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The Enigmatic Number E: a History in Verse and Its Uses in the Mathematics Classroom
To appear in MAA Loci: Convergence The Enigmatic Number e: A History in Verse and Its Uses in the Mathematics Classroom Sarah Glaz Department of Mathematics University of Connecticut Storrs, CT 06269 [email protected] Introduction In this article we present a history of e in verse—an annotated poem: The Enigmatic Number e . The annotation consists of hyperlinks leading to biographies of the mathematicians appearing in the poem, and to explanations of the mathematical notions and ideas presented in the poem. The intention is to celebrate the history of this venerable number in verse, and to put the mathematical ideas connected with it in historical and artistic context. The poem may also be used by educators in any mathematics course in which the number e appears, and those are as varied as e's multifaceted history. The sections following the poem provide suggestions and resources for the use of the poem as a pedagogical tool in a variety of mathematics courses. They also place these suggestions in the context of other efforts made by educators in this direction by briefly outlining the uses of historical mathematical poems for teaching mathematics at high-school and college level. Historical Background The number e is a newcomer to the mathematical pantheon of numbers denoted by letters: it made several indirect appearances in the 17 th and 18 th centuries, and acquired its letter designation only in 1731. Our history of e starts with John Napier (1550-1617) who defined logarithms through a process called dynamical analogy [1]. Napier aimed to simplify multiplication (and in the same time also simplify division and exponentiation), by finding a model which transforms multiplication into addition. -
Exponent and Logarithm Practice Problems for Precalculus and Calculus
Exponent and Logarithm Practice Problems for Precalculus and Calculus 1. Expand (x + y)5. 2. Simplify the following expression: √ 2 b3 5b +2 . a − b 3. Evaluate the following powers: 130 =,(−8)2/3 =,5−2 =,81−1/4 = 10 −2/5 4. Simplify 243y . 32z15 6 2 5. Simplify 42(3a+1) . 7(3a+1)−1 1 x 6. Evaluate the following logarithms: log5 125 = ,log4 2 = , log 1000000 = , logb 1= ,ln(e )= 1 − 7. Simplify: 2 log(x) + log(y) 3 log(z). √ √ √ 8. Evaluate the following: log( 10 3 10 5 10) = , 1000log 5 =,0.01log 2 = 9. Write as sums/differences of simpler logarithms without quotients or powers e3x4 ln . e 10. Solve for x:3x+5 =27−2x+1. 11. Solve for x: log(1 − x) − log(1 + x)=2. − 12. Find the solution of: log4(x 5)=3. 13. What is the domain and what is the range of the exponential function y = abx where a and b are both positive constants and b =1? 14. What is the domain and what is the range of f(x) = log(x)? 15. Evaluate the following expressions. (a) ln(e4)= √ (b) log(10000) − log 100 = (c) eln(3) = (d) log(log(10)) = 16. Suppose x = log(A) and y = log(B), write the following expressions in terms of x and y. (a) log(AB)= (b) log(A) log(B)= (c) log A = B2 1 Solutions 1. We can either do this one by “brute force” or we can use the binomial theorem where the coefficients of the expansion come from Pascal’s triangle. -
Square Series Generating Function Transformations 127
Journal of Inequalities and Special Functions ISSN: 2217-4303, URL: www.ilirias.com/jiasf Volume 8 Issue 2(2017), Pages 125-156. SQUARE SERIES GENERATING FUNCTION TRANSFORMATIONS MAXIE D. SCHMIDT Abstract. We construct new integral representations for transformations of the ordinary generating function for a sequence, hfni, into the form of a gen- erating function that enumerates the corresponding \square series" generating n2 function for the sequence, hq fni, at an initially fixed non-zero q 2 C. The new results proved in the article are given by integral{based transformations of ordinary generating function series expanded in terms of the Stirling num- bers of the second kind. We then employ known integral representations for the gamma and double factorial functions in the construction of these square series transformation integrals. The results proved in the article lead to new applications and integral representations for special function series, sequence generating functions, and other related applications. A summary Mathemat- ica notebook providing derivations of key results and applications to specific series is provided online as a supplemental reference to readers. 1. Notation and conventions Most of the notational conventions within the article are consistent with those employed in the references [11, 15]. Additional notation for special parameterized classes of the square series expansions studied in the article is defined in Table 1 on page 129. We utilize this notation for these generalized classes of square series functions throughout the article. The following list provides a description of the other primary notations and related conventions employed throughout the article specific to the handling of sequences and the coefficients of formal power series: I Sequences and Generating Functions: The notation hfni ≡ ff0; f1; f2;:::g is used to specify an infinite sequence over the natural numbers, n 2 N, where we define N = f0; 1; 2;:::g and Z+ = f1; 2; 3;:::g. -
Convolution on the N-Sphere with Application to PDF Modeling Ivan Dokmanic´, Student Member, IEEE, and Davor Petrinovic´, Member, IEEE
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 3, MARCH 2010 1157 Convolution on the n-Sphere With Application to PDF Modeling Ivan Dokmanic´, Student Member, IEEE, and Davor Petrinovic´, Member, IEEE Abstract—In this paper, we derive an explicit form of the convo- emphasis on wavelet transform in [8]–[12]. Computation of the lution theorem for functions on an -sphere. Our motivation comes Fourier transform and convolution on groups is studied within from the design of a probability density estimator for -dimen- the theory of noncommutative harmonic analysis. Examples sional random vectors. We propose a probability density function (pdf) estimation method that uses the derived convolution result of applications of noncommutative harmonic analysis in engi- on . Random samples are mapped onto the -sphere and esti- neering are analysis of the motion of a rigid body, workspace mation is performed in the new domain by convolving the samples generation in robotics, template matching in image processing, with the smoothing kernel density. The convolution is carried out tomography, etc. A comprehensive list with accompanying in the spectral domain. Samples are mapped between the -sphere theory and explanations is given in [13]. and the -dimensional Euclidean space by the generalized stereo- graphic projection. We apply the proposed model to several syn- Statistics of random vectors whose realizations are observed thetic and real-world data sets and discuss the results. along manifolds embedded in Euclidean spaces are commonly termed directional statistics. An excellent review may be found Index Terms—Convolution, density estimation, hypersphere, hy- perspherical harmonics, -sphere, rotations, spherical harmonics. in [14]. It is of interest to develop tools for the directional sta- tistics in analogy with the ordinary Euclidean. -
Leonhard Euler: His Life, the Man, and His Works∗
SIAM REVIEW c 2008 Walter Gautschi Vol. 50, No. 1, pp. 3–33 Leonhard Euler: His Life, the Man, and His Works∗ Walter Gautschi† Abstract. On the occasion of the 300th anniversary (on April 15, 2007) of Euler’s birth, an attempt is made to bring Euler’s genius to the attention of a broad segment of the educated public. The three stations of his life—Basel, St. Petersburg, andBerlin—are sketchedandthe principal works identified in more or less chronological order. To convey a flavor of his work andits impact on modernscience, a few of Euler’s memorable contributions are selected anddiscussedinmore detail. Remarks on Euler’s personality, intellect, andcraftsmanship roundout the presentation. Key words. LeonhardEuler, sketch of Euler’s life, works, andpersonality AMS subject classification. 01A50 DOI. 10.1137/070702710 Seh ich die Werke der Meister an, So sehe ich, was sie getan; Betracht ich meine Siebensachen, Seh ich, was ich h¨att sollen machen. –Goethe, Weimar 1814/1815 1. Introduction. It is a virtually impossible task to do justice, in a short span of time and space, to the great genius of Leonhard Euler. All we can do, in this lecture, is to bring across some glimpses of Euler’s incredibly voluminous and diverse work, which today fills 74 massive volumes of the Opera omnia (with two more to come). Nine additional volumes of correspondence are planned and have already appeared in part, and about seven volumes of notebooks and diaries still await editing! We begin in section 2 with a brief outline of Euler’s life, going through the three stations of his life: Basel, St. -
Calculus Terminology
AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential -
Hydraulics Manual Glossary G - 3
Glossary G - 1 GLOSSARY OF HIGHWAY-RELATED DRAINAGE TERMS (Reprinted from the 1999 edition of the American Association of State Highway and Transportation Officials Model Drainage Manual) G.1 Introduction This Glossary is divided into three parts: · Introduction, · Glossary, and · References. It is not intended that all the terms in this Glossary be rigorously accurate or complete. Realistically, this is impossible. Depending on the circumstance, a particular term may have several meanings; this can never change. The primary purpose of this Glossary is to define the terms found in the Highway Drainage Guidelines and Model Drainage Manual in a manner that makes them easier to interpret and understand. A lesser purpose is to provide a compendium of terms that will be useful for both the novice as well as the more experienced hydraulics engineer. This Glossary may also help those who are unfamiliar with highway drainage design to become more understanding and appreciative of this complex science as well as facilitate communication between the highway hydraulics engineer and others. Where readily available, the source of a definition has been referenced. For clarity or format purposes, cited definitions may have some additional verbiage contained in double brackets [ ]. Conversely, three “dots” (...) are used to indicate where some parts of a cited definition were eliminated. Also, as might be expected, different sources were found to use different hyphenation and terminology practices for the same words. Insignificant changes in this regard were made to some cited references and elsewhere to gain uniformity for the terms contained in this Glossary: as an example, “groundwater” vice “ground-water” or “ground water,” and “cross section area” vice “cross-sectional area.” Cited definitions were taken primarily from two sources: W.B. -
A5730107.Pdf
International OPEN ACCESS Journal Of Modern Engineering Research (IJMER) Applications of Bipartite Graph in diverse fields including cloud computing 1Arunkumar B R, 2Komala R Prof. and Head, Dept. of MCA, BMSIT, Bengaluru-560064 and Ph.D. Research supervisor, VTU RRC, Belagavi Asst. Professor, Dept. of MCA, Sir MVIT, Bengaluru and Ph.D. Research Scholar,VTU RRC, Belagavi ABSTRACT: Graph theory finds its enormous applications in various diverse fields. Its applications are evolving as it is perfect natural model and able to solve the problems in a unique way.Several disciplines even though speak about graph theory that is only in wider context. This paper pinpoints the applications of Bipartite graph in diverse field with a more points stressed on cloud computing. KEY WORDS: Graph theory, Bipartite graph cloud computing, perfect matching applications I. INTRODUCTION Graph theory has emerged as most approachable for all most problems in any field. In recent years, graph theory has emerged as one of the most sociable and fruitful methods for analyzing chemical reaction networks (CRNs). Graph theory can deal with models for which other techniques fail, for example, models where there is incomplete information about the parameters or that are of high dimension. Models with such issues are common in CRN theory [16]. Graph theory can find its applications in all most all disciplines of science, engineering, technology and including medical fields. Both in the view point of theory and practical bipartite graphs are perhaps the most basic of entities in graph theory. Until now, several graph theory structures including Bipartite graph have been considered only as a special class in some wider context in the discipline such as chemistry and computer science [1]. -
The Logarithmic Chicken Or the Exponential Egg: Which Comes First?
The Logarithmic Chicken or the Exponential Egg: Which Comes First? Marshall Ransom, Senior Lecturer, Department of Mathematical Sciences, Georgia Southern University Dr. Charles Garner, Mathematics Instructor, Department of Mathematics, Rockdale Magnet School Laurel Holmes, 2017 Graduate of Rockdale Magnet School, Current Student at University of Alabama Background: This article arose from conversations between the first two authors. In discussing the functions ln(x) and ex in introductory calculus, one of us made good use of the inverse function properties and the other had a desire to introduce the natural logarithm without the classic definition of same as an integral. It is important to introduce mathematical topics using a minimal number of definitions and postulates/axioms when results can be derived from existing definitions and postulates/axioms. These are two of the ideas motivating the article. Thus motivated, the authors compared manners with which to begin discussion of the natural logarithm and exponential functions in a calculus class. x A related issue is the use of an integral to define a function g in terms of an integral such as g()() x f t dt . c We believe that this is something that students should understand and be exposed to prior to more advanced x x sin(t ) 1 “surprises” such as Si(x ) dt . In particular, the fact that ln(x ) dt is extremely important. But t t 0 1 must that fact be introduced as a definition? Can the natural logarithm function arise in an introductory calculus x 1 course without the -
Differentiating Logarithm and Exponential Functions
Differentiating logarithm and exponential functions mc-TY-logexp-2009-1 This unit gives details of how logarithmic functions and exponential functions are differentiated from first principles. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. After reading this text, and/or viewing the video tutorial on this topic, you should be able to: • differentiate ln x from first principles • differentiate ex Contents 1. Introduction 2 2. Differentiation of a function f(x) 2 3. Differentiation of f(x)=ln x 3 4. Differentiation of f(x) = ex 4 www.mathcentre.ac.uk 1 c mathcentre 2009 1. Introduction In this unit we explain how to differentiate the functions ln x and ex from first principles. To understand what follows we need to use the result that the exponential constant e is defined 1 1 as the limit as t tends to zero of (1 + t) /t i.e. lim (1 + t) /t. t→0 1 To get a feel for why this is so, we have evaluated the expression (1 + t) /t for a number of decreasing values of t as shown in Table 1. Note that as t gets closer to zero, the value of the expression gets closer to the value of the exponential constant e≈ 2.718.... You should verify some of the values in the Table, and explore what happens as t reduces further. 1 t (1 + t) /t 1 (1+1)1/1 = 2 0.1 (1+0.1)1/0.1 = 2.594 0.01 (1+0.01)1/0.01 = 2.705 0.001 (1.001)1/0.001 = 2.717 0.0001 (1.0001)1/0.0001 = 2.718 We will also make frequent use of the laws of indices and the laws of logarithms, which should be revised if necessary. -
Math1414-Laws-Of-Logarithms.Pdf
Properties of Logarithms Since the exponential and logarithmic functions with base a are inverse functions, the Laws of Exponents give rise to the Laws of Logarithms. Laws of Logarithms: Let a be a positive number, with a ≠ 1. Let A > 0, B > 0, and C be any real numbers. Law Description 1. log a (AB) = log a A + log a B The logarithm of a product of numbers is the sum of the logarithms of the numbers. ⎛⎞A The logarithm of a quotient of numbers is the 2. log a ⎜⎟ = log a A – log a B ⎝⎠B difference of the logarithms of the numbers. C 3. log a (A ) = C log a A The logarithm of a power of a number is the exponent times the logarithm of the number. Example 1: Use the Laws of Logarithms to rewrite the expression in a form with no logarithm of a product, quotient, or power. ⎛⎞5x3 (a) log3 ⎜⎟2 ⎝⎠y (b) ln 5 x2 ()z+1 Solution (a): 3 ⎛⎞5x 32 log333⎜⎟2 =− log() 5xy log Law 2 ⎝⎠y 32 =+log33 5 logxy − log 3 Law 1 =+log33 5 3logxy − 2log 3 Law 3 By: Crystal Hull Example 1 (Continued) Solution (b): 1 ln5 xz22()+= 1 ln( xz() + 1 ) 5 1 2 =+5 ln()xz() 1 Law 3 =++1 ⎡⎤lnxz2 ln() 1 Law 1 5 ⎣⎦ 112 =++55lnxz ln() 1 Distribution 11 =++55()2lnxz ln() 1 Law 3 21 =+55lnxz ln() + 1 Multiplication Example 2: Rewrite the expression as a single logarithm. (a) 3log44 5+− log 10 log4 7 1 2 (b) logx −+ 2logyz2 log( 9 ) Solution (a): 3 3log44 5+−=+− log 10 log 4 7 log 4 5 log 4 10 log 4 7 Law 3 3 =+−log444 125 log 10 log 7 Because 5= 125 =⋅−log44() 125 10 log 7 Law 1 =−log4 () 1250 log4 7 Simplification ⎛⎞1250 = log4 ⎜⎟ Law 2 ⎝⎠7 By: Crystal Hull Example 2 (Continued): -
Logarithms of Integers Are Irrational
Logarithms of Integers are Irrational J¨orgFeldvoss∗ Department of Mathematics and Statistics University of South Alabama Mobile, AL 36688{0002, USA May 19, 2008 Abstract In this short note we prove that the natural logarithm of every integer ≥ 2 is an irrational number and that the decimal logarithm of any integer is irrational unless it is a power of 10. 2000 Mathematics Subject Classification: 11R04 In this short note we prove that logarithms of most integers are irrational. Theorem 1: The natural logarithm of every integer n ≥ 2 is an irrational number. a Proof: Suppose that ln n = b is a rational number for some integers a and b. Wlog we can assume that a; b > 0. Using the third logarithmic identity we obtain that the above equation is equivalent to nb = ea. Since a and b both are positive integers, it follows that e is the root of a polynomial with integer coefficients (and leading coefficient 1), namely, the polynomial xa − nb. Hence e is an algebraic number (even an algebraic integer) which is a contradiction. 2 Remark 1: It follows from the proof that for any base b which is a transcendental number the logarithm logb n of every integer n ≥ 2 is an irrational number. (It is not alwaysp true that logb n is already irrational for every integer n ≥ 2 if b is irrational as the example b = 2 and n = 2 shows!) Question: Is it always true that logb n is already transcendental for every integer n ≥ 2 if b is transcendental? The following well-known result will be needed in [1, p.