Lambert's Proof of the Irrationality of Pi
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Irrational Numbers Unit 4 Lesson 6 IRRATIONAL NUMBERS
Irrational Numbers Unit 4 Lesson 6 IRRATIONAL NUMBERS Students will be able to: Understand the meanings of Irrational Numbers Key Vocabulary: • Irrational Numbers • Examples of Rational Numbers and Irrational Numbers • Decimal expansion of Irrational Numbers • Steps for representing Irrational Numbers on number line IRRATIONAL NUMBERS A rational number is a number that can be expressed as a ratio or we can say that written as a fraction. Every whole number is a rational number, because any whole number can be written as a fraction. Numbers that are not rational are called irrational numbers. An Irrational Number is a real number that cannot be written as a simple fraction or we can say cannot be written as a ratio of two integers. The set of real numbers consists of the union of the rational and irrational numbers. If a whole number is not a perfect square, then its square root is irrational. For example, 2 is not a perfect square, and √2 is irrational. EXAMPLES OF RATIONAL NUMBERS AND IRRATIONAL NUMBERS Examples of Rational Number The number 7 is a rational number because it can be written as the 7 fraction . 1 The number 0.1111111….(1 is repeating) is also rational number 1 because it can be written as fraction . 9 EXAMPLES OF RATIONAL NUMBERS AND IRRATIONAL NUMBERS Examples of Irrational Numbers The square root of 2 is an irrational number because it cannot be written as a fraction √2 = 1.4142135…… Pi(휋) is also an irrational number. π = 3.1415926535897932384626433832795 (and more...) 22 The approx. value of = 3.1428571428571.. -
Metrical Diophantine Approximation for Quaternions
This is a repository copy of Metrical Diophantine approximation for quaternions. White Rose Research Online URL for this paper: https://eprints.whiterose.ac.uk/90637/ Version: Accepted Version Article: Dodson, Maurice and Everitt, Brent orcid.org/0000-0002-0395-338X (2014) Metrical Diophantine approximation for quaternions. Mathematical Proceedings of the Cambridge Philosophical Society. pp. 513-542. ISSN 1469-8064 https://doi.org/10.1017/S0305004114000462 Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ Under consideration for publication in Math. Proc. Camb. Phil. Soc. 1 Metrical Diophantine approximation for quaternions By MAURICE DODSON† and BRENT EVERITT‡ Department of Mathematics, University of York, York, YO 10 5DD, UK (Received ; revised) Dedicated to J. W. S. Cassels. Abstract Analogues of the classical theorems of Khintchine, Jarn´ık and Jarn´ık-Besicovitch in the metrical theory of Diophantine approximation are established for quaternions by applying results on the measure of general ‘lim sup’ sets. -
Algebraic Generality Vs Arithmetic Generality in the Controversy Between C
Algebraic generality vs arithmetic generality in the controversy between C. Jordan and L. Kronecker (1874). Frédéric Brechenmacher (1). Introduction. Throughout the whole year of 1874, Camille Jordan and Leopold Kronecker were quarrelling over two theorems. On the one hand, Jordan had stated in his 1870 Traité des substitutions et des équations algébriques a canonical form theorem for substitutions of linear groups; on the other hand, Karl Weierstrass had introduced in 1868 the elementary divisors of non singular pairs of bilinear forms (P,Q) in stating a complete set of polynomial invariants computed from the determinant |P+sQ| as a key theorem of the theory of bilinear and quadratic forms. Although they would be considered equivalent as regard to modern mathematics (2), not only had these two theorems been stated independently and for different purposes, they had also been lying within the distinct frameworks of two theories until some connections came to light in 1872-1873, breeding the 1874 quarrel and hence revealing an opposition over two practices relating to distinctive cultural features. As we will be focusing in this paper on how the 1874 controversy about Jordan’s algebraic practice of canonical reduction and Kronecker’s arithmetic practice of invariant computation sheds some light on two conflicting perspectives on polynomial generality we shall appeal to former publications which have already been dealing with some of the cultural issues highlighted by this controversy such as tacit knowledge, local ways of thinking, internal philosophies and disciplinary ideals peculiar to individuals or communities [Brechenmacher 200?a] as well as the different perceptions expressed by the two opponents of a long term history involving authors such as Joseph-Louis Lagrange, Augustin Cauchy and Charles Hermite [Brechenmacher 200?b]. -
Proofs, Sets, Functions, and More: Fundamentals of Mathematical Reasoning, with Engaging Examples from Algebra, Number Theory, and Analysis
Proofs, Sets, Functions, and More: Fundamentals of Mathematical Reasoning, with Engaging Examples from Algebra, Number Theory, and Analysis Mike Krebs James Pommersheim Anthony Shaheen FOR THE INSTRUCTOR TO READ i For the instructor to read We should put a section in the front of the book that organizes the organization of the book. It would be the instructor section that would have: { flow chart that shows which sections are prereqs for what sections. We can start making this now so we don't have to remember the flow later. { main organization and objects in each chapter { What a Cfu is and how to use it { Why we have the proofcomment formatting and what it is. { Applications sections and what they are { Other things that need to be pointed out. IDEA: Seperate each of the above into subsections that are labeled for ease of reading but not shown in the table of contents in the front of the book. ||||||||| main organization examples: ||||||| || In a course such as this, the student comes in contact with many abstract concepts, such as that of a set, a function, and an equivalence class of an equivalence relation. What is the best way to learn this material. We have come up with several rules that we want to follow in this book. Themes of the book: 1. The book has a few central mathematical objects that are used throughout the book. 2. Each central mathematical object from theme #1 must be a fundamental object in mathematics that appears in many areas of mathematics and its applications. -
Real Proofs of Complex Theorems (And Vice Versa)
REAL PROOFS OF COMPLEX THEOREMS (AND VICE VERSA) LAWRENCE ZALCMAN Introduction. It has become fashionable recently to argue that real and complex variables should be taught together as a unified curriculum in analysis. Now this is hardly a novel idea, as a quick perusal of Whittaker and Watson's Course of Modern Analysis or either Littlewood's or Titchmarsh's Theory of Functions (not to mention any number of cours d'analyse of the nineteenth or twentieth century) will indicate. And, while some persuasive arguments can be advanced in favor of this approach, it is by no means obvious that the advantages outweigh the disadvantages or, for that matter, that a unified treatment offers any substantial benefit to the student. What is obvious is that the two subjects do interact, and interact substantially, often in a surprising fashion. These points of tangency present an instructor the opportunity to pose (and answer) natural and important questions on basic material by applying real analysis to complex function theory, and vice versa. This article is devoted to several such applications. My own experience in teaching suggests that the subject matter discussed below is particularly well-suited for presentation in a year-long first graduate course in complex analysis. While most of this material is (perhaps by definition) well known to the experts, it is not, unfortunately, a part of the common culture of professional mathematicians. In fact, several of the examples arose in response to questions from friends and colleagues. The mathematics involved is too pretty to be the private preserve of specialists. -
Robert De Montessus De Ballore's 1902 Theorem on Algebraic
Robert de Montessus de Ballore’s 1902 theorem on algebraic continued fractions : genesis and circulation Hervé Le Ferrand ∗ October 31, 2018 Abstract Robert de Montessus de Ballore proved in 1902 his famous theorem on the convergence of Padé approximants of meromorphic functions. In this paper, we will first describe the genesis of the theorem, then investigate its circulation. A number of letters addressed to Robert de Montessus by different mathematicians will be quoted to help determining the scientific context and the steps that led to the result. In particular, excerpts of the correspondence with Henri Padé in the years 1901-1902 played a leading role. The large number of authors who mentioned the theorem soon after its derivation, for instance Nörlund and Perron among others, indicates a fast circulation due to factors that will be thoroughly explained. Key words Robert de Montessus, circulation of a theorem, algebraic continued fractions, Padé’s approximants. MSC : 01A55 ; 01A60 1 Introduction This paper aims to study the genesis and circulation of the theorem on convergence of algebraic continued fractions proven by the French mathematician Robert de Montessus de Ballore (1870-1937) in 1902. The main issue is the following : which factors played a role in the elaboration then the use of this new result ? Inspired by the study of Sturm’s theorem by Hourya Sinaceur [52], the scientific context of Robert de Montessus’ research will be described. Additionally, the correlation with the other topics on which he worked will be highlighted, -
The History of the Abel Prize and the Honorary Abel Prize the History of the Abel Prize
The History of the Abel Prize and the Honorary Abel Prize The History of the Abel Prize Arild Stubhaug On the bicentennial of Niels Henrik Abel’s birth in 2002, the Norwegian Govern- ment decided to establish a memorial fund of NOK 200 million. The chief purpose of the fund was to lay the financial groundwork for an annual international prize of NOK 6 million to one or more mathematicians for outstanding scientific work. The prize was awarded for the first time in 2003. That is the history in brief of the Abel Prize as we know it today. Behind this government decision to commemorate and honor the country’s great mathematician, however, lies a more than hundred year old wish and a short and intense period of activity. Volumes of Abel’s collected works were published in 1839 and 1881. The first was edited by Bernt Michael Holmboe (Abel’s teacher), the second by Sophus Lie and Ludvig Sylow. Both editions were paid for with public funds and published to honor the famous scientist. The first time that there was a discussion in a broader context about honoring Niels Henrik Abel’s memory, was at the meeting of Scan- dinavian natural scientists in Norway’s capital in 1886. These meetings of natural scientists, which were held alternately in each of the Scandinavian capitals (with the exception of the very first meeting in 1839, which took place in Gothenburg, Swe- den), were the most important fora for Scandinavian natural scientists. The meeting in 1886 in Oslo (called Christiania at the time) was the 13th in the series. -
On the Origin and Early History of Functional Analysis
U.U.D.M. Project Report 2008:1 On the origin and early history of functional analysis Jens Lindström Examensarbete i matematik, 30 hp Handledare och examinator: Sten Kaijser Januari 2008 Department of Mathematics Uppsala University Abstract In this report we will study the origins and history of functional analysis up until 1918. We begin by studying ordinary and partial differential equations in the 18th and 19th century to see why there was a need to develop the concepts of functions and limits. We will see how a general theory of infinite systems of equations and determinants by Helge von Koch were used in Ivar Fredholm’s 1900 paper on the integral equation b Z ϕ(s) = f(s) + λ K(s, t)f(t)dt (1) a which resulted in a vast study of integral equations. One of the most enthusiastic followers of Fredholm and integral equation theory was David Hilbert, and we will see how he further developed the theory of integral equations and spectral theory. The concept introduced by Fredholm to study sets of transformations, or operators, made Maurice Fr´echet realize that the focus should be shifted from particular objects to sets of objects and the algebraic properties of these sets. This led him to introduce abstract spaces and we will see how he introduced the axioms that defines them. Finally, we will investigate how the Lebesgue theory of integration were used by Frigyes Riesz who was able to connect all theory of Fredholm, Fr´echet and Lebesgue to form a general theory, and a new discipline of mathematics, now known as functional analysis. -
Download Report 2010-12
RESEARCH REPORt 2010—2012 MAX-PLANCK-INSTITUT FÜR WISSENSCHAFTSGESCHICHTE Max Planck Institute for the History of Science Cover: Aurora borealis paintings by William Crowder, National Geographic (1947). The International Geophysical Year (1957–8) transformed research on the aurora, one of nature’s most elusive and intensely beautiful phenomena. Aurorae became the center of interest for the big science of powerful rockets, complex satellites and large group efforts to understand the magnetic and charged particle environment of the earth. The auroral visoplot displayed here provided guidance for recording observations in a standardized form, translating the sublime aesthetics of pictorial depictions of aurorae into the mechanical aesthetics of numbers and symbols. Most of the portait photographs were taken by Skúli Sigurdsson RESEARCH REPORT 2010—2012 MAX-PLANCK-INSTITUT FÜR WISSENSCHAFTSGESCHICHTE Max Planck Institute for the History of Science Introduction The Max Planck Institute for the History of Science (MPIWG) is made up of three Departments, each administered by a Director, and several Independent Research Groups, each led for five years by an outstanding junior scholar. Since its foundation in 1994 the MPIWG has investigated fundamental questions of the history of knowl- edge from the Neolithic to the present. The focus has been on the history of the natu- ral sciences, but recent projects have also integrated the history of technology and the history of the human sciences into a more panoramic view of the history of knowl- edge. Of central interest is the emergence of basic categories of scientific thinking and practice as well as their transformation over time: examples include experiment, ob- servation, normalcy, space, evidence, biodiversity or force. -
European Influences in the Fine Arts: Melbourne 1940-1960
INTERSECTING CULTURES European Influences in the Fine Arts: Melbourne 1940-1960 Sheridan Palmer Bull Submitted in total fulfilment of the requirements of the degree ofDoctor ofPhilosophy December 2004 School of Art History, Cinema, Classics and Archaeology and The Australian Centre The University ofMelbourne Produced on acid-free paper. Abstract The development of modern European scholarship and art, more marked.in Austria and Germany, had produced by the early part of the twentieth century challenging innovations in art and the principles of art historical scholarship. Art history, in its quest to explicate the connections between art and mind, time and place, became a discipline that combined or connected various fields of enquiry to other historical moments. Hitler's accession to power in 1933 resulted in a major diaspora of Europeans, mostly German Jews, and one of the most critical dispersions of intellectuals ever recorded. Their relocation to many western countries, including Australia, resulted in major intellectual and cultural developments within those societies. By investigating selected case studies, this research illuminates the important contributions made by these individuals to the academic and cultural studies in Melbourne. Dr Ursula Hoff, a German art scholar, exiled from Hamburg, arrived in Melbourne via London in December 1939. After a brief period as a secretary at the Women's College at the University of Melbourne, she became the first qualified art historian to work within an Australian state gallery as well as one of the foundation lecturers at the School of Fine Arts at the University of Melbourne. While her legacy at the National Gallery of Victoria rests mostly on an internationally recognised Department of Prints and Drawings, her concern and dedication extended to the Gallery as a whole. -
Arxiv:1803.01386V4 [Math.HO] 25 Jun 2021
2009 SEKI http://wirth.bplaced.net/seki.html ISSN 1860-5931 arXiv:1803.01386v4 [math.HO] 25 Jun 2021 A Most Interesting Draft for Hilbert and Bernays’ “Grundlagen der Mathematik” that never found its way into any publi- Working-Paper cation, and 2 CVof Gisbert Hasenjaeger Claus-Peter Wirth Dept. of Computer Sci., Saarland Univ., 66123 Saarbrücken, Germany [email protected] SEKI Working-Paper SWP–2017–01 SEKI SEKI is published by the following institutions: German Research Center for Artificial Intelligence (DFKI GmbH), Germany • Robert Hooke Str.5, D–28359 Bremen • Trippstadter Str. 122, D–67663 Kaiserslautern • Campus D 3 2, D–66123 Saarbrücken Jacobs University Bremen, School of Engineering & Science, Campus Ring 1, D–28759 Bremen, Germany Universität des Saarlandes, FR 6.2 Informatik, Campus, D–66123 Saarbrücken, Germany SEKI Editor: Claus-Peter Wirth E-mail: [email protected] WWW: http://wirth.bplaced.net Please send surface mail exclusively to: DFKI Bremen GmbH Safe and Secure Cognitive Systems Cartesium Enrique Schmidt Str. 5 D–28359 Bremen Germany This SEKI Working-Paper was internally reviewed by: Wilfried Sieg, Carnegie Mellon Univ., Dept. of Philosophy Baker Hall 161, 5000 Forbes Avenue Pittsburgh, PA 15213 E-mail: [email protected] WWW: https://www.cmu.edu/dietrich/philosophy/people/faculty/sieg.html A Most Interesting Draft for Hilbert and Bernays’ “Grundlagen der Mathematik” that never found its way into any publication, and two CV of Gisbert Hasenjaeger Claus-Peter Wirth Dept. of Computer Sci., Saarland Univ., 66123 Saarbrücken, Germany [email protected] First Published: March 4, 2018 Thoroughly rev. & largely extd. (title, §§ 2, 3, and 4, CV, Bibliography, &c.): Jan. -
James Clerk Maxwell
Conteúdo Páginas Henry Cavendish 1 Jacques Charles 2 James Clerk Maxwell 3 James Prescott Joule 5 James Watt 7 Jean-Léonard-Marie Poiseuille 8 Johann Heinrich Lambert 10 Johann Jakob Balmer 12 Johannes Robert Rydberg 13 John Strutt 14 Michael Faraday 16 Referências Fontes e Editores da Página 18 Fontes, Licenças e Editores da Imagem 19 Licenças das páginas Licença 20 Henry Cavendish 1 Henry Cavendish Referência : Ribeiro, D. (2014), WikiCiências, 5(09):0811 Autor: Daniel Ribeiro Editor: Eduardo Lage Henry Cavendish (1731 – 1810) foi um físico e químico que investigou isoladamente de acordo com a tradição de Sir Isaac Newton. Cavendish entrou no seminário em 1742 e frequentou a Universidade de Cambridge (1749 – 1753) sem se graduar em nenhum curso. Mesmo antes de ter herdado uma fortuna, a maior parte das suas despesas eram gastas em montagem experimental e livros. Em 1760, foi eleito Fellow da Royal Society e, em 1803, um dos dezoito associados estrangeiros do Institut de France. Entre outras investigações e descobertas de Cavendish, a maior ocorreu em 1781, quando compreendeu que a água é uma substância composta por hidrogénio e oxigénio, uma reformulação da opinião de há milénios de que a água é um elemento químico básico. O químico inglês John Warltire (1725/6 – 1810) realizou uma Figura 1 Henry Cavendish (1731 – 1810). experiência em que explodiu uma mistura de ar e hidrogénio, descobrindo que a massa dos gases residuais era menor do que a da mistura original. Ele atribuiu a perda de massa ao calor emitido na reação. Cavendish concluiu que algum erro substancial estava envolvido visto que não acreditava que, dentro da teoria do calórico, o calor tivesse massa suficiente, à escala em análise.