10 Surprising Facts About Pi
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The Quest for Pi David H. Bailey, Jonathan M. Borwein, Peter B
The Quest for Pi David H. Bailey, Jonathan M. Borwein, Peter B. Borwein and Simon Plouffe June 25, 1996 Ref: Mathematical Intelligencer, vol. 19, no. 1 (Jan. 1997), pg. 50–57 Abstract This article gives a brief history of the analysis and computation of the mathematical constant π =3.14159 ..., including a number of the formulas that have been used to compute π through the ages. Recent developments in this area are then discussed in some detail, including the recent computation of π to over six billion decimal digits using high-order convergent algorithms, and a newly discovered scheme that permits arbitrary individual hexadecimal digits of π to be computed. D. Bailey: NASA Ames Research Center, Mail Stop T27A-1, Moffett Field, CA 94035-1000. E-mail: [email protected]. J. Borwein: Dept. of Mathematics and Statistics, Simon Fraser University Burnaby, BC V5A 1S6 Canada. Email: [email protected]. This work was supported by NSERC and the Shrum Endowment at Simon Fraser University. P. Borwein: Dept. of Mathematics and Statistics, Simon Fraser University Burnaby, BC V5A 1S6 Canada. Email: [email protected]. S. Plouffe: Dept. of Mathematics and Statistics, Simon Fraser University Burnaby, BC V5A 1S6 Canada. Email: [email protected]. 1 Introduction The fascinating history of the constant we now know as π spans several millennia, almost from the beginning of recorded history up to the present day. In many ways this history parallels the advancement of science and technology in general, and of mathematics and computer technology in particular. An overview of this history is presented here in sections one and two. -
History of the Numerical Methods of Pi
History of the Numerical Methods of Pi By Andy Yopp MAT3010 Spring 2003 Appalachian State University This project is composed of a worksheet, a time line, and a reference sheet. This material is intended for an audience consisting of undergraduate college students with a background and interest in the subject of numerical methods. This material is also largely recreational. Although there probably aren't any practical applications for computing pi in an undergraduate curriculum, that is precisely why this material aims to interest the student to pursue their own studies of mathematics and computing. The time line is a compilation of several other time lines of pi. It is explained at the bottom of the time line and further on the worksheet reference page. The worksheet is a take-home assignment. It requires that the student know how to construct a hexagon within and without a circle, use iterative algorithms, and code in C. Therefore, they will need a variety of materials and a significant amount of time. The most interesting part of the project was just how well the numerical methods of pi have evolved. Currently, there are iterative algorithms available that are more precise than a graphing calculator with only a single iteration of the algorithm. This is a far cry from the rough estimate of 3 used by the early Babylonians. It is also a different entity altogether than what the circle-squarers have devised. The history of pi and its numerical methods go hand-in-hand through shameful trickery and glorious achievements. It is through the evolution of these methods which I will try to inspire students to begin their own research. -
Regiões Circulares E O Número Pi
Universidade Federal de Goiás Instituto de Matemática e Estatística Programa de Mestrado Profissional em Matemática em Rede Nacional Regiões Circulares e o número Pi THIAGO VERÍSSIMO PEREIRA Goiânia 2013 THIAGO VERÍSSIMO PEREIRA Regiões Circulares e o número Pi Trabalho de Conclusão de Curso apresentado ao Programa de Pós–Graduação do Instituto de Matemática e Estatística da Universidade Federal de Goiás, como requisito parcial para obtenção do título de Mestre em matemática Área de concentração: Matemática do ensino básico. Orientador: Prof. Dr. José Yunier Bello Cruz Goiânia 2013 Dados Internacionais de Catalogação na Publicação (CIP) GPT/BC/UFG Pereira, Thiago Veríssimo. P436r Regiões circulares e o número pi [manuscrito] / Thiago Veríssimo Pereira. – 2013. 40 f. : il., figs., tabs. Orientador: Prof. Dr. José Yunier Bello Cruz. Dissertação (Mestrado) – Universidade Federal de Goiás, Instituto de Matemática e Estatística, 2013. Bibliografia. 1. Matemática, História da. 2. Geometria. 3. Pi. I. Título. CDU: 51:930.1 Todos os direitos reservados. É proibida a reprodução total ou parcial do trabalho sem autorização da universidade, do autor e do orientador(a). Thiago Veríssimo Pereira Licenciado em Matemática pela UnB. Professor da Secretaria de Educação do Distrito Federal e da rede particular desde 2009. A Deus e a minha família, em especial minha esposa, que me deram apoio e força para chegar até aqui, e a meus professores que me inspiraram. Agradecimentos A Deus, minha família, em especial a minha esposa, aos professores, tutores e coordenadores do IME-UFG pelo empenho e dedicação mostrados ao longo do curso, em especial ao Prof. Dr. José Yunier Bello Cruz e aos colegas de turma pelo apoio e compreensão nos momentos difíceis. -
36 Surprising Facts About Pi
36 Surprising Facts About Pi piday.org/pi-facts Pi is the most studied number in mathematics. And that is for a good reason. The number pi is an integral part of many incredible creations including the Pyramids of Giza. Yes, that’s right. Here are 36 facts you will love about pi. 1. The symbol for Pi has been in use for over 250 years. The symbol was introduced by William Jones, an Anglo-Welsh philologist in 1706 and made popular by the mathematician Leonhard Euler. 2. Since the exact value of pi can never be calculated, we can never find the accurate area or circumference of a circle. 3. March 14 or 3/14 is celebrated as pi day because of the first 3.14 are the first digits of pi. Many math nerds around the world love celebrating this infinitely long, never-ending number. 1/8 4. The record for reciting the most number of decimal places of Pi was achieved by Rajveer Meena at VIT University, Vellore, India on 21 March 2015. He was able to recite 70,000 decimal places. To maintain the sanctity of the record, Rajveer wore a blindfold throughout the duration of his recall, which took an astonishing 10 hours! Can’t believe it? Well, here is the evidence: https://twitter.com/GWR/status/973859428880535552 5. If you aren’t a math geek, you would be surprised to know that we can’t find the true value of pi. This is because it is an irrational number. But this makes it an interesting number as mathematicians can express π as sequences and algorithms. -
Slides from These Events
The wonderπ of March 14th, 2019 by Chris H. Rycroft Harvard University The circle Radius r Area = πr2 Circumference = 2πr The wonder of pi • The calculation of pi is perhaps the only mathematical problem that has been of continuous interest from antiquity to present day • Many famous mathematicians throughout history have considered it, and as such it provides a window into the development of mathematical thought itself Assumed knowledge 3 + = Hindu–Arabic Possible French origin Welsh origin 1st–4th centuries 14th century 16th century • Many symbols and thought processes that we take for granted are the product of millennia of development • Early explorers of pi had no such tools available An early measurement • Purchased by Henry Rhind in 1858, in Luxor, Egypt • Scribed in 1650BCE, and copied from an earlier work from ~ 2000BCE • One of the oldest mathematical texts in existence • Consists of fifty worked problems, the last of which gives a value for π Problem 24 (An example of Egyptian mathematical logic) A heap and its 1/7 part become 19. What is the heap? Then 1 heap is 7. (Guess an answer and see And 1/7 of the heap is 1. if it works.) Making a total of 8. But this is not the right answer, and therefore we must rescale 7 by the (Re-scale the answer to obtain the correct solution.) proportion of 19/8 to give 19 5 7 × =16 8 8 Problem 24 A heap and its 1/7 part become 19. What is the heap? • Modern approach using high-school algebra: let x be the size of the heap. -
Appendix I: on the Early History of Pi
Appendix I: On the Early History of Pi Our earliest source on 7r is a problem from the Rhind mathematical papyrus, one of the few major sources we have for the mathematics of ancient Egypt. Although the material on the papyrus in its present form comes from about 1550 B.C. during the Middle Kingdom, scholars believe that the mathematics of the document originated in the Old Kingdom, which would date it perhaps to 1900 B.C. The Egyptian source does not state an explicit value for 7r, but tells, instead, how to compute the area of a circle as the square of eight-ninths of the diameter. (One is reminded of the 'classical' problem of squaring the circle.) Quite different is the passage in I Kings 7, 23 (repeated in II Chronicles 4, 21), which clearly implies that one should multiply the diameter of a circle by 3 to find its circumference. It is worth noting, however, that the Egyptian and the Biblical texts talk about what are, a priori, different concepts: the diameter being regarded as known in both cases, the first text gives a factor used to produce the area of the circle and the other gives (by implication) a factor used to produce the circumference.1 Also from the early second millenium are Babylonian texts from Susa. In the mathematical cuneiform texts the standard factor for multiplying the diameter to produce the circumference is 3, but, according to some interpre tations, the Susa texts give a value of 31.2 Another source of ancient values of 7r is the class of Indian works known as the Sulba Sfitras, of which the four most important, and oldest are Baudhayana, .Apastamba, Katyayana and Manava (resp. -
Progress in Computing Pi, 250 BCE to the Present
Progress in Computing Pi, 250 BCE to the Present From Polygons to Elliptic Integrals Donald Byrd Woodrow Wilson Indiana Teaching Fellow and Research Technologies and School of Informatics Indiana University Bloomington rev. late April 2014 The history of computing π offers a high-level view of several aspects of mathematics and its history. It also says something about aspects of computing and the history of computers. And, since the practical value of more accurate approximations to π was exhausted by 1706, if not earlier, and the mathematical value of high-precision approximations is almost zero, it says something about human culture. In this article, I give background for and outline the four fundamentally different techniques (measurement and three methods of calculation) of finding approximations that have been used, and I discuss some surprising facts about what it really takes, other than infinite series that converge with astounding rapidity, to compute π to the trillions of decimal places that are now known. This article assumes mathematical background about the level of what used to be called “college algebra” but is now known as precalculus. Two non-obvious conventions it uses are: (1) Numbers of decimal places or approximate values for π in boldface italics indicate values that were world records when they were found. (2) References giving only a page number are to Beckmann [2]. The record for accuracy in computing approximations to π now stands at 12.1 trillion decimal places [15]. There’s a great deal of history behind this remarkable feat. Indeed, Beckmann comments in his fascinating book A History of Pi [2], “The history of π is a quaint little mirror of the history of man.” (p. -
Mathematical Amazements and Surprises on Pi-Day [Mathematik - - -Mag Man Eben!] O
Mathematical Amazements and Surprises on Pi-Day [Mathematik - - -mag man eben!] o. Univ. Prof. Dr. Alfred S. Posamentier Executive Director for Internationalization and Sponsored Programs Long Island University, New York Former Dean, School of Education Professor Emeritus of Mathematics Education The City College of The City University of New York This Is the ultimate special -Day 3.14-15 9:26:53 3.1415926535897932384626433832... George Washington was born on February 22, 1732. The string 02221732 occurs at position 9,039,149. This string occurs 3 times in the first 200M digits of , counting from the first digit after the decimal point. (The 3. is not counted. ) The string and surrounding digits: 3.14159265... 478149013493242974180222173249756451826 61284136 Where the symbol came from: • In 1706 William Jones (1675 – 1749) published his book, Synopsis Palmariorum Matheseos, where he used to represent the ratio of the circumference of a circle to its diameter. • In 1736, Leonhard Euler began using to represent the ratio of the circumference of a circle to its diameter. But not until he used the symbol in 1748 in his famous book Introductio in analysin infinitorum did the use of to represent the ratio of the circumference of a circle to its diameter become widespread. Diameter of circumcircle = 1 A A B C E B C B A D D C A F A B A B a L C 1/2 K x K D F C E B O J E E D I F 1 360 180 H G x D C 2 nn a The perimeter of the n-sided regular polygon sinxa 2 1 is then n times the side length, which makes 2 this perimeter 180 180 180 One -
Edray H Goins: Indiana Pi Bill.Html 1/6/14, 4:33 PM
Edray H Goins: Indiana Pi Bill.html 1/6/14, 4:33 PM Department of Mathematics search About Us Academic Programs Courses Resources Research Calendar and Events > Home Edray H Goins Home Dessins d'Enfants Indiana Pi Bill Notes and Publications Number Theory Seminar PRiME 2013 SUMSRI In Celebration of Pi Day: The History of the Indiana Pi Bill During the Spring of 2012, I taught an experimental course in the mathematics department at Purdue University called "Great Issues in Mathematics." Purdue established the Great Issues courses in the 1950's as "[having] the general purpose of serving as a 'bridge' between formal college courses and the continuing study and consideration of major issues by responsible college graduates." My class served nearly 100 students majoring in mathematics, physics, chemistry, and biology. In celebration of Pi Day, I'd like to present to you, as I did to my students a year ago, one of the Indiana legislature's most embarrassing moments: the year we nearly made the rationality of π a state law. What is π? The number π is defined as the ratio of the circumference of a circle to its diameter. The Old Testament of the Bible even discusses the number π. For example, I Kings 7:23 reads: "[King Solomon] made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high. It took a line of thirty cubits to measure around it." Solomon's Molten Sea Since a cubit is about 1.50 feet (about 0.46 meters), it's easy to see that π is approximately 3. -
Indiana Pols Forced to Eat Humble Pi: the Curious History of an Irrational Number
The History of the Number \π" On Lambert's Proof of the Irrationality of π House Bill #246: the \Indiana Pi Bill" Indiana Pols Forced to Eat Humble Pi: The Curious History of an Irrational Number Edray Herber Goins D´ıade Pi Instituto de Matem´aticas Pontificia Universidad Cat´olica de Valpara´ıso(PUCV) Department of Mathematics Purdue University March 14, 2018 PUCV Pi Day Celebration Indiana Pols Forced to Eat Humble Pi The History of the Number \π" On Lambert's Proof of the Irrationality of π House Bill #246: the \Indiana Pi Bill" Outline of Talk 1 The History of the Number \π" Introduction What is \π"? Approximations to π 2 On Lambert's Proof of the Irrationality of π Bessel Functions Laczkovich's Theorem Lambert's Theorem 3 House Bill #246: the \Indiana Pi Bill" Lindemann's Theorem Squaring the Circle A Bill for an Act Introducing a New Mathematical Truth PUCV Pi Day Celebration Indiana Pols Forced to Eat Humble Pi The History of the Number \π" On Lambert's Proof of the Irrationality of π House Bill #246: the \Indiana Pi Bill" Abstract In 1897, Indiana physician Edwin J. Goodwin believed he had discovered a way to square the circle, and proposed a bill to Indiana Representative Taylor I. Record which would secure Indiana's the claim to fame for his discovery. About the time the debate about the bill concluded, Purdue University professor Clarence A. Waldo serendipitously came across the claimed discovery, and pointed out its mathematical impossibility to the lawmakers. It had only be shown just 15 years before, by the German mathematician Ferdinand von Lindemann, that it was impossible to square the circle because π is an irrational number. -
Progress in Computing Pi, 250 BCE to the Present
Progress in Computing Pi, 250 BCE to the Present From Polygons to Elliptic Integrals Donald Byrd Research Technologies and School of Informatics Indiana University Bloomington and Woodrow Wilson Indiana Teaching Fellow rev. late September 2014 The history of computing π offers a high-level view of several aspects of mathematics and its history. It also says something about aspects of computing and the history of computers. Beyond that, since the practical value of more accurate approximations to π was exhausted by the turn of the 18th century and the mathematical value of extremely high-precision approximations is very limited, it says something about human culture. In this article, I give background for and outline the four fundamentally different techniques (measurement and three methods of calculation) of finding approximations that have been used, and I discuss both the mathematics and the computer science behind computing π to the trillions of decimal places that are now known. I assume mathematical background about the level of what used to be called “college algebra” but is now known as precalculus. Two non-obvious conventions are: (1) Numbers of decimal places or approximate values for π in boldface italics indicate values that were world records when they were found. (2) References giving only a page number are to Beckmann [2]. The record for accuracy in computing approximations to π now stands at 12.1 trillion decimal places [16]. There’s a great deal of history behind this remarkable feat. Indeed, Beckmann comments in his fascinating book A History of Pi [2], “The history of π is a quaint little mirror of the history of man.” (p. -
Circular Reasoning: Who First Proved That $ C/D $ Is a Constant?
CIRCULAR REASONING: WHO FIRST PROVED THAT C=d IS A CONSTANT? DAVID RICHESON Abstract. We answer the question: who first proved that C=d is a con- stant? We argue that Archimedes proved that the ratio of the circumfer- ence of a circle to its diameter is a constant independent of the circle and that the circumference constant equals the area constant (C=d = A=r2). He stated neither result explicitly, but both are implied by his work. His proof required the addition of two axioms beyond those in Euclid's Elements; this was the first step toward a rigorous theory of arc length. We also discuss how Archimedes's work coexisted with the 2000-year belief|championed by scholars from Aristotle to Descartes|that it is impossible to find the ratio of a curved line to a straight line. For a long time I was too embarrassed to ask the question in the title. The expression C = πd, which gives the relationship between the circumference and the diameter of a circle, is one of the few formulas known to almost all children and adults, regardless of how long they have been out of school. But who first proved it? I was sure that merely asking the question would tarnish my reputation as a mathematician and as a student of the history of mathematics. Surely the answer is well-known and easy to find. But I could not locate it|at least none of the standard reference books discussed the matter. When I finally summoned the courage to ask others I received two types of replies: either \It is obvious; all circles are similar" or \I don't know." The first of these is intriguing and has some merit; it may be why the invariance of C=d was discovered in so many different cultures.