as the long-run frequency of an event occurring or as a subjective measure of uncertainty - are also introduced. The chapter then covers probability distributions, conditional probability, independence, Bayes' rule, and sequences of events.
Add math project 2018 kuala lumpur (simultaneous and swing)Anandraka
This document is a student's additional mathematics project work. It includes an introduction on the history of equations and their applications. It discusses how ancient Babylonians, Arabs, and Europeans contributed to solving quadratic equations over time. It also provides examples of how quadratic equations are used in various areas like determining the length of the hypotenuse and establishing the proportions of paper sizes. The student expresses frustration with lack of help from their teacher and tuition teacher in completing the project, but was able to finish it through their own research and discussion with friends.
This document is the preface to a textbook on number theory. It discusses the goals of the textbook, which are to encourage independent thinking and problem solving rather than rote memorization. Number theory is well-suited for this purpose as patterns in the natural numbers can be discerned through observation and experimentation, but proving theorems requires rigorous demonstration. The textbook was originally written for a course at Brown University designed to attract non-science majors to mathematics. The prerequisites are few, requiring only high school algebra and a willingness to experiment, make mistakes, learn from them, and persevere.
This document provides an overview and history of "Proofs Without Words" (PWWs), which are visual proofs of mathematical theorems without accompanying words or equations. The document begins by discussing the ancient Chinese visual proof of the Pythagorean theorem. It then discusses how PWWs gained recognition in the 1970s when mathematics journals began publishing them regularly. The document presents examples of PWWs from geometry, calculus, and integer sums, providing traditional proofs alongside for comparison. It analyzes how PWWs convey information visually and whether they satisfy the definition of a formal proof. Overall, the document explores the value and philosophical debate around PWWs as a form of visual mathematical reasoning.
Comparative Analysis of Different Numerical Methods of Solving First Order Di...ijtsrd
A mathematical equation which relates various function with its derivatives is known as a differential equation.. It is a well known and popular field of mathematics because of its vast application area to the real world problems such as mechanical vibrations, theory of heat transfer, electric circuits etc. In this paper, we compare some methods of solving differential equations in numerical analysis with classical method and see the accuracy level of the same. Which will helpful to the readers to understand the importance and usefulness of these methods. Chandrajeet Singh Yadav"Comparative Analysis of Different Numerical Methods of Solving First Order Differential Equation" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-4 , June 2018, URL: http://www.ijtsrd.com/papers/ijtsrd13045.pdf http://www.ijtsrd.com/mathemetics/applied-mathematics/13045/comparative-analysis-of-different-numerical-methods-of-solving-first-order-differential-equation/chandrajeet-singh-yadav
Theory of Knowledge - mathematics philosophiesplangdale
This document discusses different philosophical views of mathematics:
- Formalism views mathematics as true or false based only on definitions, but this cannot explain unproven conjectures.
- Platonism sees mathematical truths existing independently and discovered through reason, but new geometries challenged this.
- Empiricism holds mathematics is generalized from experience, but math has greater certainty than science.
Godel's incompleteness theorems showed that for any axiom system, there are true statements not provable within the system, implying mathematics involves more than just applying rules to axioms. Whether mathematics is discovered or created is debated, with some arguing both discovery and creation are involved.
This course introduces students to the nature of mathematics by exploring how it describes patterns found in nature and its applications in daily life. Students will learn that mathematics is more than just formulas, but also a means of understanding aesthetics, logic, and science. The course will survey how mathematics provides tools for managing finances, making choices, appreciating designs, and dividing resources. Students will complete exercises applying mathematical concepts to gain a broader understanding of its dimensions and test their comprehension.
This document discusses the correlation between mathematics and other disciplines. It defines correlation as the relationship between two or more variables where a change in one variable creates a corresponding change in the other. It provides examples of the correlation between different branches of mathematics like algebra and geometry. It also explains how mathematics is correlated with sciences by expressing scientific laws with mathematical equations, with social sciences by using math for maps, dates, and geography, with language by using math terms and definitions, and with fine arts by applying mathematical concepts of ratio, proportion, symmetry and rhythm. It concludes that mathematics provides relationships and applications across many fields of study.
This document provides an introduction to sets and functions for cryptography students. It covers basic concepts like sets and members, set notation, empty and number sets, subsets and power sets, equality of sets, universal sets and Venn diagrams, set operations, and set identities. The document is divided into multiple sections and includes examples and review questions to help explain the topics.
K TO 12 GRADE 7 LEARNING MODULE IN MATHEMATICS (Quarter 3)LiGhT ArOhL
This document provides a lesson plan on solving linear equations and inequalities in one variable algebraically. It begins with reviewing translating between verbal and mathematical phrases and evaluating expressions. The main focus is on introducing and applying the properties of equality, including reflexive, symmetric, transitive, and substitution properties, to solve equations algebraically. Word problems involving equations in one variable are also discussed. The objectives are to identify and apply the properties of equality to find solutions to equations and solve word problems involving one variable equations.
This document provides an introduction to calculus and integration. It states that integration allows us to determine the area under curves and functions. As an example, it notes that if we are given the line f(x) = x, integration can be used to find the area under the line. It then briefly explains that differentiation is the counterpart of integration and allows us to determine the slope or rate of change of a curve or function.
* There are 6 guests besides Alice who will sit in the chairs around the table
* The first chair can be filled by any of the 6 guests.
* Once the first chair is filled, the second chair can be filled by any of the remaining 5 guests.
* Continuing in this way, the number of ways to seat the guests is 6 * 5 * 4 * 3 * 2 * 1 = 720
Since they change seating every half hour and there are 720 possible seatings, it will take 720 / 2 = 360 hours or 15 days for every possible seating to occur.
Benno artmann (auth.) euclid—the creation of mathematics-springer-verlag new ...Vidi Al Imami
This document provides background information on Euclid's Elements, a seminal work in mathematics from ancient Greece. It discusses the contents and organization of the 13 books that make up the Elements. The books cover topics in plane and solid geometry, numbers, proportions, commensurability, constructions, and the Platonic solids. The document also provides context on the origins of mathematics in ancient Greece and key figures like Pythagoras who influenced Euclid's work. It aims to help readers understand and appreciate Euclid's work, which set standards for deductive reasoning in mathematics that are still followed today.
The document summarizes a research paper on the Euler line theorem in geometry. It provides four different proofs of the theorem, which states that the orthocenter, centroid, and circumcenter of any triangle lie on a single line, called the Euler line, with the centroid always lying halfway between the orthocenter and circumcenter. It then gives examples of applying the theorem to solve geometry problems. The paper aims to shed light on the properties and uses of the Euler line, which is an important theorem that is not widely known.
This document provides a preface and table of contents for a book on the theory of polynomials. The preface outlines the book's contents, which include discussions of roots of polynomials, irreducible polynomials, special classes of polynomials, properties of polynomials, Galois theory, Hilbert's theorems, and Hilbert's 17th problem on representing nonnegative polynomials as sums of squares. The table of contents provides further details on the chapters and sections.
Mortad-Mohammed-Hichem-Introductory-topology-exercises-and-solutions-World-Sc...Bui Loi
This document is the preface to the book "Introductory Topology: Exercises and Solutions" by Mohammed Hichem Mortad. The book aims to provide students with a good introduction to basic general topology through solved exercises. It contains over 500 exercises divided into 8 chapters covering topics such as sets, metric spaces, topological spaces, continuity, compactness, connectedness, complete metric spaces, and function spaces. The preface explains that topology can be a difficult subject for students to grasp initially, but that working through exercises is key to learning. It encourages students to attempt exercises on their own before checking the solutions.
The document discusses how mathematicians have historically expanded mathematical systems and concepts to overcome limitations. It provides examples of how the number system was expanded to include complex numbers to allow taking square roots of negative numbers. Geometry was expanded through projective geometry to allow for parallel lines to intersect at a point at infinity. Recently, mathematicians defined new elements for division by zero to fill a gap in algebraic structures. The thesis will present the development of these concepts and discuss properties that fail when moving from fields to newer structures.
Here are the answers to the exercises:
1. The 2007th digit after the period in the decimal expansion of 1/7 is 7, since the expansion repeats with a period of 7 digits (142857...).
2. a) and b) have finite decimal expansions, while c) does not.
3. A = [-1, 2], B = (-∞, -1] ∪ (2, ∞). C = (-∞, 1) ∪ (2, ∞). D = (-∞, 1) ∪ (3, ∞). E = [-1, 2].
The finite sets are A and E.
3. Functions
3
This document provides the preface to a textbook on abstract algebra. It discusses how abstract algebra differs from elementary algebra in its emphasis on concepts and principles over problem solving. The preface aims to make abstract algebraic concepts more meaningful and intuitive for students by tracing their origins and exploring connections to other areas of mathematics. It also outlines changes made for the second edition, including additional exercises and applications.
The Fascinating World of Real Number Sequences.pdfDivyanshu Ranjan
The study of sequences of real numbers is a fascinating and important part of mathematics. It is a central topic in analysis, a branch of mathematics that deals with continuous functions and their properties. A sequence is simply a function whose domain is the set of positive integers, and whose range is a set of real numbers. Sequences of real numbers have many interesting and useful properties, and they are used to model a wide range of mathematical and real-world phenomena.
In this book, we will delve into the properties of sequences of real numbers and explore their connections with other areas of mathematics. We will start with a gentle introduction to sequences, including the definitions and notation used in the study of sequences. We will then move on to more advanced topics, such as convergence and divergence of sequences, Cauchy sequences, and subsequences.
One of the central ideas in the study of sequences is the concept of convergence. A sequence is said to converge if its terms become arbitrarily close to a fixed real number as the index of the sequence increases. We will explore the different types of convergence and their properties, as well as the notion of limit, which is a fundamental concept in analysis.
In addition to convergence, we will also study the properties of divergent sequences, which are sequences that do not converge. We will examine the relationship between convergence and divergence and their connection with real-valued functions.
Throughout the book, we will use a friendly and engaging tone, making the material accessible to a wide range of readers, including students, mathematicians, and anyone with an interest in mathematics. Whether you are a beginner or an expert, you will find something of interest in this book.
Chapter 1: Introduction to Sequences
In this chapter, we will introduce the basic concepts and notation used in the study of sequences of real numbers. A sequence is simply a function whose domain is the set of positive integers, and whose range is a set of real numbers. We will start by defining sequences and their terms, and we will explore some basic properties of sequences, such as their limits and bounds.
We will also introduce the notation used to represent sequences, including the use of the Greek letter n to represent the index of a sequence. This is a crucial piece of notation, as it allows us to express the properties of a sequence in a concise and easily understood way.
Finally, we will look at some examples of sequences, including arithmetic and geometric sequences, and we will explore some of their properties. This will provide a foundation for the more advanced topics we will study later in the book.
Definition of Sequence
A sequence of real numbers is a function whose domain is the set of positive integers, and whose range is a set of real numbers. The elements of the sequence are referred to as terms, and they are denoted by a variable (usually "a_n") with a subscrip
This document is the introduction chapter of a textbook on probability. It introduces fundamental probability concepts like outcome spaces, events, and probability as a function of events. Probability is first discussed in the context of equally likely outcomes, where each outcome has the same probability of occurring. Two common interpretations of probability - as the long-run frequency of an event occurring or as a subjective measure of uncertainty - are also introduced. The chapter then covers probability distributions, conditional probability, independence, Bayes' rule, and sequences of events.
Add math project 2018 kuala lumpur (simultaneous and swing)Anandraka
This document is a student's additional mathematics project work. It includes an introduction on the history of equations and their applications. It discusses how ancient Babylonians, Arabs, and Europeans contributed to solving quadratic equations over time. It also provides examples of how quadratic equations are used in various areas like determining the length of the hypotenuse and establishing the proportions of paper sizes. The student expresses frustration with lack of help from their teacher and tuition teacher in completing the project, but was able to finish it through their own research and discussion with friends.
This document is the preface to a textbook on number theory. It discusses the goals of the textbook, which are to encourage independent thinking and problem solving rather than rote memorization. Number theory is well-suited for this purpose as patterns in the natural numbers can be discerned through observation and experimentation, but proving theorems requires rigorous demonstration. The textbook was originally written for a course at Brown University designed to attract non-science majors to mathematics. The prerequisites are few, requiring only high school algebra and a willingness to experiment, make mistakes, learn from them, and persevere.
This document discusses how the work of mathematician George Lakatos can inform mathematics teaching. While directly applying Lakatos' concepts in the classroom has proven difficult, his emphasis on proof as a heuristic, experimental process rather than an authoritarian presentation of finished theorems is inspiring. Both Lakatos and mathematician George Pólya viewed mathematics as an inductive science in the making, rather than just a deductive one. They both stressed the importance of teaching the heuristic process behind mathematical concepts rather than just presenting finished results. While Lakatos and Pólya shared many commonalities in their views, the document argues that Pólya's approach may be more pedagogically useful for teaching heuristics than directly applying Lakatos' philosophical framework.
Stefano Gentili - Measure, Integration and a Primer on Probability Theory_ Vo...BfhJe1
This document provides information about a textbook titled "Measure, Integration and a Primer on Probability Theory" by Stefano Gentili.
The textbook is published as Volume 125 of the UNITEXT series. It presents advanced topics in real analysis, measure theory, and integration in a thorough manner, accessible to novices. It references the historical and scientific context of these topics' development.
The focus is the Lebesgue integral, which was created to overcome limitations of the Riemann integral. Developing a general integration theory required strengthening the understanding of measure. The text covers important contributions from mathematicians like Lebesgue, Borel, Cantor, and others.
The textbook is intended for STEM and
A combinatorial miscellany by Anders BJ Orner and Richard P. StanleyPim Piepers
This document provides an introduction to bijective proofs in enumerative combinatorics. It begins by explaining that bijective proofs involve finding a bijection between the set being counted and another set whose size is known, thereby establishing that the two sets have equal sizes. It then gives some simple examples of bijective proofs, including proving that the number of subsets of a set of size n is 2n by giving a bijection to the set of n-bit binary strings. The document also introduces the concept of compositions of integers and defines what they are.
A combinatorial miscellany by anders bj ¨orner and richard p. stanleyPim Piepers
This document summarizes bijective proofs in combinatorics. It explains that a bijective proof determines the number of elements in a set by finding a one-to-one correspondence between that set and another set with a known number of elements. This correspondence pairs up each element of one set with exactly one element of the other set, proving they have the same number of elements. The document gives examples of using bijective proofs to enumerate words, partitions, and tilings.
1. Assume that an algorithm to solve a problem takes f(n) microse.docxSONU61709
1. Assume that an algorithm to solve a problem takes f(n) microseconds for some function f of the input size n. For each time t labeled across the top, determine the exact largest value of n which can be solved in time f(n) where f(n) ≤ t. Use a calculator! You will find it helpful to convert the t values to microseconds, and you may find it helpful to insert a row for n. Note that “lg n” is the log2 n. Note that the only row you can’t write out the values for fully is the “lg n” row—only there may you write 2x for the appropriate value of x. Use the Windows built-in scientific calculator (under Accessories menu) as necessary. A couple values are filled in to get you started. Important: “exact values” means precisely that. Check your answers with values above and below!
Time t =
f(n) =
1 second
1 hour
1 day
1 month
=30 days
n2
1,609,968
lg n
n3
2n
n lg n
2,755,147,513
2. Use loop counting to give a O( ) characterization of each of the following loops basing each upon the size of its input:
a. Algorithm Loop1(n):
s ← 0
for i ← 1 to n do
s ← s + i
b. Algorithm Loop2(p):
p ← 1
for i ← 1 to 2n do
p ← p * i
c. Algorithm Loop3(n):
p ← 1
for j ← 1 to n2 do
p ← p * i
d. Algorithm Loop4(n):
s ← 0
for j ← 1 to 2n do
for k ← 1 to j do
s ← s + j
e. Algorithm Loop5(n):
k ← 0
for r ← 1 to n2 do
for s ← 1 to r do
k ← k + r
3. Order the following functions from smallest to largest by their big-O notation—you can use the letters in your answer rather than copying each formula. Be clear which is smallest and which is largest, and which functions are asymptotically equivalent. For example, if g, h, and m are all O(n lg n), you would write g = h = m = O(n lg n).
a. 562 log3 108
b. n3
c. 2n lg n
d. lg nn
e. n3 lg n
f. (n3 lg n3)/2
g. nn
h. 56n
i. log5 (n!)
j. ncos n
k. n / lg n
l. lg* n
m.
4. a. Which of these equations is true, and why?
b. Which of these is smaller for very large n?
Trisecting the Circle: A Case for Euclidean Geometry
Author(s): Alfred S. Posamentier
Source: The Mathematics Teacher, Vol. 99, No. 6 (FEBRUARY 2006), pp. 414-418
Published by: National Council of Teachers of Mathematics
Stable URL: http://www.jstor.org/stable/27972006
Accessed: 09-02-2018 18:19 UTC
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The document analyzes a function f(x) to demonstrate that π is irrational. It defines f(x) and studies its properties, showing that its kth derivative at 0 and 1 are integers. It also shows the integral of f(x) from 0 to 1 is bounded between 0 and 1. The document then defines another function G(x) related to f(x) and derives two important results about G(x). This leads to a contradiction when assuming π is rational, demonstrating it must be irrational.
John Stillwell-Numerele reale-An Introduction to Set Theory and Analysis ( PD...LobontGheorghe
This document provides context about the book "The Real Numbers: An Introduction to Set Theory and Analysis" by John Stillwell. It discusses how typical real analysis courses fail to properly define or address the foundations of the real number system. The book aims to fill this gap by constructing the real numbers rigorously while also making the material accessible through historical context, examples, and explanatory remarks. It argues that analysis is fundamentally based on studying sets, particularly uncountable sets like the real numbers and countable ordinals, and their relationship is central to the continuum problem.
John Stillwell-Numerele reale-An Introduction to Set Theory and Analysis ( PD...
new report
1. Indian Institute of Technology
Kanpur
SUMMER PROJECT REPORT 2015
Book : Euler The Master Of Us All
Jitendra Thoury
Student of IITK
Roll No. -14290
Date : May29,2015
3. Summer Project
1 Introduction
”Mathematics lacks the tactile solidity of architecture.It is intangible,existing not in
stone and morter but in human imagination.Yet like architecture it is real and it has
its masters.”
Once the great mathematician of his era Laplace said that ”Read Euler,read Euler, Euler is the
master of us all.”
Why such a great mathematician said such statement? Because There is nearly no branch of
mathematics among them in which Euler doesn not give his valuable contribution. The book ”Euler
The Master Of Us All” contains some selected branches of mathematics and then gives us basic
introduction to Euler’s work on these branches of mathematics.The Book also have small biography of
Euler.He born in land of great mathematicians of his time ’Basel’(Switzerland) ;the land of Bernoulli
familyJakob Bernoulli, Johan Bernoulli and Denial Bernoulli.His full name is Leonhard Euler.
There are few things that we must note about Euler during reading this book. Euler was far from
infallible and his proofs are not so rigorous and do not follow recent mathematical boundations but
nobody can argue that his proof was wrong beacause Euler gave more than one proofs of most of the
results that he deduced and these were more rigorous.Most of his proofs were of analytical type.Euler
use infinite serieses as his best analysis tool and specially the logarithms.One of the interesting fact
about Euler is that “whenever Euler saw products he took logarithms.” Author of the book Prof.
Dolcini writes that “Euler wrote mathematics faster than most people can absorb it.”
The way of writting the book is so nice for a begginers and experinced all type of readers.The
way of explaining a topic is this that it first gives us the knowledge of mathematics that Euler got
from his predecessors about that topic and then expresses Euler way of resoning and demonstrates
his coclusions and results.And as the summary of the topic it introduces us from further research
on that topic and leaves some unsolved problems for mathematicians community.
There is no great prior knowledge required for reading this book except some basic courses of
mathematics that covered in first year by a student of mathematics or any engineering. So this book
can be read and learn by any student of mathematics or who have very basic knowledge of it.And
the author goes very cleverly in sequencing the chapters that we do not have to worry about such
things.
The book divided into eight chapters each of which related to a different branch of mathe-
matics.Each of the chapter have three subsections ’Prologue’-research before the Euler came,’Enter
Euler’-research in the field by Euler and last is ’Epilogue’-research in the field after Euler and further
challanges for mathematicians in the field.So this nice arranged book in this pattern.
The very first chapter of the book is from number theory mostly focuses on prefect numbers.The
study of prefect numbers starts from ancient greeks whose motivation was to find number which is
equal to sum of its all proper divisors.They found that these numbers are rare.Euclid gave a theorem
or say method for generating these numbers but that was limited to even prefect numbers.Ancient
Greek mathematicians could not find any odd prefect numbers.Euler improved the method of Euclid
by adding necessary condition.Chapter ends with introducing the reader by further research on the
odd prefect numbers.
Chapter second of the book is logarithms which have the series expansion of logarithms and their use
as a analysis tool.This was Euler’s favorite tool of analysis so chapter have some proofs from various
branches of mathematics.The third chapter is about the surprising infinite serieses.In this chapter
some open problems of Euler’s time is solved and introduces by some challanges in theory.Chapter
consists a very important theorem about relation of coefficients of polynomial with its roots.Next
chapter of the book is the Analytic number theory.Euler said it as the most difficult and profund
subject.Chapter mainly focuses on prime numbers and on some infinite serieses having prime numbers
.This gives us the good introduction to techniques of proving results.Complex Variables is the fifth
chapter of the book.In this we will see Euler’s infinite series expansion of sin(x) and cos(x) using
De Moivre’s Theorem.In the end we come in the touch of solution of some strange equations in
1 IIT KANPUR
4. Summer Project
the form of complex numbers.In the chapter Euler and Algebra author tries to see us the effort
of mathematicians in proving the fundamental theorem of algebra.In the beginning ,we touch the
thought of Japanese mathematicians aim to find out the roots of a polynomial in the terms of its
coefficients.But that proved after 200 years that fifth degree polynomial’s roots can not be found in
terms of their coefficients in general.Ends with the proof of the fundamental theorem of algebra.In
between the chapter we will find few tricks and method of finding roots for 3rd
and 4th degree
polynomials in terms of their coefficients.Geometry of Euler was of mixture of synthetic and analytic
type.Synthetic geometry is said to be pure geometry which uses only the ancients Greeks way or this
does not superimpose the triangle on a plane paper.But the analytic type of geometry superimposes
the triangle on a paper plane or use coordinate system to represent a geometric object.Euler gave the
proof of Heron formula (for triangle area) in two different ways ,one by using synthetic geometry and
other one by analytic geometry.Further we see the advantages of analytic geometry in reaching on
the Euler’s line.The last chapter of the book is combinatorics.There author makes changes slightly
in his pattern and he teaches us combinatorics by taking a problem and then by solving it.He gives
us the proof of formula for calculating number of derrangments of n items. And here the content of
the book ends but author also gives us the further study sources in the appendix.
2 IIT KANPUR
5. Summer Project
2 Prefect Numbers
2.1 Proper divisors :-
All divisors of a natural number (except that number) are called as proper divisors of the that
number.Ex.- All divisors of number 6 is 1,2,3,6 but proper divisors are 1,2,3.
Number of proper divisors :- If a number n can be written in the form
N = Pα1
1 .Pα2
2 ...Pαk
k where P1, P2...Pk are prime numbers and α1, α2...αk are non negative integers.
This is obvious that all the proper divisors will be generated by multiplication of any combination
of P1, P2...Pk because any number that not formed by these does not divides N.So the total number
of divisors can be found by multiplication theorem of permutation theory.
So any of the divisor will take the form Pβ1
1 .Pβ2
2 ...Pβk
k s.t. βi ∈ {0, 1, 2...αi } for all i ∈ {0, 1...k} So
now the total number of divisors = (α1 + 1).(α2 + 1)...(αk + 1).
That’s why proper divisors=(α1 + 1).(α2 + 1)...(αk + 1) -1.
Sum of all divisors(σ(N))=
(P
α1+1
1 −1)
(P1−1) .
(P
α2+1
2 −1)
(P2−1) ...
(P
αk+1
k
−1)
(Pk−1)
This follows by simply the geometric series sum.
The standard notation is σ.
So the sum of proper divisors = σ(N) - N.
2.2 Definition -
A whole number whose sum of all proper divisors equal to that number is called a prefect num-
ber.These prefect are rare.Ex.6,28.
So mathematically a number is prefect if and only if σ(N) − N = N ⇒ σ(N) = 2N.
Theorem 1.1 :- Euclid gave a way of generating prefect numbers.
Statement : If 2p
− 1 is a prime number then 2p−1
.(2p
− 1) will be a prefect number.
Such numbers are named as Merssene primes.
Proof : The proof follows simply by checking that this number is prefect or not.So for checking so
say N = 2p−1
.(2p
− 1). Then sum of its divisors,
σ(N) = (2p
− 1).(1 + 21
+ 22
+ ...2p−1
) = (2p
− 1).2p
= 2 × N
So N is a prefect number.
Note : 1.There is so much wastage of time in getting Merssene primes.So we want to get some
conditions on such p.By simply checking we can say if p is not a prime number then (2p
− 1) will
also not be prime number.
2. Two numbers m and n are said to be amicable if sum of the proper divisors of m is n and vice
versa.
σ(m) − m = n , σ(n) − n = m
σ(m) = σ(n) = m + n
Euler’s study of prefect numbers started from here.
3.Some properties of σ-
If a and b are two relative primes(no common factor except 1) then
σ(a × b) = σ(a) × σ(b).
Euler proved that Euclid’s sufficiency condition is also necessary condition for even prefect numbers.
Theorem 1.2(Euclid-Euler Theorem) :-
Statement : A number N is even prefect number if and only if N can be written in form N =
2p−1
.(2p
− 1) where 2p
− 1 is prime number.
3 IIT KANPUR
6. 2.3 Challanges in Theory :- Summer Project
Proof of this theorem follows from above property of σ.
2.3 Challanges in Theory :-
Prefect numbers known to today’s date are only even numbers and they are in finite numbers.So
there are two profound problems in theory of prefect numbers -
(i)Whether there are any odd prefect number or not.
(ii)Are there finite number of prefect numbers or they are infinitely many?
Also mathematicians could not prove the infinitude or finitude of Merssene primes.Problem
(i) would immediately follow from infinitude of Merssene primes.These problems proposed many
centuries ago and till now not answered.Mathematicians admitted to being stumped on the topic
for a large time but now they are running for finding conditions for odd prefect numbers assuming
that they exist.Great mathematicians J. J. Sylvester proved that if any odd prefect number exists
then it must have 3 different prime factors.
2.4 Further Research :-
Theorem 1.3 :- An odd prefect number must have at least 3 different prime factors.
Proof : case(1)(one prime factor)
Say N = Pα
(P is a prime number)
Then σ(N) = 2 × N
P α+1
P −1 = 2 × Pα
⇒ 2 × Pα
− Pα+1
= 1
Here LHS is divisible by Pα
but RHS is not .So such odd prefect is not possible.
Case(2)(two prime factors)
Say N = Pα
.Qβ
(P and Q are prime numbers)
Then, σ(N) = 2 × N = 2 × Pα
.Qβ
.
(1 + P + P2
+ ...Pα
).(1 + Q + Q2
+ ...Qβ
) = 2 × Pα
.Qβ
.
2 = (1 + 1
P + 1
P 2 + ... 1
P α ).(1 + 1
Q + 1
Q2 + ... 1
Qβ )
But RHS ≤ (1 + 1
3 + 1
9 + ... 1
3α ).(1 + 1
5 + ... 1
5β )
RHS ≤ (1 + 1
3 + 1
9 + ...).(1 + 1
5 + 1
25 + ...) ≤ 3
2 .5
4 < 2
So RHS = LHS
So such odd prefect number is not possible.
Recent Research - 1.An odd prefect can not be divisible by 105 = 3 × 5 × 7.
2. It must contain at least 8 different prime factors.
3. Smallest odd prefect number must exceed 103
00.
4. The second largest prime factor of an odd prefect number must exceeds 1000.
5. The sum of reciprocals of all odd prefect number is finite.
4 IIT KANPUR
7. Summer Project
3 Logarithms
Logarithms is Euler’s most favorite chapter and one of the newest subject of study included in this
book.Euler used logarithms in as a most important analysis tool in calculas,analytic number theory
and algebra etc.
Euler’s Definition of a Function -
A function of a variable quantity is an analytic expression composed in any way whatsoever of
the variable quantity and number or constant quantities.
−→ This is not a modern concept.Euler seemed to equate function with a formula. Before
Euler people Henry Briggs generated a log table of base 10 logarithm and that calculated the value
of base 10 log. of any number by linear interpolation.He calculated the values of logarirhms of
√
10,
√
10, ...8192 th square root of 10 and then approximated any desired value by these.
3.1 Logarithm and exponential series :-
Euler obtained the series expansions of exponentials and logarithms by using infinitesimal small and
infinitely large number concept.At that time no one could believe that such a concept could do such
a great research.
Series expansion for ax
- Let ω(> 0) be an infinitesimal small number so that,
aω
= 1 + (such that is also infinitesimal small)
So connect and ω by , = k × ω
Take j = x
ω (j is infinitely large and x is finite)
ax
= (aω
)j
= (1 + k · ω)j
So by binomial expansion ax
= 1 + j.k.x
j + j.(j−1)
2
k.x
j
2
+ ...
Now using approximations j.(j−1)
j2 = 1, j(j−1)(j−2)
j3 = 1 and so on. So ax
= 1 + k · x + k·x2
2.1 + ...
We can choose a base ‘a such that k = 1 and call that base is ‘e .
So now ex
= 1 + x
1! + x2
2! + x3
3! + ...
Series expansion of ln(1 + x) :- By previous result , ejω
= (1 + ω)j
ln(1 + ω)j
= j · ω
we can choose (1 + ω)j
=(1+x) so ln(1 + x) = j · ((1 + x)
1
j − 1)
Then by similar reasoning used above we get,
ln(1 + x) = x − x2
2 + x3
3 + ...
Clearly this series is convergent for x > 1 but it is not so easy to visualize that this series is also
convergent for 0 < x ≤ 1 because terms like x2
2 , x3
3 are growing rapidly.
So for visualizing this -
ln(1 − x) = −(x + x2
2 + x3
3 + ...)
so ln(1+x)
(1−x) = 2 × (x + x3
3 + x5
5 + ...)
Cosider f : (−1, 1) −→ (0, ∞)
So we can find that this series is convergent for all x ∈ (−1, 1).
Divergence of Harmonic Series - This is the Euler’s proof of divergence of harmonic series.
Putting X=1 in the expansion of ln(1 − x) we get-
ln(0) = −(1 + 1/2 + 1/3 + 1/4 + ...) ⇒ 1 + 1
2 + 1
3 + ... = ln(1
0 ) = ∞.
Hence proved that harmonic series diverges.
Euler-Mascheroni constant - Euler proved by repeating use of expansion of ln(1 + 1/n) that the
following limit exists,
γ(Euler-Mascheroni constant) = limn→∞ (
n
k=1(1/k) − ln(n + 1)) = 0.577218.
5 IIT KANPUR
8. Summer Project
This constant appears in many places in mathematics like pi.
4 Euler and Infinite Series
Jakob Bernoulli contributed in the study of infinite series as he gave exact sum of many infinite series
and prove some other’s divergence and convergence.Euler used these infinite series in his analysis.
He proposed a series called p-series -
Sp =
∞
k=1
1
kp = 1 + 1
2p + 1
3p + ...
For p=1 this series become harmonic series which diverges and for other values the sum of the series
is converging but not known at that time.Specially for p=2 the problem was called basel’s problem.
4.1 Euler’s solution of Basel’s problem -
He first approxiamted this series to a rapidly converging sequence.He used his favorite tool of analysis,
Consider, I =
1/2
0
−ln(1−t)
t dt
By doing series expansion and afterthat integrating we get,
I = 1
2 + 1
22·22 + 1
23·32 + ... =
∞
k=1
1
2k−1.k2 .
Now put z=1-t in I, I =
1/2
1
ln(z)
1−z dz
Now by doing binomial expansion of (1 − t)−1
and afterthat solving the integration we get,
I =
∞
k=1 1/k2
=
∞
k=1
1
2k−1.k2 + [ln(2)2
].
Exact sum of the series - Euler’s Assumption - Euler represented the infinite polynomial in
terms of their roots in same way as we do for finite polynomial.
P(x) = sinx
x = 1 − x2
3! + x4
5! − ...
As it is clear that ±nπ(n is any natural number) is a general solution of the P(x).
So P(x) = 1 − x2
π2 . 1 − x2
4π2 ....
By comparing coefficient of x2
in both the expansions of P(x) we get that ,
∞
k=1 1/k2
= π2
6 .
Proof of Wallis formula is very simple by using this representation,
simply by putting x=π/2 in the P(x) we get that-
2/π = 1.3.3.5.5.7.7....
2.2.4.4.6.6.8.8....
The sum of the p-series was still unknown for p > 2.However Euler found the expressions for exact
sum of series for even p’s.
Theorem 3.1
Statement :- If the nth
polynomial ,
f(y) = yn
− A · yn−1
+ B · yn−2
− C · yn−3
+ ... ± N
is factored as
f(y) = (y − r1).(y − r2)...(y − rn)
then
n
k=1
rk = A ,
n
k=1
(rk)2
= A ·
n
k=1
rk − 2B,
n
k=1
= A ·
n
k=1
(rk)2
− B ·
n
k=1
rk + 3C...and so on.
6 IIT KANPUR
9. 4.1 Euler’s solution of Basel’s problem - Summer Project
Observation - Consider a polynomial ,
R(x) = 1 − A · x2
+ B · x4
− ... ± N · x2n
can be written in the form
(1 − r1 · x2
) · (1 − r2 · x2
)...(1 − rn · x2
)
substitute y = 1
x2 so
1 − A.
1
y
+ B.
1
y2
− C.
1
y3
... ±
1
y2n
Now by multiplying both sides by yn
we get the same equation as in above theorem.So above results
can also be applied to R(x).
So again considering P(x),
A =
1
3!
, B =
1
5!
, C =
1
7!
... so on.
So
n
k=1
rk =
n
k=1
(kπ)2
=
1
3!
,
n
k=1
(rk)2
=
n
k=1
(kπ)4
= (
1
3!
)2
−
2
5!
⇒
∞
k=1
1
k4
=
π4
90
and so on.
This proof of Basel’s problem was not much acceptable at that time because of Euler’s assumption
so Euler gave two more rigorous proof of the problem.
Challanges : Sum of the p-series for odd p’s is still unknown and mathematicians are stumped on
this problem.
7 IIT KANPUR
10. Summer Project
5 Euler and Analytic Number Theory
Analytic number theory is said to be most difficult and profound branch of study in mathematics.This
is one of the most ancient branch of mathematics.Out of 13 volumes of the book Elements of Euclid
three volumes are devoted to number theory.In these volumes he described basic definitions and
basic theorems of number theory.
Euclid’s theorem 4.1 : No finite collection of prime numbers include them all.
Remarks : 1.There are infinitely many prime numbers.
2. There is at least one prime number between P(n) and 2 · 3 · 5 · 7...P(n) + 1 (here P(k) denotes
kth
prime).
All prime numbers are odd (except 2) so there are two families of primes,
4 × k + 1series − 5, 13, 17, 29...
4 × k − 1series − 3, 7, 11, 19, ...
Theorem 4.2 : There are infinitely many primes of 4 · k − 1 series.
Theorem 4.3 : A prime from 4 · k + 1 series can be uniquely written in the form a2
+ b2
(sum of
two prefect squares) but a prime from 4 · k − 1 series can not.
5.1 Enter Euler
Euler found sum of the many infinite series using his armours of analysis and properties of numbers.
He used his previous results to reach on conclusions.Ex.-
S =
1
15
+
1
63
+
1
80
+
1
255
+ ... =
7
4
−
π2
6
Proof - Ovserve that terms in the series are those reciprocal whose denominators are one less than
all prefect squares which simultaneously of other powers.Ex.16 = 42
= 24
, 81 = 34
= 92
Take the series,
π2
6
= 1 +
1
4
+
1
9
+
1
16
+
1
25
+
1
36
+
1
49
+
1
64
+
1
81
+
1
100
+
1
121
.
π2
6
−1 =
1
4
+
1
16
+
1
64
+ ... +
1
9
+
1
81
+
1
729
+ ... +
1
25
+
1
625
+ ... +
1
36
+
1
1296
+ ... +
1
7
+
1
49
+ ...
So ,
π2
6
− 1 =
1
3
+
1
8
+
1
24
+
1
35
+
1
48
+
1
49
... =
∞
k=1
1
k2 − 1
− S =
3
4
− S
S =
3
4
+ 1 −
π2
6
=
7
4
−
π2
6
Theorem 4.4 :
∞
k=1
1
ks
=
∞
p=2
1
1 − 1
ps
for s > 1 and p is prime.
8 IIT KANPUR
11. 5.1 Enter Euler Summer Project
This theorem states a great relation between primes numbers and p-series.We can prove this theorem
by continuosly eliminating the prime factors from p-series.
Take x = 1 +
1
2s
+
1
3s
+
1
4s
+
1
5s
+ ... (1)
x
2s
=
1
2s
+
1
4s
+
1
6s
+ ... (2)
By dividing eqn (1) and eqn (2), x · 1 −
1
2s
= 1 +
1
3s
+
1
5s
+
1
7s
+ ... (3)
x
3s
· 1 −
1
2s
=
1
3s
+
1
9s
+
1
15s
+ ... (4)
By dividing eqn (3) and eqn (4), x · 1 −
1
3s
(1 −
1
2s
) = 1 +
1
5S
+
1
7s
+
1
11s
+
1
13s
+ ... (5)
By continuing this process we will get
x ·
P
1 −
1
ps
= 1
So
x =
∞
k=1
1
ks
=
∞
p=2
1
1 − 1
ps
What about the density and distribution of prime numbers in integers ??This can be understand by
following theorem.
Theorem 4.5 : Statement -
p
1
p diverges.
Euler’s proof : Let M =
∞
k=1
1
k =
p
1
1− 1
p
ln(M) = −[ln 1 −
1
p
+ ln 1 −
1
3
+ ln 1 −
1
5
+ ...]
ln(M) =
1
2
+
1
2
×
1
22
+
1
3
×
1
23
+ ...
+
1
3
+
1
2
×
1
32
+
1
3
×
1
33
+ ...
+ 1
5 + 1
2 × 1
52 + 1
3 × 1
53 + ...
ln(M) = 1
2 + 1
3 + 1
5 + ... + 1
2 × 1
22 + 1
32 + 1
52 + ...
ln(M) = p
1
p + p
1
p2
2 + p
1
p3
3 + ... A = p
1
p ; B = p
1
p2 ; C = p
1
p3 ; and so on.
Lemma : L = B
2 + C
3 + D
4 + ... converges.
Observation-
∞
k=2
1
kn ≤
∞
1
1
xn dx
This observation simply follows the fact that
l+2
k=l+1
1
kn ≤
l+1
l
1
xn dx
So now -
∞
k=2
1
kn ≤ 1
n−1
Note that
p
1
pn ≤
p
1
p2 ≤
k=2
1
k2 ≤ 1.
9 IIT KANPUR
12. Summer Project
By using above results,
L = p
1
p2
2 + p
1
p3
3 + ... ≤ 1
2 ·
∞
k=2
1
k2 + 1
3 ·
∞
k=2
1
k3 + ... ≤ 1
2 + 1
3 · 1
2 + 1
4 · 1
3 + ...
So now L ≤ 1 , using this lemma Euler reached on the conclusion that if series M diverges then
series A would also diverge.Hence the sum of reciprocals of primes diverge to infinity.
6 Euler and Complex Variables
Journey of complex numbers begin with the problem of finding roots for equation x2
+ 1 = 0.If we
try to solve it we get x = ±
√
−1 .But the problem was that there is no such real number exists which
square is negative number.So the problem was discarded by saying that this equation is unsolvable
like other equation ex
+ 1 = 0 , cos(x) = 2 .But after some decades mathematician seemed that it
is unavoidable when dealing with real solution of cubic equation.
Theorem 5.1 : A real solution to the depressed cubic x3
= mx + n is given by
x =
n
2
+
n2
4
−
m3
27
1
3
+
n
2
−
n2
4
−
m3
27
1
3
Proof : Proof follows from letting the solution x = p
1
3 + q
1
3 and putting this in equation.By getting
the values of p and q from equation we can find solution.
Remarks : 1. A general cubic equation z3
+ az2
+ bZ + c = 0 can be tranformed into depressed
polynomial by putting z = x − a
3 .So now we can find the solutions of any cubic equation.
But the problem occured when mathematicians applied this theorem to equation x3
= 6x + 4.
By applying above theorem, x = 2 + 2
√
−1
1
3
+ 2 − 2
√
−1
1
3
.
But it was known that equation have three real roots.So mathematicians went deeper in the
subject and write theory for complex numbers and these number are used in solving many problems
in a very simple way.They said that this imaginary number
√
−1 can be treated as real number
while applying binary operations.
Theorem 5.1(De Moivre’s Theorem) : (cosθ ± isinθ)n
= cos(nθ) ± isin(nθ) .
Proof : Proof of the above theorem can be done simply by using mathematical induction.
Euler found nth
roots of any real and complex number using this theorem.Euler also used this the-
orem in finding the very famous series expansions of sin(x) and cos(x).
Theorem 5.2 :
cos(x) = 1 − x2
2! + x4
4! − x6
6! + ... and sin(x) = x − x3
3! + x5
5! − x7
7! + ...
Proof :
(cosθ + isinθ)n
= cos(nθ) + isin(nθ) (6)
(cosθ − isinθ)n
= cos(nθ) − isin(nθ) (7)
eqn(6) + eqn(7) ⇒ cos(nθ) =
(cosθ + isinθ)n
+ (cosθ − isinθ)n
2
. (8)
Take θ is very small and n is very large such that x = nθ is finite.
lim
θ→0
cos(θ) = 1 lim
θ→0
sin(θ)
θ = 1
So using binomial expansion in equation(3) we get,
10 IIT KANPUR
13. Summer Project
cos(x) = 1 −
n.(n − 1)
2!
·
x2
n2
+
n(n − 1)(n − 2)(n − 3)
4!
·
x4
4!
− ...
so
cos(x) = 1 −
x2
2!
+
x4
4!
−
x6
6!
+ ...
similarly we can obtain expression for sin(x).
Theorem 5.3 : For any real x,
eix
= cos(x) + isin(x)
. This is called Euler’s identity .
Proof : One proof simply follows from series expansions of sin(x) and cos(x).
Alternative proof by using calculas :
x = sin−1
y =
y
0
dv
√
1 − v2
put v = iz ,
x = i
0
−iy
dz
√
1 + z2
= iln( a + z2 + z)
y = sin(x) = iz so
√
1 + z2 = cos(x) Then x = iln(cos(x) − isin(x))
⇒ cos(x) + isin(x) = eix
.
Euler also gave solution of complex arguments of trigonometeric and logarithm functions.Euler found
out the logarithm of negative values of its argument.He found that there are infinitely many roots
exists for logarithm of any number as its argument.Further research makes complex numbers more
important to study.Using complex number we can prove the Fundamental Theorem of Algebra.
7 Euler and Algebra
According to Euler Algebra is the science which teaches how to determine unknown quantities by
means of those that are known.
Algebra is a branch of mathematics which uses complex number so frequently in its theory.
7.1 Eighteenth century and Algebra
There were two open problems in Algebra in eighteenth century.
1.First was to find out solutions of any nth
degree polynomial.
2.Second was to give a Fundamental theorem of Algebra (to prove or disprove it).
But there was no major progress in both of the problems.First one is remaining unsolved yet
today’s date and second one is solved in nineteenth century.Euler gave a method for solving quartry
equations but he could not give general method for nth
degree.
7.2 Euler and Fundamental theorem of Algebra
This was conjecture at that time that every polynomial with real coefficients can be written in the
product of real linear and real quadratic factors.(Fundamental theorem of Algebra)
11 IIT KANPUR
14. 7.3 Further research Summer Project
7.2.1 Theorem 6.1
Any quartic polynomial
P(x) = x4
+ Ax3
+ Bx2
+ Cx + D
where A,B,C,D are real ,can be decomposed in two real factors of second degree.
Proof : Transform P(x) in depressed quartic by tansformation x = y − A
4 .
So now cosider the new polynomia,
R(x) = x4
+ bx2
+ cx + d
case 1. c = 0
Now R(x) = x4
+ bx2
+ d .This is a quadratic in x2
.
(i)If b2
− 4d ≥ 0
then R(x) will have two real roots in x2
.So R(x) = x2
+ b−
√
b2−4d
2 · x2
+ b+
√
b2−4d
2 (ii)If b2
−4d ≤
0
4d ≥ b2
⇒ d ≥ 0 ⇒ 2
√
d − b ≥ 0 .
Observe that R(x) = (x2
+
√
d)2
− (x · 2
√
d − b)2
R(x) = (x2
+
√
d − x · 2
√
d − b)(x2
+
√
d + x · 2
√
d − b)
case 2. c = 0
Assume that we can write R(x) = (x2
+ ux + α)(x2
− ux + β)
If we succed in finding real u, α, β then we are done.
By comparing coefficints in R(x) we get,
b = α + β − u2
, c = u(β − α), d = αβ
as c = 0 ⇒ u = 0, (β − α) = 0
By eliminating α and β we get,
4d = (u2
+
c
u
+ b)(u2
−
c
u
+ b) ⇒ f(u) = u6
+ 2bu4
+ (b2
− 4d)u2
− c2
= 0
as u → ∞f(u) → ∞ and f(0) = −c2
≤ 0
So by IVP f(u) must have one real solution of u and from u we will get one pair of real values of
α β.
Remarks - 1.If P(x) is an odd degree polynomial then it must have one real solution.
P(x) = (x − a)Q(x) where a is a real number.
Now Q(x) is an even degree polynomial(degree one less from P(x)).
So if fourth degree polynomial decomposed in two quadratic factors and fifth degree also decomposed.
2.If we could prove decomposition for real polynomial of degree 2,4,8,...2n
... then we could prove for
any.
7.3 Further research
In nineteenth century Niels Abel proved that general fifth degree equation can not be solvable in the
form of algebric operations on the coefficients of equation.But the Fundamental theorem of Algebra
proved in nineteenth century by joint work of many mathematicians. Liouville’s Theorem - An
entire ,bounded complex function is constant.
Entire function - A function differentiable on all complex space.
Bounded function - If there ∃M ∈ such that |f(z)| ≤ M ∀Z ∈ Domain
12 IIT KANPUR
15. Summer Project
Lemma - If P(z) is not a constant complex polynomial then the equation P(z)=0 have at least one
solution.
Fundamental theorem of Algebra - Any nth
degree polynomial
P(z) = cnzn
+ cn−1zn−1
+ ...c0
can be factored into n complex linear factors.
Proof - By above lemma ,∃α1 ∈ C(complex space) such that
P(z) = (z − α1)Q(z)
∃α2 ∈ C such that
P(z) = (z − α1)(z − α2)R(z).
By repeating this process n times we get,
P(z) = cn(z − α1)(z − α2)...(z − αn)
. Hence the Fundamental theorem of Algebra has proved.
8 Euler and Combinatorics
Combinatorics is branch of Discrete Mathematics which primary obective is to count finite collection
of items.Ancient Indian mathematicians had a developed combinatorics theory than others.They had
a multiplication rule for counting the ways of doing any work. Multiplication Rule : If there
is a task which have two steps and first steps have n ways of completing it and second step have m
number of ways of completing it then the total number of ways in which the task can be completed
will be m × n. In most of the problems we use its more general case for n steps task for which the
formula can be obtaoned by induction formula
(k) = Nk × (k − 1)
here (k)= number of ways of completing first k steps of task and Nk=number of ways of doing
nth
step.
Problem 1.Numbers of subsets of r different items chosen from n distinguishable items?
Solution - Say C(n,r) are the number of ways of doing so.
We know number of permutations of r items chosen from n,= n!
r! (by multiplication rule).Now think
about it as a two step process.We can say that we first make a set of r objects and then arrange
them ,so by multiplication rule -
n(n − 1)(n − 2)...(n − r + 1) = C(n, r) · r!
Thus
C(n, r) =
n!
r!(n − r)!
.
Problem 2. Number of ways of choosing r items from n different type of items if there are plentiful
item of each type(basically we allow n items to be reused)?
Solution - We will struck this problem by an example.
Suppose there are four boxes which are different.They contain Red(R),Yellow(Y),White(W) and
Brown(B) balls plentiful.
Now we want to take 3 balls out of them.
One way is Red ,Red and White.We represent this by RR||W|. Here vertical bars represent the
divisions of boxes.
13 IIT KANPUR
16. Summer Project
Actually we do not have any connection with this that which color ball puts out because we are
sepearating boxes by order that first is Red and then Yellow and then White and last Brown.
So RR||W| ≡ XX||X|.
So we can move these 6 pieces randomly such that three of them are different from three other,so
number of ways =C(6,3) because if we choose any three then we are done.So by same reasoning we
can extend it to general problem such that there are n+r-1 total pieces and r X’s.Thus, the number
of ways=
C(n + r − 1, r).
Euler’s curious question : What is the total number of derangements of n different things?
Solution : Euler said,consider (n) is the number of derangements of n items.Ex.- (1) = 0, (2) =
1, (3) = 2, (4) = 9, (5) = 44etc.
Euler observed a relation between these numbers. Theorem 7.1 For n ≥ 3
(n) = (n − 1)[ (n − 1) + (n − 2)].
Proof : Suppose a,b,c,d... is correct arrangement.We can choose first item by (n-1) ways for de-
rangement. For a moment assume first item is b for simplicity.
case 1.If sequence starts with b,a... then we are obliged to derrange (n-2) items c,d,e...
(n − 2)
ways.
case 2.If second letter of sequence is a thenour aim is ultimately to derrange (n-1) letters a,c,d,e...
So
(n − 1)
ways.
So now total number of ways
(n) = (n − 1)[ (n − 1) + (n − 2)].
Theorem 7.2
(n) = n (n − 1) + (−1)n
.
Proof : Again Euler observed this pattern by his bright eyes.
(r) − r (r − 1) = −[ (r − 1) − (r − 1) (r − 2)].
Use repeatedly with r=n-1,n-2,n-3,...3 to get :-
(n) − n (n − 1) = (−1)n
.
Theorem 7.3
(n) = [1 −
1
1!
+
1
2!
−
1
3!
+ ...
(−1)n
n!
]
(n) = n (n − 1) + (−1)n−2
n (n − 1) = n(n − 1) (n − 2) − n(−1)n−3
...
n(n − 1)(n − 2)(n − 3)...3 (2) = n! (1) + n(n − 1)(n − 2)...3(−1)2
By adding them all we get the result.
14 IIT KANPUR
17. 8.1 Partition of a whole number Summer Project
8.1 Partition of a whole number
Representation of a number as the sum of other whole numbers.There are few special cases.
(i)Partition into different summands.
(ii)Partition into odd summands.
Theorem 7.4The number of partitions of a whole number as the sum of different summands are
same as the number of partition as the number of odd summands.
Proof : Let
Q(x) = (1 + x)(1 + x2
)(1 + x3
)...
The coefficients of xn
in the Q(x) represents the total number of ways in which n can be written as
the sum of different summands.Consider
R(x) =
1
(1 − x)(1 − x3)(1 − x5)...
R(x) = (1 + x + x2
+ x3
+ ...)(1 + x3
+ x6
+ x9
+ ...)(1 + x5
+ x1
0 + ...)...
Coefficient of xn
in R(x) is the total number of ways in which n can be written as the sum of odd
summands.
P(x) = (1 − x)(1 − x2
)(1 − x3
)(1 − x4
)...
P(x)Q(x) = (1 − x2
)(1 − x4
)(1 − x6
)...
1
Q(x)
=
P(x)
P(x)Q(x)
= (1 − x)(1 − x3
)(1 − x5
)...
Q(x) =
1
(1 − x)(1 − x3)(1 − x5)...
= R(x).
Hence proved.
15 IIT KANPUR