Pi and E, and the Most Beautiful Theorem in Mathematics
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Mathematics Is a Gentleman's Art: Analysis and Synthesis in American College Geometry Teaching, 1790-1840 Amy K
Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 2000 Mathematics is a gentleman's art: Analysis and synthesis in American college geometry teaching, 1790-1840 Amy K. Ackerberg-Hastings Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Higher Education and Teaching Commons, History of Science, Technology, and Medicine Commons, and the Science and Mathematics Education Commons Recommended Citation Ackerberg-Hastings, Amy K., "Mathematics is a gentleman's art: Analysis and synthesis in American college geometry teaching, 1790-1840 " (2000). Retrospective Theses and Dissertations. 12669. https://lib.dr.iastate.edu/rtd/12669 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margwis, and improper alignment can adversely affect reproduction. in the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. -
Logarithms – a Journey of Their Tables to All Over the World 1
Logarithms – a Journey of their Tables to all over the World 1 Klaus Kühn, Nienstädt (Germany) 2 1 Introduction These days, nobody expects anything really new on the subject of logarithms. Surprisingly, however, there is still something to be written, something to be reviewed or something to be looked at again with fresh eyes. The history of logarithms has been described and dealt with on many occasions and is very clear. Nevertheless there are unnecessary discussions from time to time about who invented logarithms. Undeniably, a 7-place logarithmic table called "Mirifici Logarithmorum Canonis Descriptio" was published in 1614 for the very first time by John Napier (1550 – 1617). 6 years later "Aritmetische und Geometrische Progreß Tabulen / sambt gründlichem Unterricht / wie solche nützlich in allerley Rechnungen zugebrauchen / und verstanden werden soll " (translation see 3) was published by Jost Bürgi (1552 - 1632) – also known as Joost or Jobst Byrgius or Byrg. He published logarithms to 8 places; however this table failed to catch on because there were no instructions for its use included as mentioned. In the eyes of some of today’s contemporaries Bürgi is regarded as the inventor of logarithms, because he had been calculating his logarithms years before Napier, but he only published them on the insistence of Johannes Kepler (1571 – 1630). There is no information on the time it took John Napier to finish calculating his logarithms but in those times it would have been a matter of many years. The theories of the Babylonians (about 1600 B.C.), Euclid (365 – 300 B.C.), and Archimedes (287 – 212 B.C.) had already begun to move in the logarithmic direction. -
Finding Pi Project
Name: ________________________ Finding Pi - Activity Objective: You may already know that pi (π) is a number that is approximately equal to 3.14. But do you know where the number comes from? Let's measure some round objects and find out. Materials: • 6 circular objects Some examples include a bicycle wheel, kiddie pool, trash can lid, DVD, steering wheel, or clock face. Be sure each object you choose is shaped like a perfect circle. • metric tape measure Be sure your tape measure has centimeters on it. • calculator It will save you some time because dividing with decimals can be tricky. • “Finding Pi - Table” worksheet It may be attached to this page, or on the back. What to do: Step 1: Choose one of your circular objects. Write the name of the object on the “Finding Pi” table. Step 2: With the centimeter side of your tape measure, accurately measure the distance around the outside of the circle (the circumference). Record your measurement on the table. Step 3: Next, measure the distance across the middle of the object (the diameter). Record your measurement on the table. Step 4: Use your calculator to divide the circumference by the diameter. Write the answer on the table. If you measured carefully, the answer should be about 3.14, or π. Repeat steps 1 through 4 for each object. Super Teacher Worksheets - www.superteacherworksheets.com Name: ________________________ “Finding Pi” Table Measure circular objects and complete the table below. If your measurements are accurate, you should be able to calculate the number pi (3.14). Is your answer name of circumference diameter circumference ÷ approximately circular object measurement (cm) measurement (cm) diameter equal to π? 1. -
Year XIX, Supplement Ethnographic Study And/Or a Theoretical Survey of a Their Position in the Article Should Be Clearly Indicated
III. TITLES OF ARTICLES DRU[TVO ANTROPOLOGOV SLOVENIJE The journal of the Slovene Anthropological Society Titles (in English and Slovene) must be short, informa- SLOVENE ANTHROPOLOGICAL SOCIETY Anthropological Notebooks welcomes the submis- tive, and understandable. The title should be followed sion of papers from the field of anthropology and by the name of the author(s), their position, institutional related disciplines. Submissions are considered for affiliation, and if possible, by e-mail address. publication on the understanding that the paper is not currently under consideration for publication IV. ABSTRACT AND KEYWORDS elsewhere. It is the responsibility of the author to The abstract must give concise information about the obtain permission for using any previously published objective, the method used, the results obtained, and material. Please submit your manuscript as an e-mail the conclusions. Authors are asked to enclose in English attachment on [email protected] and enclose your contact information: name, position, and Slovene an abstract of 100 – 200 words followed institutional affiliation, address, phone number, and by three to five keywords. They must reflect the field of e-mail address. research covered in the article. English abstract should be placed at the beginning of an article and the Slovene one after the references at the end. V. NOTES A N T H R O P O L O G I C A L INSTRUCTIONS Notes should also be double-spaced and used sparingly. They must be numbered consecutively throughout the text and assembled at the end of the article just before references. VI. QUOTATIONS Short quotations (less than 30 words) should be placed in single quotation marks with double marks for quotations within quotations. -
Evaluating Fourier Transforms with MATLAB
ECE 460 – Introduction to Communication Systems MATLAB Tutorial #2 Evaluating Fourier Transforms with MATLAB In class we study the analytic approach for determining the Fourier transform of a continuous time signal. In this tutorial numerical methods are used for finding the Fourier transform of continuous time signals with MATLAB are presented. Using MATLAB to Plot the Fourier Transform of a Time Function The aperiodic pulse shown below: x(t) 1 t -2 2 has a Fourier transform: X ( jf ) = 4sinc(4π f ) This can be found using the Table of Fourier Transforms. We can use MATLAB to plot this transform. MATLAB has a built-in sinc function. However, the definition of the MATLAB sinc function is slightly different than the one used in class and on the Fourier transform table. In MATLAB: sin(π x) sinc(x) = π x Thus, in MATLAB we write the transform, X, using sinc(4f), since the π factor is built in to the function. The following MATLAB commands will plot this Fourier Transform: >> f=-5:.01:5; >> X=4*sinc(4*f); >> plot(f,X) In this case, the Fourier transform is a purely real function. Thus, we can plot it as shown above. In general, Fourier transforms are complex functions and we need to plot the amplitude and phase spectrum separately. This can be done using the following commands: >> plot(f,abs(X)) >> plot(f,angle(X)) Note that the angle is either zero or π. This reflects the positive and negative values of the transform function. Performing the Fourier Integral Numerically For the pulse presented above, the Fourier transform can be found easily using the table. -
February 2009
How Euler Did It by Ed Sandifer Estimating π February 2009 On Friday, June 7, 1779, Leonhard Euler sent a paper [E705] to the regular twice-weekly meeting of the St. Petersburg Academy. Euler, blind and disillusioned with the corruption of Domaschneff, the President of the Academy, seldom attended the meetings himself, so he sent one of his assistants, Nicolas Fuss, to read the paper to the ten members of the Academy who attended the meeting. The paper bore the cumbersome title "Investigatio quarundam serierum quae ad rationem peripheriae circuli ad diametrum vero proxime definiendam maxime sunt accommodatae" (Investigation of certain series which are designed to approximate the true ratio of the circumference of a circle to its diameter very closely." Up to this point, Euler had shown relatively little interest in the value of π, though he had standardized its notation, using the symbol π to denote the ratio of a circumference to a diameter consistently since 1736, and he found π in a great many places outside circles. In a paper he wrote in 1737, [E74] Euler surveyed the history of calculating the value of π. He mentioned Archimedes, Machin, de Lagny, Leibniz and Sharp. The main result in E74 was to discover a number of arctangent identities along the lines of ! 1 1 1 = 4 arctan " arctan + arctan 4 5 70 99 and to propose using the Taylor series expansion for the arctangent function, which converges fairly rapidly for small values, to approximate π. Euler also spent some time in that paper finding ways to approximate the logarithms of trigonometric functions, important at the time in navigation tables. -
MATLAB Examples Mathematics
MATLAB Examples Mathematics Hans-Petter Halvorsen, M.Sc. Mathematics with MATLAB • MATLAB is a powerful tool for mathematical calculations. • Type “help elfun” (elementary math functions) in the Command window for more information about basic mathematical functions. Mathematics Topics • Basic Math Functions and Expressions � = 3�% + ) �% + �% + �+,(.) • Statistics – mean, median, standard deviation, minimum, maximum and variance • Trigonometric Functions sin() , cos() , tan() • Complex Numbers � = � + �� • Polynomials = =>< � � = �<� + �%� + ⋯ + �=� + �=@< Basic Math Functions Create a function that calculates the following mathematical expression: � = 3�% + ) �% + �% + �+,(.) We will test with different values for � and � We create the function: function z=calcexpression(x,y) z=3*x^2 + sqrt(x^2+y^2)+exp(log(x)); Testing the function gives: >> x=2; >> y=2; >> calcexpression(x,y) ans = 16.8284 Statistics Functions • MATLAB has lots of built-in functions for Statistics • Create a vector with random numbers between 0 and 100. Find the following statistics: mean, median, standard deviation, minimum, maximum and the variance. >> x=rand(100,1)*100; >> mean(x) >> median(x) >> std(x) >> mean(x) >> min(x) >> max(x) >> var(x) Trigonometric functions sin(�) cos(�) tan(�) Trigonometric functions It is quite easy to convert from radians to degrees or from degrees to radians. We have that: 2� ������� = 360 ������� This gives: 180 � ������� = �[�������] M � � �[�������] = �[�������] M 180 → Create two functions that convert from radians to degrees (r2d(x)) and from degrees to radians (d2r(x)) respectively. Test the functions to make sure that they work as expected. The functions are as follows: function d = r2d(r) d=r*180/pi; function r = d2r(d) r=d*pi/180; Testing the functions: >> r2d(2*pi) ans = 360 >> d2r(180) ans = 3.1416 Trigonometric functions Given right triangle: • Create a function that finds the angle A (in degrees) based on input arguments (a,c), (b,c) and (a,b) respectively. -
Maty's Biography of Abraham De Moivre, Translated
Statistical Science 2007, Vol. 22, No. 1, 109–136 DOI: 10.1214/088342306000000268 c Institute of Mathematical Statistics, 2007 Maty’s Biography of Abraham De Moivre, Translated, Annotated and Augmented David R. Bellhouse and Christian Genest Abstract. November 27, 2004, marked the 250th anniversary of the death of Abraham De Moivre, best known in statistical circles for his famous large-sample approximation to the binomial distribution, whose generalization is now referred to as the Central Limit Theorem. De Moivre was one of the great pioneers of classical probability the- ory. He also made seminal contributions in analytic geometry, complex analysis and the theory of annuities. The first biography of De Moivre, on which almost all subsequent ones have since relied, was written in French by Matthew Maty. It was published in 1755 in the Journal britannique. The authors provide here, for the first time, a complete translation into English of Maty’s biography of De Moivre. New mate- rial, much of it taken from modern sources, is given in footnotes, along with numerous annotations designed to provide additional clarity to Maty’s biography for contemporary readers. INTRODUCTION ´emigr´es that both of them are known to have fre- Matthew Maty (1718–1776) was born of Huguenot quented. In the weeks prior to De Moivre’s death, parentage in the city of Utrecht, in Holland. He stud- Maty began to interview him in order to write his ied medicine and philosophy at the University of biography. De Moivre died shortly after giving his Leiden before immigrating to England in 1740. Af- reminiscences up to the late 1680s and Maty com- ter a decade in London, he edited for six years the pleted the task using only his own knowledge of the Journal britannique, a French-language publication man and De Moivre’s published work. -
Fundamental Theorems in Mathematics
SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635]. -
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. -
Napier's Ideal Construction of the Logarithms
Napier’s ideal construction of the logarithms Denis Roegel To cite this version: Denis Roegel. Napier’s ideal construction of the logarithms. [Research Report] 2010. inria-00543934 HAL Id: inria-00543934 https://hal.inria.fr/inria-00543934 Submitted on 6 Dec 2010 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Napier’s ideal construction of the logarithms∗ Denis Roegel 6 December 2010 1 Introduction Today John Napier (1550–1617) is most renowned as the inventor of loga- rithms.1 He had conceived the general principles of logarithms in 1594 or be- fore and he spent the next twenty years in developing their theory [108, p. 63], [33, pp. 103–104]. His description of logarithms, Mirifici Logarithmorum Ca- nonis Descriptio, was published in Latin in Edinburgh in 1614 [131, 161] and was considered “one of the very greatest scientific discoveries that the world has seen” [83]. Several mathematicians had anticipated properties of the correspondence between an arithmetic and a geometric progression, but only Napier and Jost Bürgi (1552–1632) constructed tables for the purpose of simplifying the calculations. Bürgi’s work was however only published in incomplete form in 1620, six years after Napier published the Descriptio [26].2 Napier’s work was quickly translated in English by the mathematician and cartographer Edward Wright3 (1561–1615) [145, 179] and published posthu- mously in 1616 [132, 162]. -
Π and Its Computation Through the Ages
π and its computation through the ages Xavier Gourdon & Pascal Sebah http://numbers.computation.free.fr/Constants/constants.html August 13, 2010 The value of π has engaged the attention of many mathematicians and calculators from the time of Archimedes to the present day, and has been computed from so many different formulae, that a com- plete account of its calculation would almost amount to a history of mathematics. - James Glaisher (1848-1928) The history of pi is a quaint little mirror of the history of man. - Petr Beckmann 1 Computing the constant π Understanding the nature of the constant π, as well as trying to estimate its value to more and more decimal places has engaged a phenomenal energy from mathematicians from all periods of history and from most civilizations. In this small overview, we have tried to collect as many as possible major calculations of the most famous mathematical constant, including the methods used and references whenever there are available. 1.1 Milestones of π's computation • Ancient civilizations like Egyptians [15], Babylonians, China ([29], [52]), India [36],... were interested in evaluating, for example, area or perimeter of circular fields. Of course in this early history, π was not yet a constant and was only implicit in all available documents. Perhaps the most famous is the Rhind Papyrus which states the rule used to compute the area of a circle: take away 1/9 of the diameter and take the square of the remainder therefore implicitly π = (16=9)2: • Archimedes of Syracuse (287-212 B.C.). He developed a method based on inscribed and circumscribed polygons which will be of practical use until the mid seventeenth.