DOCSLIB.ORG

E Is Everywhere

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
E Is Everywhere measure for measure e is everywhere From determining the compound interest on borrowed money to gauging chances at the roulette wheel in Monte Carlo, Stefanie Reichert explains that there’s no way around Euler’s number. ven outside school or university, Euler’s constant, also called the Euler– we cannot escape Euler’s number. Mascheroni constant, γ ≈ 0.57721, defined EJacob Bernoulli is credited with as the limit of the difference between the discovering e while thinking about matters harmonic series and the natural logarithm. of continuous compound interest in 1683. The Euler–Mascheroni constant appears, He realized that when the compounding for example, in the Bessel function of the period became smaller and smaller and second kind, and has not been proven to more and more periods were considered, be irrational or transcendental. Another the amount of money would converge tricky case are Euler numbers (also known towards a limit that was later found to be as zig or secant numbers), referring to the one of the representations of e. Since then, number of odd alternating permutations in use of Euler’s number has become more expressions for the secant and hyperbolic widespread and now it appears in many secant (https://go.nature.com/2N0G3tc). branches of science and in everyday life. To complicate things further, at least three For example, Euler’s number shows Credit: GL Archive / Alamy Stock Photo other mathematical terminologies are up in probability theory. Imagine you are in use denoting the Euler number of a in Monte Carlo enjoying a few games of finite complex, Euler primes or the Euler roulette, which is a Bernoulli trial process. It took a while before scholars connected characteristics, a topological invariant. If you place a bet on a single number, your the dots and realized that the base of the And in fluid dynamics, the Euler number chances are 1/37 to win that game. For logarithm introduced by Napier and the characterizes the energy loss in a flow. 37 games, the probability that you will limit discovered by Bernoulli were closely We have all encountered Euler’s lose every single time is — maybe related and settled on a common notation. number in more ways than one — from surprisingly — close to 1/e. Or, pretend Gottfried Leibniz referred to what is now natural logarithms to the definition of the you are at the theatre, where you — along known as Euler’s number as b in discussions exponential function, which relies on the with everybody else — leave your coat with Christiaan Huygens, whereas others series expansion of e discovered by Euler in the cloak room, which has one hook such as Jean-Baptiste le Rond d’Alembert himself in 1748. The constant e appears per guest, and receive a number. However, preferred to use the notation c instead. practically everywhere in science: popping your coat is placed on a random hook. This dispute was eventually settled when up in the definition of the standard normal The probability that none of the coats are the Swiss mathematician Leonhard Euler distribution; allowing us to decompose a on the correct hook for a large number of (pictured) used the letter e in an early essay time-dependent signal into its frequencies guests approaches, again, 1/e. The number on the firing of cannons — and his choice via Fourier transformation; telling us how of practical examples is endless. became increasingly popular. to calculate the half-life of radioactive The history of e reads like the Who’s Similar to π, Euler’s number e ≈ 2.71828 elements; playing a crucial role in the growth Who of mathematics and physics. It all is irrational and also transcendental — of bacteria; and governing temperature- started with the discovery of the logarithm meaning it doesn’t form a solution of activated chemical reactions. by John Napier: Euler’s number is hidden a non-zero polynomial equation with Not for nothing, e counts among the deep in the many pages of the appendix integer coefficients. Whether e (or π) most important constants in mathematics tabulating natural logarithms in his 1614 is a normal number remains to be and physics, along with 0, 1, i and π that all work Mirifici Logarithmorum Canonis determined. A normal number consists of show up in Euler’s identity eiπ + 1 = 0. It is Descriptio. Later, when Bernoulli studied a sequence of digits in which single digits truly a constant in everyone’s life. ❐ the case of continuous compound interest, between 0 and 9 occur with a frequency he concluded that the limit must converge of 10%, whereas each pair of digits between Stefanie Reichert to a number between 2 and 3. As it turned 00 and 99 occurs with a frequency of 1%, Associate Editor, Nature Physics. out, this limit equals Euler’s number (less and so on. e-mail: [email protected] commonly known as Napier’s constant), Euler is credited with a whole bunch and Bernoulli came up with its first of constants besides e, so one should be Published online: 3 September 2019 approximation. careful not to mix Euler’s number up with https://doi.org/10.1038/s41567-019-0655-9 982 NATURE PHYSICS | VOL 15 | SEPTEMBER 2019 | 982 | www.nature.com/naturephysics.
Load more

Recommended publications
  • 18.01A Topic 5: Second Fundamental Theorem, Lnx As an Integral. Read
    18.01A Topic 5: Second fundamental theorem, ln x as an integral. Read: SN: PI, FT. 0 R b b First Fundamental Theorem: F = f ⇒ a f(x) dx = F (x)|a Questions: 1. Why dx? 2. Given f(x) does F (x) always exist? Answer to question 1. Pn R b a) Riemann sum: Area ≈ 1 f(ci)∆x → a f(x) dx In the limit the sum becomes an integral and the finite ∆x becomes the infinitesimal dx. b) The dx helps with change of variable. a b 1 Example: Compute R √ 1 dx. 0 1−x2 Let x = sin u. dx du = cos u ⇒ dx = cos u du, x = 0 ⇒ u = 0, x = 1 ⇒ u = π/2. 1 π/2 π/2 π/2 R √ 1 R √ 1 R cos u R Substituting: 2 dx = cos u du = du = du = π/2. 0 1−x 0 1−sin2 u 0 cos u 0 Answer to question 2. Yes → Second Fundamental Theorem: Z x If f is continuous and F (x) = f(u) du then F 0(x) = f(x). a I.e. f always has an anit-derivative. This is subtle: we have defined a NEW function F (x) using the definite integral. Note we needed a dummy variable u for integration because x was already taken. Z x+∆x f(x) proof: ∆area = f(x) dx 44 4 x Z x+∆x Z x o area = ∆F = f(x) dx − f(x) dx 0 0 = F (x + ∆x) − F (x) = ∆F. x x + ∆x ∆F But, also ∆area ≈ f(x) ∆x ⇒ ∆F ≈ f(x) ∆x or ≈ f(x).
    [Show full text]
  • Generalizations of Euler Numbers and Polynomials 1
    GENERALIZATIONS OF EULER NUMBERS AND POLYNOMIALS QIU-MING LUO AND FENG QI Abstract. In this paper, the concepts of Euler numbers and Euler polyno- mials are generalized, and some basic properties are investigated. 1. Introduction It is well-known that the Euler numbers and polynomials can be defined by the following definitions. Definition 1.1 ([1]). The Euler numbers Ek are defined by the following expansion t ∞ 2e X Ek = tk, |t| ≤ π. (1.1) e2t + 1 k! k=0 In [4, p. 5], the Euler numbers is defined by t/2 ∞ n 2n 2e t X (−1) En t = sech = , |t| ≤ π. (1.2) et + 1 2 (2n)! 2 n=0 Definition 1.2 ([1, 4]). The Euler polynomials Ek(x) for x ∈ R are defined by xt ∞ 2e X Ek(x) = tk, |t| ≤ π. (1.3) et + 1 k! k=0 It can also be shown that the polynomials Ei(t), i ∈ N, are uniquely determined by the following two properties 0 Ei(t) = iEi−1(t),E0(t) = 1; (1.4) i Ei(t + 1) + Ei(t) = 2t . (1.5) 2000 Mathematics Subject Classification. 11B68. Key words and phrases. Euler numbers, Euler polynomials, generalization. The authors were supported in part by NNSF (#10001016) of China, SF for the Prominent Youth of Henan Province, SF of Henan Innovation Talents at Universities, NSF of Henan Province (#004051800), Doctor Fund of Jiaozuo Institute of Technology, China. This paper was typeset using AMS-LATEX. 1 2 Q.-M. LUO AND F. QI Euler polynomials are related to the Bernoulli numbers. For information about Bernoulli numbers and polynomials, please refer to [1, 2, 3, 4].
    [Show full text]
  • Laws of Similarity in Fluid Mechanics 21
    Laws of similarity in fluid mechanics B. Weigand1 & V. Simon2 1Institut für Thermodynamik der Luft- und Raumfahrt (ITLR), Universität Stuttgart, Germany. 2Isringhausen GmbH & Co KG, Lemgo, Germany. Abstract All processes, in nature as well as in technical systems, can be described by fundamental equations—the conservation equations. These equations can be derived using conservation princi- ples and have to be solved for the situation under consideration. This can be done without explicitly investigating the dimensions of the quantities involved. However, an important consideration in all equations used in fluid mechanics and thermodynamics is dimensional homogeneity. One can use the idea of dimensional consistency in order to group variables together into dimensionless parameters which are less numerous than the original variables. This method is known as dimen- sional analysis. This paper starts with a discussion on dimensions and about the pi theorem of Buckingham. This theorem relates the number of quantities with dimensions to the number of dimensionless groups needed to describe a situation. After establishing this basic relationship between quantities with dimensions and dimensionless groups, the conservation equations for processes in fluid mechanics (Cauchy and Navier–Stokes equations, continuity equation, energy equation) are explained. By non-dimensionalizing these equations, certain dimensionless groups appear (e.g. Reynolds number, Froude number, Grashof number, Weber number, Prandtl number). The physical significance and importance of these groups are explained and the simplifications of the underlying equations for large or small dimensionless parameters are described. Finally, some examples for selected processes in nature and engineering are given to illustrate the method. 1 Introduction If we compare a small leaf with a large one, or a child with its parents, we have the feeling that a ‘similarity’ of some sort exists.
    [Show full text]
  • 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.
    [Show full text]
  • Discovery of E Name: Compound Interest – Interest Paid on The
    Discovery of E Name: Compound Interest – interest paid on the principal of an investment and any previously earned interest. A -> P-> r-> n-> t-> 1) An investment account pays 4.2% annual interest compounded monthly. If $2500 is invested in this account, what will be the balance after 15 years? Set up the equation: A = 2) Find the balance of an account after 7 years if $700 is deposited into an account paying 4.3% interest compounded monthly. John Bernoulli discovered something by studying a question about compound interest. (Who is John Bernoulli? Johann Bernoulli (27 July 1667 – 1 January 1748) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infinitesimal calculus and educated Leonhard Euler in his youth. (Source: Wikipedia) Ready to be John Bernoulli? An account starts with $1.00 and pays 100 percent interest per year. If the interest is credited once, at the end of the year, the value of the account at year-end will be $2.00. What happens if the interest is computed and credited more frequently during the year? Write your prediction here: ______________________________________________________ ___________________________________________________________________________ Let’s do solve this mathematically using compound interest rate formula. Frequency (number of compounding Equation Total periods each year.) Once a year $2.00 Twice a year $2.25 Yay! We made $0.25 more! Compound bimonthly Compound monthly Compound weekly Compound daily Compound hourly Compound secondly What did you observe from the above table? Can we ever reach $3? _________________________ ________________________________________________________________________________ Bernoulli noticed that this sequence approaches a limit with larger n and, thus, smaller compounding intervals.
    [Show full text]
  • 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.
    [Show full text]
  • HW 3.1.2 Compound Interest
    HW 3.1.2: Compound Interest n Compounds per year Compound Continuously A= Amount nt P = Principal (initial investment) ⎛⎞r rt AP=+1 A = Pe r = APR (annual percent rate, decimal) ⎜⎟n ⎝⎠ t = number of years n = number of compounds per year Do each of the bullets for Exercises 1-6. • Find the amount, A(t) , in the account as a function of the term of the investment t in years. • Determine how much is in the account after 5 years, 10 years, and 30 years. Round your answers to the nearest cent. • Determine how long will it take for the initial investment to double. Round your answer to the nearest year. 1. $500 is invested in an account, which offers 0.75%, compounded monthly. 2. $500 is invested in an account, which offers 0.75%, compounded continuously. 3. $1000 is invested in an account, which offers 1.25%, compounded quarterly. 4. $1000 is invested in an account, which offers 1.25%, compounded continuously. 5. $5000 is invested in an account, which offers 2.125%, compounded daily. 6. $5000 is invested in an account, which offers 2.125%, compounded continuously. 7. Look back at your answers to Exercises 1-6. What can be said about the differences between monthly, daily, or quarterly compounding and continuously compounding the interest in those situations? 8. How much money needs to be invested now to obtain $2000 in 3 years if the interest rate in a savings account is 0.25%, compounded continuously? Round your answer to the nearest cent. 9. How much money needs to be invested now to obtain $5000 in 10 years if the interest rate in a CD is 2.25%, compounded monthly? Round your answer to the nearest cent.
    [Show full text]
  • An Identity for Generalized Bernoulli Polynomials
    1 2 Journal of Integer Sequences, Vol. 23 (2020), 3 Article 20.11.2 47 6 23 11 An Identity for Generalized Bernoulli Polynomials Redha Chellal1 and Farid Bencherif LA3C, Faculty of Mathematics USTHB Algiers Algeria [email protected] [email protected] [email protected] Mohamed Mehbali Centre for Research Informed Teaching London South Bank University London United Kingdom [email protected] Abstract Recognizing the great importance of Bernoulli numbers and Bernoulli polynomials in various branches of mathematics, the present paper develops two results dealing with these objects. The first one proposes an identity for the generalized Bernoulli poly- nomials, which leads to further generalizations for several relations involving classical Bernoulli numbers and Bernoulli polynomials. In particular, it generalizes a recent identity suggested by Gessel. The second result allows the deduction of similar identi- ties for Fibonacci, Lucas, and Chebyshev polynomials, as well as for generalized Euler polynomials, Genocchi polynomials, and generalized numbers of Stirling. 1Corresponding author. 1 1 Introduction Let N and C denote, respectively, the set of positive integers and the set of complex numbers. (α) In his book, Roman [41, p. 93] defined generalized Bernoulli polynomials Bn (x) as follows: for all n ∈ N and α ∈ C, we have ∞ tn t α B(α)(x) = etx. (1) n n! et − 1 Xn=0 The Bernoulli numbers Bn, classical Bernoulli polynomials Bn(x), and generalized Bernoulli (α) numbers Bn are, respectively, defined by (1) (α) (α) Bn = Bn(0), Bn(x)= Bn (x), and Bn = Bn (0). (2) The Bernoulli numbers and the Bernoulli polynomials play a fundamental role in various branches of mathematics, such as combinatorics, number theory, mathematical analysis, and topology.
    [Show full text]
  • Applications of the Exponential and Natural Logarithm Functions
    M06_GOLD7774_14_SE_C05.indd Page 254 09/11/16 7:31 PM localadmin /202/AW00221/9780134437774_GOLDSTEIN/GOLDSTEIN_CALCULUS_AND_ITS_APPLICATIONS_14E1 ... FOR REVIEW BY POTENTIAL ADOPTERS ONLY chapter 5SAMPLE Applications of the Exponential and Natural Logarithm Functions 5.1 Exponential Growth and Decay 5.4 Further Exponential Models 5.2 Compound Interest 5.3 Applications of the Natural Logarithm Function to Economics n Chapter 4, we introduced the exponential function y = ex and the natural logarithm Ifunction y = ln x, and we studied their most important properties. It is by no means clear that these functions have any substantial connection with the physical world. How- ever, as this chapter will demonstrate, the exponential and natural logarithm functions are involved in the study of many physical problems, often in a very curious and unex- pected way. 5.1 Exponential Growth and Decay Exponential Growth You walk into your kitchen one day and you notice that the overripe bananas that you left on the counter invited unwanted guests: fruit flies. To take advantage of this pesky situation, you decide to study the growth of the fruit flies colony. It didn’t take you too FOR REVIEW long to make your first observation: The colony is increasing at a rate that is propor- tional to its size. That is, the more fruit flies, the faster their number grows. “The derivative is a rate To help us model this population growth, we introduce some notation. Let P(t) of change.” See Sec. 1.7, denote the number of fruit flies in your kitchen, t days from the moment you first p.
    [Show full text]
  • Chapter 5 Dimensional Analysis and Similarity
    Chapter 5 Dimensional Analysis and Similarity Motivation. In this chapter we discuss the planning, presentation, and interpretation of experimental data. We shall try to convince you that such data are best presented in dimensionless form. Experiments which might result in tables of output, or even mul- tiple volumes of tables, might be reduced to a single set of curves—or even a single curve—when suitably nondimensionalized. The technique for doing this is dimensional analysis. Chapter 3 presented gross control-volume balances of mass, momentum, and en- ergy which led to estimates of global parameters: mass flow, force, torque, total heat transfer. Chapter 4 presented infinitesimal balances which led to the basic partial dif- ferential equations of fluid flow and some particular solutions. These two chapters cov- ered analytical techniques, which are limited to fairly simple geometries and well- defined boundary conditions. Probably one-third of fluid-flow problems can be attacked in this analytical or theoretical manner. The other two-thirds of all fluid problems are too complex, both geometrically and physically, to be solved analytically. They must be tested by experiment. Their behav- ior is reported as experimental data. Such data are much more useful if they are ex- pressed in compact, economic form. Graphs are especially useful, since tabulated data cannot be absorbed, nor can the trends and rates of change be observed, by most en- gineering eyes. These are the motivations for dimensional analysis. The technique is traditional in fluid mechanics and is useful in all engineering and physical sciences, with notable uses also seen in the biological and social sciences.
    [Show full text]
  • International Conference Euler's Equations: 250 Years on Program
    International Conference Euler’s Equations: 250 Years On Program of Lectures and Discussions (Dated: June 14, 2007) Schedule of the meeting Tuesday 19 June Wednesday 20 June Thursday 21 June Friday 22 June 08:30-08:40 Opening remarks 08:30-09:20 08:30-09:20 G. Eyink 08:30-09:20 F. Busse 08:40-09:10 E. Knobloch O. Darrigol/U. Frisch 09:30-10:00 P. Constantin 09:30-10:00 Ph. Cardin 09:20-10:10 J. Gibbon 09:20-09:50 M. Eckert 10:00-10:30 T. Hou 10:00-10:30 J.-F. Pinton 10:20-10:50 Y. Brenier 09:50-10:30 Coffee break 10:30-11:00 Coffee break 10:30-11:00 Coffee break 10:50-11:20 Coffee break 10:30-11:20 P. Perrier 11:00-11:30 K. Ohkitani 11:00-11:30 Discussion D 11:20-11:50 K. Sreenivasan 11:20-12:30 Poster session II 11:30-12:00 P. Monkewitz 11:30-12:30 Discussion C 12:00-12:30 C.F. Barenghi 12:00-12:30 G.J.F. van Heijst 12:30-14:00 Lunch break 12:30-14:00 Lunch break 12:30-14:00 Lunch break 12:30-14:00 Lunch break 14:00-14:50 I. Procaccia 14:00-14:30 G. Gallavotti 14:00-14:50 M. Ghil 14:00-14:30 N. Mordant 14:50-15:20 L. Biferale 14:40-15:10 L. Saint Raymond 14:50-15:20 A. Nusser 14:30-15:00 J.
    [Show full text]
  • Compounding-Interest-Notes.Pdf
    AP US Government/Econ Compounding Interest NAME_________ The two most common methods of calculating interest use the simple interest and the compound interest formulas. Simple interest: Simple interest is the dollar cost of borrowing money. This cost is based on three elements: the amount borrowed, which is called the principal; the rate of interest; and the time for which the principal is borrowed. The formula used to find simple interest is: interest = principal x rate of interest x amount of time the loan is outstanding or I = P x R x T Example Suppose you borrow $1,000 at 10 percent simple annual interest and repay it in one lump sum at the end of 3 years. To find the amount that must be repaid, calculate the interest: Interest = $1,000 x .10 x 3 = $300 This computes to $100 of interest each year. The amount that must be repaid is the $1,000 principal plus the $300 interest or a total of $1,300. Compound interest: Unlike simple interest, compound interest calculates interest not only on the principal, but also on the prior period's interest. The formula for calculating compound interest is: Future repayment value = principal x (1 + rate of interest)amount of time or F = P x (1 + R)T The factor (1+ R)T can be obtained easily using pencil and paper, a calculator, or a compound interest table. Most consumer loans use monthly or, possibly, daily compounding. Thus, if you are dealing with monthly compounding, the "R" term in the above formula relates to the monthly interest rate, and "T" equals the number of months in the loan term.
    [Show full text]