DOCSLIB.ORG

The Essence of Mathematics Through Elementary Problems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Essence of Mathematics Through Elementary Problems The Essence of Mathematics Through Elementary Problems ALEXANDRE BOROVIK AND TONY GARDINER To access digital resources including: blog posts videos online appendices and to purchase copies of this book in: hardback paperback ebook editions Go to: https://www.openbookpublishers.com/product/979 Open Book Publishers is a non-profit independent initiative. We rely on sales and donations to continue publishing high-quality academic works. THE ESSENCE OF MATHEMATICS THROUGH ELEMENTARY PROBLEMS The Essence of Mathematics Through Elementary Problems Alexandre Borovik and Tony Gardiner http://www.openbookpublishers.com c 2019 Alexandre Borovik and Tony Gardiner This work is licensed under a Creative Commons Attribution 4.0 International license (CC BY 4.0). This license allows you to share, copy, distribute and transmit the work; to adapt the work and to make commercial use of the work providing attribution is made to the author (but not in any way that suggests that they endorse you or your use of the work). Attribution should include the following information: Alexandre Borovik and Tony Gardiner, The Essence of Mathematics through Elementary Problems. Cambridge, UK: Open Book Publishers, 2019. http://dx.doi.org/10.11647/OBP.0168 Further details about CC BY licenses are available at http://creativecommons.org/licenses/by/4.0/ All external links were active on 10/04/2019 and archived via the Internet Archive Wayback Machine: https://archive.org/web/ Every effort has been made to identify and contact copyright holders and any omission or error will be corrected if notification is made to the publisher. Digital material and resources associated with this volume are available at http://www.openbookpublishers.com/product/979#resources This is the third volume of the OBP Series in Mathematics: ISSN 2397-1126 (Print) ISSN 2397-1134 (Online) ISBN Paperback 9781783746996 ISBN Hardback: 9781783747009 ISBN Digital (PDF): 9781783747016 DOI: 10.11647/OBP.0168 Cover photo: Abstract Spiral Pattern (2015) by Samuel Zeller, https://unsplash.com/photos/j0g8taxHZa0 Cover design by Anna Gatti. Contents Preface v About this text xi I. Mental Skills 1 1.1 Mental arithmetic and algebra 2 1.1.1 Times tables. 2 1.1.2 Squares, cubes, and powers of 2. 2 1.1.3 Primes 5 1.1.4 Common factors and common multiples 5 1.1.5 The Euclidean algorithm 6 1.1.6 Fractions and ratio 7 1.1.7 Surds 9 1.2 Direct and inverse procedures 9 1.2.1 Factorisation 12 1.3 Structural arithmetic 12 1.4 Pythagoras' Theorem 13 1.4.1 Pythagoras' Theorem, trig for special angles, and CAST 14 1.4.2 Converses and Pythagoras' Theorem 16 1.4.3 Pythagorean triples 17 1.4.4 Sums of two squares 19 1.5 Visualisation 20 1.6 Trigonometry and radians 22 1.6.1 Sine Rule 22 1.6.2 Radians and spherical triangles 23 1.6.3 Polar form and sin(A+B) 27 1.7 Regular polygons and regular polyhedra 27 1.7.1 Regular polygons are cyclic 28 1.7.2 Regular polyhedra 28 1.8 Chapter 1: Comments and solutions 29 II. Arithmetic 51 2.1 Place value and decimals: basic structure 51 2.2 Order and factors 53 2.3 Standard written algorithms 53 2.4 Divisibility tests 54 2.5 Sequences 56 2.5.1 Triangular numbers 56 2.5.2 Fibonacci numbers 56 2.6 Commutative, associative and distributive laws 60 2.7 Infinite decimal expansions 61 2.8 The binary numeral system 64 2.9 The Prime Number Theorem 66 2.10 Chapter 2: Comments and solutions 69 III. Word Problems 91 3.1 Twenty problems which embody \3 ´ 1 “ 2" 93 3.2 Some classical examples 94 3.3 Speed and acceleration 95 3.4 Hidden connections 96 3.5 Chapter 3: Comments and solutions 97 IV. Algebra 111 4.1 Simultaneous linear equations and symmetry 112 4.2 Inequalities and modulus 115 4.2.1 Geometrical interpretation of modulus, of inequalities, and of modulus inequalities 115 4.2.2 Inequalities 117 4.3 Factors, roots, polynomials and surds 119 4.3.1 Standard factorisations 119 4.3.2 Quadratic equations 123 4.4 Complex numbers 126 4.5 Cubic equations 131 4.6 An extra 133 4.7 Chapter 4: Comments and solutions 134 V. Geometry 169 5.1 Comparing geometry and arithmetic 171 5.2 Euclidean geometry: a brief summary 173 5.3 Areas, lengths and angles 194 5.4 Regular and semi-regular tilings in the plane 196 5.5 Ruler and compasses constructions for regular polygons 199 5.6 Regular and semi-regular polyhedra 201 5.7 The Sine Rule and the Cosine Rule 206 5.8 Circular arcs and circular sectors 211 5.9 Convexity 217 5.10 Pythagoras' Theorem in three dimensions 217 5.11 Loci and conic sections 220 5.12 Cubes in higher dimensions 227 5.13 Chapter 5: Comments and solutions 230 VI. Infinity: recursion, induction, infinite descent 283 6.1 Proof by mathematical induction I 286 6.2 `Mathematical induction' and `scientific induction' 287 6.3 Proof by mathematical induction II 290 6.4 Infinite geometric series 297 6.5 Some classical inequalities 299 6.6 The harmonic series 304 6.7 Induction in geometry, combinatorics and number theory 311 6.8 Two problems 313 6.9 Infinite descent 314 6.10 Chapter 6: Comments and solutions 317 Simon Phillips Norton 1952{2019 In memoriam Preface Understanding mathematics cannot be transmitted by painless entertainment . actual contact with the content of living mathematics is necessary. The present book . is not a concession to the dangerous tendency toward dodging all exertion. Richard Courant (1888{1972) and Herbert Robbins (1915{2001) Preface to the first edition of What is mathematics? Interested students of mathematics, who seek insight into the \essence of the discipline", and who read more widely with a view to discovering what the subject is really about, may emerge with the justifiable impression of serious mathematics as an austere, but distant mountain range { accessible only to those who devote their lives to its exploration. And they may conclude that the beginner can only appreciate its rough outline through a haze of unbridgeable distance. The best popularisers sometimes manage to convey more than this { including hints of the human story behind recent developments, and the way different branches and results interact in unexpected ways; but the essence of mathematics still tends to remain elusive, and the picture they paint is inevitably a broad brush substitute for the detail of living mathematics. This collection takes a different approach. We start out by observing that mathematics is not a fixed entity { as one might unconsciously infer from the metaphor of an \austere mountain range". Mathematics is a mental universe, a work-in-progress in our collective imagination, which grows dramatically over time, and whose eventual extent would seem to be unconstrained { without any obvious limits. This boundlessness also works in reverse, when applied to small details: features which we thought we had understood are repeatedly filled in, or reinterpreted, in new ways to reveal finer and finer micro-structures. Hence whatever the essence of the discipline may be, it is clearly not something which can only be accessed through the complete exploration of some fixed corpus of knowledge. Rather the essential character of mathematics seems to be related to viii Essence of Mathematics • the kind of material that counts as mathematical, • the way this material is addressed, • the changes in perspective that occur as our understanding grows and deepens, and • the unexpected connections that regularly emerge between separate strands and layers. There are a number of books giving excellent general advice to prospective students about how university mathematics differs from school mathematics. In contrast, this collection { which we hope will be enjoyed by interested high school students and their teachers, by undergraduates and postgraduates, and by many others is more like a messy workshop than a polished exposition. Here the reader is asked to tackle a sequence of problems, to reflect on what they discover, and mostly to draw their own conclusions (though some key messages are explicitly discussed in the text, or in the solutions at the end of each chapter). This attempt to engage the reader as an active participant along the way is inevitably untidy { and may sometimes prove frustrating. In particular, whereas a polished exposition would break up the text with eye-catching diagrams, an untidy workshop will usually leave the reader to draw their own figures as an essential part of the struggle. This temporary untidiness and frustration is an integral part of \the essence" that we seek to capture { provided it leads to occasional glimpses of the power, and the elegance of mathematics. Young children and students of all ages regularly experience the power, the economy, the beauty, and the elegance of mathematics and of mathematical thinking on a small scale, through struggling with certain elementary results and problems (or groups of problems). For example, one of the problems we have included in Chapter 3 was mentioned explicitly in an interview1 with the leading Russian mathematician Vladimir Arnold (1937{2010): Interviewer: Please tell us a little bit about your early education. Were you already interested in mathematics as a child? Arnold: [. ] The first real mathematical experience I had was when our schoolteacher I.V. Morotzkin gave us the following problem [VA then formulated Problem 89 in Chapter 3]. I spent a whole day thinking on this oldie, and the solution (based on what are now called scaling arguments, dimensional analysis, or toric variety theory, depending on your taste) came as a revelation.
Load more

Recommended publications
  • Feasibility Study for Teaching Geometry and Other Topics Using Three-Dimensional Printers
    Feasibility Study For Teaching Geometry and Other Topics Using Three-Dimensional Printers Elizabeth Ann Slavkovsky A Thesis in the Field of Mathematics for Teaching for the Degree of Master of Liberal Arts in Extension Studies Harvard University October 2012 Abstract Since 2003, 3D printer technology has shown explosive growth, and has become significantly less expensive and more available. 3D printers at a hobbyist level are available for as little as $550, putting them in reach of individuals and schools. In addition, there are many “pay by the part” 3D printing services available to anyone who can design in three dimensions. 3D graphics programs are also widely available; where 10 years ago few could afford the technology to design in three dimensions, now anyone with a computer can download Google SketchUp or Blender for free. Many jobs now require more 3D skills, including medical, mining, video game design, and countless other fields. Because of this, the 3D printer has found its way into the classroom, particularly for STEM (science, technology, engineering, and math) programs in all grade levels. However, most of these programs focus mainly on the design and engineering possibilities for students. This thesis project was to explore the difficulty and benefits of the technology in the mathematics classroom. For this thesis project we researched the technology available and purchased a hobby-level 3D printer to see how well it might work for someone without extensive technology background. We sent designed parts away. In addition, we tried out Google SketchUp, Blender, Mathematica, and other programs for designing parts. We came up with several lessons and demos around the printer design.
    [Show full text]
  • Elementary Math Program
    North Shore Schools Elementary Mathematics Curriculum Our math program, Math in Focus, is based upon Singapore Math, which places an emphasis on developing proficiency with problem solving while fostering conceptual understanding and fundamental skills. Online resources are available at ThinkCentral using the username and password provided by the classroom teacher. The mathematics curriculum is aligned with the Common Core Learning Standards (CCLS). There are eight Mathematical Practices that set an expectation of understanding of mathematics and are the same for grades K-12. Mathematical Practices 1. Make sense of problems and persevere in 4. Model with mathematics. solving them. 5. Use appropriate tools strategically. 2. Reason abstractly and quantitatively. 6. Attend to precision. 3. Construct viable arguments and critique the 7. Look for and make use of structure. reasoning of others. 8. Look for and express regularity in repeated reasoning. The content at each grade level focuses on specific critical areas. Kindergarten Representing and Comparing Whole Numbers Students use numbers, including written numerals, to represent quantities and to solve quantitative problems, such as counting objects in a set; counting out a given number of objects; comparing sets or numerals; and modeling simple joining and separating situations with sets of objects, or eventually with equations such as 5 + 2 = 7 and 7 – 2 = 5. (Kindergarten students should see addition and subtraction equations, and student writing of equations in kindergarten is encouraged, but it is not required.) Students choose, combine, and apply effective strategies for answering quantitative questions, including quickly recognizing the cardinalities of small sets of objects, counting and producing sets of given sizes, counting the number of objects in combined sets, or counting the number of objects that remain in a set after some are taken away.
    [Show full text]
  • Math for Elementary Teachers
    Math for Elementary Teachers Math 203 #76908 Name __________________________________________ Spring 2020 Santiago Canyon College, Math and Science Division Monday 10:30 am – 1:25 pm (with LAB) Wednesday 10:30 am – 12:35 pm Instructor: Anne Hauscarriague E-mail: [email protected] Office: Home Phone: 714-628-4919 Website: www.sccollege.edu/ahauscarriague (Grades will be posted here after each exam) Office Hours: Mon: 2:00 – 3:00 Tues/Thurs: 9:30 – 10:30 Wed: 2:00 – 4:00 MSC/CraniumCafe Hours: Mon/Wed: 9:30 – 10:30, Wed: 4:00 – 4:30 Math 203 Student Learning Outcomes: As a result of completing Mathematics 203, the student will be able to: 1. Analyze the structure and properties of rational and real number systems including their decimal representation and illustrate the use of a representation of these numbers including the number line model. 2. Evaluate the equivalence of numeric algorithms and explain the advantages and disadvantages of equivalent algorithms. 3. Analyze multiple approaches to solving problems from elementary to advanced levels of mathematics, using concepts and tools from sets, logic, functions, number theory and patterns. Prerequisite: Successful completion of Math 080 (grade of C or better) or qualifying profile from the Math placement process. This class has previous math knowledge as a prerequisite and it is expected that you are comfortable with algebra and geometry. If you need review work, some resources are: School Zone Math 6th Grade Deluxe Edition, (Grade 5 is also a good review of basic arithmetic skills); Schaum’s Outline series Elementary Mathematics by Barnett Rich; www.math.com; www.mathtv.com; and/or www.KhanAcademy.com.
    [Show full text]
  • Elementary Mathematics Framework
    2015 Elementary Mathematics Framework Mathematics Forest Hills School District Table of Contents Section 1: FHSD Philosophy & Policies FHSD Mathematics Course of Study (board documents) …….. pages 2-3 FHSD Technology Statement…………………………………...... page 4 FHSD Mathematics Calculator Policy…………………………….. pages 5-7 Section 2: FHSD Mathematical Practices Description………………………………………………………….. page 8 Instructional Guidance by grade level band …………………….. pages 9-32 Look-for Tool……………………………………………………….. pages 33-35 Mathematical Practices Classroom Visuals……………………... pages 36-37 Section 3: FHSD Mathematical Teaching Habits (NCTM) Description………………………………………………………….. page 38 Mathematical Teaching Habits…………………...……………….. pages 39-54 Section 4: RtI: Response to Intervention RtI: Response to Intervention Guidelines……………………….. pages 55-56 Skills and Scaffolds………………………………………………... page 56 Gifted………………………………………………………………... page 56 1 Section 1: Philosophy and Policies Math Course of Study, Board Document 2015 Introduction A team of professional, dedicated and knowledgeable K-12 district educators in the Forest Hills School District developed the Math Course of Study. This document was based on current research in mathematics content, learning theory and instructional practices. The Ohio’s New Learning Standards and Principles to Actions: Ensuring Mathematical Success for All were the main resources used to guide the development and content of this document. While the Ohio Department of Education’s Academic Content Standards for School Mathematics was the main source of content, additional sources were used to guide the development of course indicators and objectives, including the College Board (AP Courses), Achieve, Inc. American Diploma Project (ADP), the Ohio Board of Regents Transfer Assurance Guarantee (TAG) criteria, and the Ohio Department of Education Program Models for School Mathematics. The Mathematics Course of Study is based on academic content standards that form an overarching theme for mathematics study.
    [Show full text]
  • The What and Why of Whole Number Arithmetic: Foundational Ideas from History, Language and Societal Changes
    Portland State University PDXScholar Mathematics and Statistics Faculty Fariborz Maseeh Department of Mathematics Publications and Presentations and Statistics 3-2018 The What and Why of Whole Number Arithmetic: Foundational Ideas from History, Language and Societal Changes Xu Hu Sun University of Macau Christine Chambris Université de Cergy-Pontoise Judy Sayers Stockholm University Man Keung Siu University of Hong Kong Jason Cooper Weizmann Institute of Science SeeFollow next this page and for additional additional works authors at: https:/ /pdxscholar.library.pdx.edu/mth_fac Part of the Science and Mathematics Education Commons Let us know how access to this document benefits ou.y Citation Details Sun X.H. et al. (2018) The What and Why of Whole Number Arithmetic: Foundational Ideas from History, Language and Societal Changes. In: Bartolini Bussi M., Sun X. (eds) Building the Foundation: Whole Numbers in the Primary Grades. New ICMI Study Series. Springer, Cham This Book Chapter is brought to you for free and open access. It has been accepted for inclusion in Mathematics and Statistics Faculty Publications and Presentations by an authorized administrator of PDXScholar. Please contact us if we can make this document more accessible: [email protected]. Authors Xu Hu Sun, Christine Chambris, Judy Sayers, Man Keung Siu, Jason Cooper, Jean-Luc Dorier, Sarah Inés González de Lora Sued, Eva Thanheiser, Nadia Azrou, Lynn McGarvey, Catherine Houdement, and Lisser Rye Ejersbo This book chapter is available at PDXScholar: https://pdxscholar.library.pdx.edu/mth_fac/253 Chapter 5 The What and Why of Whole Number Arithmetic: Foundational Ideas from History, Language and Societal Changes Xu Hua Sun , Christine Chambris Judy Sayers, Man Keung Siu, Jason Cooper , Jean-Luc Dorier , Sarah Inés González de Lora Sued , Eva Thanheiser , Nadia Azrou , Lynn McGarvey , Catherine Houdement , and Lisser Rye Ejersbo 5.1 Introduction Mathematics learning and teaching are deeply embedded in history, language and culture (e.g.
    [Show full text]
  • Title Studies on Hardware Algorithms for Arithmetic Operations
    Studies on Hardware Algorithms for Arithmetic Operations Title with a Redundant Binary Representation( Dissertation_全文 ) Author(s) Takagi, Naofumi Citation Kyoto University (京都大学) Issue Date 1988-01-23 URL http://dx.doi.org/10.14989/doctor.r6406 Right Type Thesis or Dissertation Textversion author Kyoto University Studies on Hardware Algorithms for Arithmetic Operations with a Redundant Binary Representation Naofumi TAKAGI Department of Information Science Faculty of Engineering Kyoto University August 1987 Studies on Hardware Algorithms for Arithmetic Operations with a Redundant Binary Representation Naofumi Takagi Abstract Arithmetic has played important roles in human civilization, especially in the area of science, engineering and technology. With recent advances of IC (Integrated Circuit) technology, more and more sophisticated arithmetic processors have become standard hardware for high-performance digital computing systems. It is desired to develop high-speed multipliers, dividers and other specialized arithmetic circuits suitable for VLSI (Very Large Scale Integrated circuit) implementation. In order to develop such high-performance arithmetic circuits, it is important to design hardware algorithms for these operations, i.e., algorithms suitable for hardware implementation. The design of hardware algorithms for arithmetic operations has become a very attractive research subject. In this thesis, new hardware algorithms for multiplication, division, square root extraction and computations of several elementary functions are proposed. In these algorithms a redundant binary representation which has radix 2 and a digit set {l,O,l} is used for internal computation. In the redundant binary number system, addition can be performed in a constant time i independent of the word length of the operands. The hardware algorithms proposed in this thesis achieve high-speed computation by using this property.
    [Show full text]
  • Survey of Modern Mathematical Topics Inspired by History of Mathematics
    Survey of Modern Mathematical Topics inspired by History of Mathematics Paul L. Bailey Department of Mathematics, Southern Arkansas University E-mail address: [email protected] Date: January 21, 2009 i Contents Preface vii Chapter I. Bases 1 1. Introduction 1 2. Integer Expansion Algorithm 2 3. Radix Expansion Algorithm 3 4. Rational Expansion Property 4 5. Regular Numbers 5 6. Problems 6 Chapter II. Constructibility 7 1. Construction with Straight-Edge and Compass 7 2. Construction of Points in a Plane 7 3. Standard Constructions 8 4. Transference of Distance 9 5. The Three Greek Problems 9 6. Construction of Squares 9 7. Construction of Angles 10 8. Construction of Points in Space 10 9. Construction of Real Numbers 11 10. Hippocrates Quadrature of the Lune 14 11. Construction of Regular Polygons 16 12. Problems 18 Chapter III. The Golden Section 19 1. The Golden Section 19 2. Recreational Appearances of the Golden Ratio 20 3. Construction of the Golden Section 21 4. The Golden Rectangle 21 5. The Golden Triangle 22 6. Construction of a Regular Pentagon 23 7. The Golden Pentagram 24 8. Incommensurability 25 9. Regular Solids 26 10. Construction of the Regular Solids 27 11. Problems 29 Chapter IV. The Euclidean Algorithm 31 1. Induction and the Well-Ordering Principle 31 2. Division Algorithm 32 iii iv CONTENTS 3. Euclidean Algorithm 33 4. Fundamental Theorem of Arithmetic 35 5. Infinitude of Primes 36 6. Problems 36 Chapter V. Archimedes on Circles and Spheres 37 1. Precursors of Archimedes 37 2. Results from Euclid 38 3. Measurement of a Circle 39 4.
    [Show full text]
  • Math 3010 § 1. Treibergs First Midterm Exam Name
    Math 3010 x 1. First Midterm Exam Name: Solutions Treibergs February 7, 2018 1. For each location, fill in the corresponding map letter. For each mathematician, fill in their principal location by number, and dates and mathematical contribution by letter. Mathematician Location Dates Contribution Archimedes 5 e β Euclid 1 d δ Plato 2 c ζ Pythagoras 3 b γ Thales 4 a α Locations Dates Contributions 1. Alexandria E a. 624{547 bc α. Advocated the deductive method. First man to have a theorem attributed to him. 2. Athens D b. 580{497 bc β. Discovered theorems using mechanical intuition for which he later provided rigorous proofs. 3. Croton A c. 427{346 bc γ. Explained musical harmony in terms of whole number ratios. Found that some lengths are irrational. 4. Miletus D d. 330{270 bc δ. His books set the standard for mathematical rigor until the 19th century. 5. Syracuse B e. 287{212 bc ζ. Theorems require sound definitions and proofs. The line and the circle are the purest elements of geometry. 1 2. Use the Euclidean algorithm to find the greatest common divisor of 168 and 198. Find two integers x and y so that gcd(198; 168) = 198x + 168y: Give another example of a Diophantine equation. What property does it have to be called Diophantine? (Saying that it was invented by Diophantus gets zero points!) 198 = 1 · 168 + 30 168 = 5 · 30 + 18 30 = 1 · 18 + 12 18 = 1 · 12 + 6 12 = 3 · 6 + 0 So gcd(198; 168) = 6. 6 = 18 − 12 = 18 − (30 − 18) = 2 · 18 − 30 = 2 · (168 − 5 · 30) − 30 = 2 · 168 − 11 · 30 = 2 · 168 − 11 · (198 − 168) = 13 · 168 − 11 · 198 Thus x = −11 and y = 13 .
    [Show full text]
  • CSU200 Discrete Structures Professor Fell Integers and Division
    CSU200 Discrete Structures Professor Fell Integers and Division Though you probably learned about integers and division back in fourth grade, we need formal definitions and theorems to describe the algorithms we use and to very that they are correct, in general. If a and b are integers, a ¹ 0, we say a divides b if there is an integer c such that b = ac. a is a factor of b. a | b means a divides b. a | b mean a does not divide b. Theorem 1: Let a, b, and c be integers, then 1. if a | b and a | c then a | (b + c) 2. if a | b then a | bc for all integers, c 3. if a | b and b | c then a | c. Proof: Here is a proof of 1. Try to prove the others yourself. Assume a, b, and c be integers and that a | b and a | c. From the definition of divides, there must be integers m and n such that: b = ma and c = na. Then, adding equals to equals, we have b + c = ma + na. By the distributive law and commutativity, b + c = (m + n)a. By closure of addition, m + n is an integer so, by the definition of divides, a | (b + c). Q.E.D. Corollary: If a, b, and c are integers such that a | b and a | c then a | (mb + nc) for all integers m and n. Primes A positive integer p > 1 is called prime if the only positive factor of p are 1 and p.
    [Show full text]
  • History of Mathematics Log of a Course
    History of mathematics Log of a course David Pierce / This work is licensed under the Creative Commons Attribution–Noncommercial–Share-Alike License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ CC BY: David Pierce $\ C Mathematics Department Middle East Technical University Ankara Turkey http://metu.edu.tr/~dpierce/ [email protected] Contents Prolegomena Whatishere .................................. Apology..................................... Possibilitiesforthefuture . I. Fall semester . Euclid .. Sunday,October ............................ .. Thursday,October ........................... .. Friday,October ............................. .. Saturday,October . .. Tuesday,October ........................... .. Friday,October ............................ .. Thursday,October. .. Saturday,October . .. Wednesday,October. ..Friday,November . ..Friday,November . ..Wednesday,November. ..Friday,November . ..Friday,November . ..Saturday,November. ..Friday,December . ..Tuesday,December . . Apollonius and Archimedes .. Tuesday,December . .. Saturday,December . .. Friday,January ............................. .. Friday,January ............................. Contents II. Spring semester Aboutthecourse ................................ . Al-Khw¯arizm¯ı, Th¯abitibnQurra,OmarKhayyám .. Thursday,February . .. Tuesday,February. .. Thursday,February . .. Tuesday,March ............................. . Cardano .. Thursday,March ............................ .. Excursus.................................
    [Show full text]
  • The History of the Primality of One—A Selection of Sources
    THE HISTORY OF THE PRIMALITY OF ONE|A SELECTION OF SOURCES ANGELA REDDICK, YENG XIONG, AND CHRIS K. CALDWELLy This document, in an altered and updated form, is available online: https:// cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.html. Please use that document instead. The table below is a selection of sources which address the question \is the number one a prime number?" Whether or not one is prime is simply a matter of definition, but definitions follow use, context and tradition. Modern usage dictates that the number one be called a unit and not a prime. We choose sources which made the author's view clear. This is often difficult because of language and typographical barriers (which, when possible, we tried to reproduce for the primary sources below so that the reader could better understand the context). It is also difficult because few addressed the question explicitly. For example, Gauss does not even define prime in his pivotal Disquisitiones Arithmeticae [45], but his statement of the fundamental theorem of arithmetic makes his stand clear. Some (see, for example, V. A. Lebesgue and G. H. Hardy below) seemed ambivalent (or allowed it to depend on the context?)1 The first column, titled `prime,' is yes when the author defined one to be prime. This is just a raw list of sources; for an evaluation of the history see our articles [17] and [110]. Any date before 1200 is an approximation. We would be glad to hear of significant additions or corrections to this list. Date: May 17, 2016. 2010 Mathematics Subject Classification.
    [Show full text]
  • Mathematical Constants and Sequences
    Mathematical Constants and Sequences a selection compiled by Stanislav Sýkora, Extra Byte, Castano Primo, Italy. Stan's Library, ISSN 2421-1230, Vol.II. First release March 31, 2008. Permalink via DOI: 10.3247/SL2Math08.001 This page is dedicated to my late math teacher Jaroslav Bayer who, back in 1955-8, kindled my passion for Mathematics. Math BOOKS | SI Units | SI Dimensions PHYSICS Constants (on a separate page) Mathematics LINKS | Stan's Library | Stan's HUB This is a constant-at-a-glance list. You can also download a PDF version for off-line use. But keep coming back, the list is growing! When a value is followed by #t, it should be a proven transcendental number (but I only did my best to find out, which need not suffice). Bold dots after a value are a link to the ••• OEIS ••• database. This website does not use any cookies, nor does it collect any information about its visitors (not even anonymous statistics). However, we decline any legal liability for typos, editing errors, and for the content of linked-to external web pages. Basic math constants Binary sequences Constants of number-theory functions More constants useful in Sciences Derived from the basic ones Combinatorial numbers, including Riemann zeta ζ(s) Planck's radiation law ... from 0 and 1 Binomial coefficients Dirichlet eta η(s) Functions sinc(z) and hsinc(z) ... from i Lah numbers Dedekind eta η(τ) Functions sinc(n,x) ... from 1 and i Stirling numbers Constants related to functions in C Ideal gas statistics ... from π Enumerations on sets Exponential exp Peak functions (spectral) ..
    [Show full text]