Π and Its Computation Through the Ages
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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. -
Section 3.3: Circles 3
Section 3.3: Circles 3 Contents at a Glance Pretest 167 Section Challenge 172 Lesson A: What’s in a Circle? 173 Lesson B: What is Pi? 187 Lesson C: Circumference of a Circle 197 Lesson D: Area of a Circle 211 Section Summary 225 Learning Outcomes By the end of this section you will be better able to: • identify the components and characteristics of circles. • describe the relationships between the parts of a circle. • construct circles using geometric tools. • apply a formula to fi nd the circumference and area of a circle. • solve problems that involve fi nding the circumference or area of a circle. • solve problems involving the area of triangles, parallelograms, and/or circles. © Open School BC MATH 7 eText 165 Module 3, Section 3 166 MATH 7 eText © Open School BC Module 3, Section 3 Pretest 3.3 Pretest 3.3 Complete this pretest if you think that you already have a strong grasp of the topics and concepts covered in this section. Mark your answers using the key found at the end of the module. If you get all the answers correct (100%), you may decide that you can omit the lesson activities. 1. Fill in the blanks: a. The ____________________ is a point inside a circle from which all points on the circumference are the same distance. b. The ________________ is the longest chord that can be drawn inside a circle. c. The diameter is ____________________ times as large as the radius. d. The ____________________ is the distance around the circle. e. A diameter is a line segment drawn from one point on a circle to another point that passes through the ____________________. -
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. -
Daf Ditty Eruvin 14 Pi, Revised X2
Daf Ditty Eruvin 14: Pi, the Ratio of God “Human beings, vegetables, or cosmic dust, we all dance to a mysterious tune intoned in the distance by an invisible player.” – Albert Einstein “The world isn't just the way it is. It is how we understand it, no? And in understanding something, we bring something to it, no? Doesn't that make life a story?” "Love is hard to believe, ask any lover. Life is hard to believe, ask any scientist. God is hard to believe, ask any believer... be excessively reasonable and you risk throwing out the universe with the bathwater." Yann Martel, Life of Pi 1 The mishna continues: If the cross beam is round, one considers it as though it were square. The Gemara asks: Why do I need this clause as well? Similar cases were already taught in the mishna. The Gemara answers: It was necessary to teach the last clause of this section, i.e., the principle that any circle with a circumference of three handbreadths is a handbreadth in diameter. 2 The Gemara asks: From where are these matters, this ratio between circumference and diameter, derived? Rabbi Yoḥanan said that the verse said with regard to King Solomon: And he made the molten sea of ten cubits from 23 גכ את ַַַויּﬠֶשׂ - ָוּ,מצםיַּה ָ:ק ֶֶﬠשׂ ר אָ בּ מַּ הָ הָ מַּ אָ בּ ר brim to brim, round in compass, and the height thereof תוֹפדשּׂﬠ ְִָמַ - ﬠוֹתְשׂפ סלָגָ ִ,יבָבֹ בּשְׁמחו ַהמֵָּאָ ָ ַהמֵָּאָ בּשְׁמחו ִ,יבָבֹ סלָגָ ﬠוֹתְשׂפ was five cubits; and a line of thirty cubits did compass וֹקמ ,וֹתָ וקוה ו( קְ ָ )ו םיִשׁhְ שׁ אָ בּ מַּ ,הָ סָ י בֹ וֹתֹ א וֹתֹ א בֹ סָ י ,הָ מַּ אָ בּ םיִשׁhְ שׁ )ו ָ קְ ו( וקוה ,וֹתָ וֹקמ .it round about ִי.בָסב I Kings 7:23 “And he made a molten sea, ten cubits from the one brim to the other: It was round all about, and its height was five cubits; and a line of thirty cubits did circle it roundabout” The Gemara asks: But isn’t there its brim that must be taken into account? The diameter of the sea was measured from the inside, and if its circumference was measured from the outside, this ratio is no longer accurate. -
Approximation of Π
Lund University/LTH Bachelor thesis FMAL01 Approximation of π Abstract The inverse tangent function can be used to approximate π. When approximating π the convergence of infinite series or products are of great importance for the accuracy of the digits. This thesis will present historical formulas used to approximate π, especially Leibniz–Gregory and Machin’s. Also, the unique Machin two-term formulas will be presented and discussed to show that Machin’s way of calculate π have had a great impact on calculating this number. Madeleine Jansson [email protected] Mentor: Frank Wikstr¨om Examinator: Tomas Persson June 13, 2019 Contents 1 Introduction 2 2 History 2 2.1 – 500 Anno Domini . .2 2.2 1500 – 1800 Anno Domini . .5 2.3 1800 – Anno Domini . .6 2.4 Proofs . .7 2.4.1 Vi`ete’sformula . .7 2.4.2 Wallis’s formula . .9 2.4.3 Irrationality of π ............................. 11 3 Mathematical background 13 3.1 Precision and accuracy . 13 3.2 Convergence of series . 14 3.3 Rounding . 15 4 Gregory–Leibniz formula 17 4.1 History . 18 4.2 Proof . 18 4.3 Convergence . 20 4.4 x = √1 ....................................... 21 3 5 Machin’s formula 22 5.1 History . 22 5.2 Proof . 23 5.3 Convergence . 24 6 Machin-like formulas 26 6.1 Machin-like formula . 27 6.2 Two-term formula . 27 6.3 Complex factors . 28 6.3.1 Euler’s formula . 30 6.3.2 Machin’s formula . 30 6.3.3 Hermann’s formula . 30 6.3.4 Hutton’s formula . -
Hronologija Izračunavanja Π Prije 1400
Hronologija izračunavanja π Prije 1400 Decimalna mjesta Datum Ko Formulacija Vrijednost π (svijetski rekord podebljan) 2000? 2 Stari Egipćani 3.16045... 1 BCЕ 4 × (8 / 9) 2000? Babilonci 3 + 1 / 8 3.125 1 BCЕ 1200? Kina 3 0 BCЕ 550? Biblija "...rastopljeno more, deset lakata od jednog kraja do 3 0 BCЕ (1 Kraljevi 7:23) drugog: bilo je sveobuhvatno, ... linija od trideset lakata ga je obuhvatila u cijelosti..." Prvi Helen koji se 434 Anaksagora nije dao nikakvo bavio kvadraturom Kompas i lenjir 0 BCE rešenje kruga je Anaksagora Sulbasutras, 350? (6/(2 + 2 )) 3.088311 … 0 BCЕ indijski sveti tekst 2 √ c. 250 223 / 71 < π < 22 / 7 3.140845...<π< 3.142857... 2 BCE Arhimed 15 25 / 8 3.125 1 BCE Marko Vitruvije 5 Liu Xin Tačna metoda je nepoznata 3.1457 2 130 Zhang Heng 10 = 3.162277…. 3.146551... 2 730/232 √ 150 Ptolomej 377 / 120 3.141666... 3 250 Wang Fan 142 / 45 3.155555... 1 3.141024 < π < 3.142074 263 Liu Hui 3927 / 1250 3.14159 5 400 He Chengtia 111035 / 35329 3.142885... 2 3.1415926 < π < 3.1415927 Zuov 480 Zu Chongzhi odnos 355 / 113 3.1415926 7 499 Aryabhata 62832 / 20000 3.1416 3 640 Brahmagupta 10 3.162277... 1 800 Al Khwarizmi 3.1416 3 √ 1150 Bhāskara II 3927/1250 i 754/240 3.1416 3 1220 Fibonači 3.141818 3 1320 Zhao Youqin 3.1415926 7 Od 1400 pa nadalje Decimalna mjesta (svijetski Datum Ko Bilješka rekord podebljan) Svizapisiod1400.godinenavodetačanbrojdecimalnihmesta. Madhava Vjerovatno je otkrivena pomoću beskonačnog reda π. -
Computer Oral History Collection, 1969-1973, 1977
Computer Oral History Collection, 1969-1973, 1977 Interviewee: John W. Wrench Interviewe: Richard R. Mertz Date of Interview: November 17, 1970 Mertz: This is an interview with Dr. John Wrench at the Naval Ship Research and Development Center, formerly the David Taylor Model Basin, conducted on November the seventeenth 1970 in Dr. Wrench's office. The interviewer is Richard Mertz. Dr. Wrench would you like to describe for us your early years, early training and development, what led you into mathematics as a field of endeavor? Wrench: My interest in mathematics was essentially motivated by an excellent teacher that I had in high school in one of the suburbs of Buffalo, Hamburg, New York. I was encouraged to go beyond the regular course work at that time and when I entered the University at Buffalo as a freshman, I found that I had sufficient preparation in the college subjects that I could anticipate sophomore courses and actually pass the freshman courses without going to formal class. In the University of Buffalo, I was particularly fortunate in having access to the files in the library stacks and files as a freshman--and I subsequently found that that was not a regularly approved procedure--so I did have the advantage of reading widely even from the first year that I was in college. Mertz: May I ask you to go back one step before this. In your family have there been any mathematicians before or scientists? Wrench: Not to my knowledge. I did have an uncle, my mother's brother, who was mathematically inclined; however, never formally in the field. -
Computers and Mathematics with Applications the Empirical Quest for Π
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Computers and Mathematics with Applications 56 (2008) 2772–2778 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa The empirical quest for π Terence Tai-Leung Chong Department of Economics, The Chinese University of Hong Kong, Shatin, Hong Kong article info a b s t r a c t Article history: This paper proposes a statistical method to obtain the value of π from empirical economic Received 28 February 2008 and financial data. It is the first study ever to link π to human behavior. It is shown that Received in revised form 19 June 2008 virtually any economic time series, such as stock indices, exchange rates and the GDP data Accepted 10 July 2008 can be used to retrieve the value of π. © 2008 Elsevier Ltd. All rights reserved. Keywords: π Bivariate normal random variables Random walk 1. Introduction The quest for π has a long history.1 The fact that the ratio of a circle’s circumference to its diameter is a constant has long been known. The ratio first enters human consciousness in Egypt. The Egyptian Rhind Papyrus, dating back to 1650 B.C., provides good evidence that π equals 4(8/9)2 ≈ 3.16. In the Bible, there are also records of the value of π. The biblical value2 of π is 3. In ancient Greece, the first approximation of π was obtained by Archimedes (287–212 B.C.), who showed 223 22 that 71 < π < 7 . -
Approximating Pi
Approximating Pi Problems from the History of Mathematics Lecture 15 | March 14, 2018 Brown University Introduction The oldest approximations to π appear in the preserved works of the ancient Babylonians and Egyptians. One example problem (from the Rhind Papyrus) follows: Problem (RMP 50): What is the area of a field with a diameter of 9 khet? 1 Solution: Take 9 from the diameter, leaving 8. Multiply this number by itself. The answer is 64 setjat (square khet). We conclude that 9 2 256 64 ≈ π( 2 ) =) π ≈ 81 ≈ 3:16049: Caution: It's easy to look at approximations like these and assume too much of the state of mathematics at the time. At this point in history there had been no attempt to codify a single value of π for use in future problems. Associating these cultures with their best guesses is misleading. 1 The Era of Polygonal Approximations Archimedes and the Method of Exhaustion Earlier in the course, we discussed Archimedes approximation 10 1 3:14085 ≈ 3 + 71 < π < 3 + 7 ≈ 3:14286 using the perimeters of inscribed and circumscribed 96-gons. Archimedes' approach (and a related one using areas instead of perimeters) is significant for two reasons: 1. It is effective: we obtain upper and lower bounds and can estimate the error in our approximation. 2. It is algorithmic: it can be used to prove estimates of any precision. 2 Liu Hui's Formula Polygonal methods continued to be used until the seventeenth century. In this time, the greatest advance was due to Liu Hui (c. 225 - c. -
The Nanjinger 93
Your Provider of International Standard Healthcare in Nanjing ~ DEPARTMENTS ~ Internal Medicine, Surgery, Gynaecology, Pediatrics, TCM, Sports medicine, Dermatology, Ophthalmology, Plastic Surgery (Botox/Hyaluronic acid/Filorga injection, Autologous fat granules filling), Psychology, Radiology, Pharmacy, Ultrasound, ECG, Laboratory ~ DIRECT BILLING SERVICES ~ ~ PAY US A VISIT ~ CLINIC HOURS Mon-Sun; 0900-1800 1680 Shuanglong Avenue, Baijiahu, (Jiangning 1912) ENGLISH HOTLINE www.guzeclinic.com 17372253440 www.thenanjinger.com Contents THETHE APRIL 2020 Volume#10/Issue#5 6 Contributors 7 Nanjing Nomads 8 Editorial #93 9 Poem; April 4; A New Tradition 10 Draw Loom Enlightenment How China Helped Dress The Beatles 14 Hume, Hope & Happiness; A Tradition of Human Survival 18 How WeChat, Urbanites & Consumerism Destroyed China’s Traditions 22 Strainer Next Time, Baby, I’ll be Bullet Proof 23 Great Nanjingers (1) The 5th Century’s Human Computer; Zu Chongzhi 24 The Trip Yunnan; Tie Dye’s Blue & White Spiritual Home 26 For Art’s Sake Nanjing Mapped Out; From City Walls to Psychedelic Spectrograms 27 Our Space 30 The Gavel Civil Code of PRC; A Crossroads of Tradition and Innovation 31 Metro Map 4 Introducing some of our www.thenanjinger.com contributors, editors & THE APRIL 2020 designers Our Editor-in-chief and Music Critic, Frank Sponsor 办单位 Hossack, has been a radio host and producer for SinoConnexion 贺福传媒 the past 35 years, the past 27 of which working in media in China, in the process winning four New 编辑出版 York Festivals awards for his work, in the Publisher categories Best Top 40 Format, Best Editing, Best The Nanjinger《南京⼈》杂志社 Director and Best Culture & The Arts. -
China in Classroom
CHINA IN CLASSROOM CONTENTE China ABC 1. National Flag and National Emblem 2. Physical Geography 3. Population, Ethnic Groups and Language 4. Brief History 5. Administration Divison Chinese Culture 1. Public Holidays and Most Popular Traditional Festivals in China 2. Chinese Zodiac 3. The Chinese Dragon 4. Historical Sites and Scenery in China 5. Beijing Opera 6. Calligraphy and Chinese Paintings 7. Chinese Traditional Papercuts China ABC 1. National Flag And National Emblem 国旗 The national flag of China The national flag of China is red in color which symbolizes revolution; the five stars on the flag symbolize the great unity of the Chinese people under the leadership of the Communist Party of China(CPC). 国徽 The national emblem of China The national emblem of China is Tian'anmen in the center illuminated by five stars and encircled by ears of grain and a cogwheel. Tian'anmen symbolizes the Chinese nation and the ears of grain and the cogwheel represent the working class and peasantry. 2. Physical Geography Position and Area China is situated in the eastern part of Asia, on the west coast of the Pacific Ocean. China has a total land area of 9.6 million square kilometres, next only to Russia and Canada in size. The nation is bordered by Korea in the east; Mongolia in the north; Russia in the northeast; Kazakhstan, Kirghizia and Tadzhikistan in the northwest; Afghanistan, Pakistan, India, Nepal, Sikkim and Bhutan in the west and southwest; and Myanmar, Laos and Viet Nam in the south. Across the seas to the east and southeast are the Republic of Korea, Japan, the Philippines, Brunei, Malaysia, and Indonesia. -
How to Compute Π to 1'000 Decimals Walter Gander
1/26 How to Compute π to 1'000 Decimals Walter Gander ETH and HKBU Qian Weichang College Shanghai University April 2016 2/26 circumference Why. this talk? π = of a circle diameter Using a computer algebra system e.g. Maple we get Digits:=1000; 1000 evalf(Pi); 3.141592653589793238462643383279502884197169399375105820974944 59230781640628620899862803482534211706798214808651328230664709 38446095505822317253594081284811174502841027019385211055596446 22948954930381964428810975665933446128475648233786783165271201 90914564856692346034861045432664821339360726024914127372458700 66063155881748815209209628292540917153643678925903600113305305 48820466521384146951941511609433057270365759591953092186117381 93261179310511854807446237996274956735188575272489122793818301 19491298336733624406566430860213949463952247371907021798609437 02770539217176293176752384674818467669405132000568127145263560 82778577134275778960917363717872146844090122495343014654958537 10507922796892589235420199561121290219608640344181598136297747 71309960518707211349999998372978049951059731732816096318595024 45945534690830264252230825334468503526193118817101000313783875 28865875332083814206171776691473035982534904287554687311595628 63882353787593751957781857780532171226806613001927876611195909 216420199 3/26 .Questions • How does Maple do this? • Can we develop a program? • What kind of algorithms are available? 4/26 Student Programming Problem • Matlab-course Fall 2014 at Qian Weichang College • Script has become a book: Walter Gander, Learning Matlab - A Problem Solving Approach.