11 Permutations, Combinations, and the Binomial Theorem
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An Appreciation of Euler's Formula
Rose-Hulman Undergraduate Mathematics Journal Volume 18 Issue 1 Article 17 An Appreciation of Euler's Formula Caleb Larson North Dakota State University Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Larson, Caleb (2017) "An Appreciation of Euler's Formula," Rose-Hulman Undergraduate Mathematics Journal: Vol. 18 : Iss. 1 , Article 17. Available at: https://scholar.rose-hulman.edu/rhumj/vol18/iss1/17 Rose- Hulman Undergraduate Mathematics Journal an appreciation of euler's formula Caleb Larson a Volume 18, No. 1, Spring 2017 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 [email protected] a scholar.rose-hulman.edu/rhumj North Dakota State University Rose-Hulman Undergraduate Mathematics Journal Volume 18, No. 1, Spring 2017 an appreciation of euler's formula Caleb Larson Abstract. For many mathematicians, a certain characteristic about an area of mathematics will lure him/her to study that area further. That characteristic might be an interesting conclusion, an intricate implication, or an appreciation of the impact that the area has upon mathematics. The particular area that we will be exploring is Euler's Formula, eix = cos x + i sin x, and as a result, Euler's Identity, eiπ + 1 = 0. Throughout this paper, we will develop an appreciation for Euler's Formula as it combines the seemingly unrelated exponential functions, imaginary numbers, and trigonometric functions into a single formula. To appreciate and further understand Euler's Formula, we will give attention to the individual aspects of the formula, and develop the necessary tools to prove it. -
Newton and Leibniz: the Development of Calculus Isaac Newton (1642-1727)
Newton and Leibniz: The development of calculus Isaac Newton (1642-1727) Isaac Newton was born on Christmas day in 1642, the same year that Galileo died. This coincidence seemed to be symbolic and in many ways, Newton developed both mathematics and physics from where Galileo had left off. A few months before his birth, his father died and his mother had remarried and Isaac was raised by his grandmother. His uncle recognized Newton’s mathematical abilities and suggested he enroll in Trinity College in Cambridge. Newton at Trinity College At Trinity, Newton keenly studied Euclid, Descartes, Kepler, Galileo, Viete and Wallis. He wrote later to Robert Hooke, “If I have seen farther, it is because I have stood on the shoulders of giants.” Shortly after he received his Bachelor’s degree in 1665, Cambridge University was closed due to the bubonic plague and so he went to his grandmother’s house where he dived deep into his mathematics and physics without interruption. During this time, he made four major discoveries: (a) the binomial theorem; (b) calculus ; (c) the law of universal gravitation and (d) the nature of light. The binomial theorem, as we discussed, was of course known to the Chinese, the Indians, and was re-discovered by Blaise Pascal. But Newton’s innovation is to discuss it for fractional powers. The binomial theorem Newton’s notation in many places is a bit clumsy and he would write his version of the binomial theorem as: In modern notation, the left hand side is (P+PQ)m/n and the first term on the right hand side is Pm/n and the other terms are: The binomial theorem as a Taylor series What we see here is the Taylor series expansion of the function (1+Q)m/n. -
Mathematics (MATH)
Course Descriptions MATH 1080 QL MATH 1210 QL Mathematics (MATH) Precalculus Calculus I 5 5 MATH 100R * Prerequisite(s): Within the past two years, * Prerequisite(s): One of the following within Math Leap one of the following: MAT 1000 or MAT 1010 the past two years: (MATH 1050 or MATH 1 with a grade of B or better or an appropriate 1055) and MATH 1060, each with a grade of Is part of UVU’s math placement process; for math placement score. C or higher; OR MATH 1080 with a grade of C students who desire to review math topics Is an accelerated version of MATH 1050 or higher; OR appropriate placement by math in order to improve placement level before and MATH 1060. Includes functions and placement test. beginning a math course. Addresses unique their graphs including polynomial, rational, Covers limits, continuity, differentiation, strengths and weaknesses of students, by exponential, logarithmic, trigonometric, and applications of differentiation, integration, providing group problem solving activities along inverse trigonometric functions. Covers and applications of integration, including with an individual assessment and study inequalities, systems of linear and derivatives and integrals of polynomial plan for mastering target material. Requires nonlinear equations, matrices, determinants, functions, rational functions, exponential mandatory class attendance and a minimum arithmetic and geometric sequences, the functions, logarithmic functions, trigonometric number of hours per week logged into a Binomial Theorem, the unit circle, right functions, inverse trigonometric functions, and preparation module, with progress monitored triangle trigonometry, trigonometric equations, hyperbolic functions. Is a prerequisite for by a mentor. May be repeated for a maximum trigonometric identities, the Law of Sines, the calculus-based sciences. -
The Discovery of the Series Formula for Π by Leibniz, Gregory and Nilakantha Author(S): Ranjan Roy Source: Mathematics Magazine, Vol
The Discovery of the Series Formula for π by Leibniz, Gregory and Nilakantha Author(s): Ranjan Roy Source: Mathematics Magazine, Vol. 63, No. 5 (Dec., 1990), pp. 291-306 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2690896 Accessed: 27-02-2017 22:02 UTC JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://about.jstor.org/terms Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to Mathematics Magazine This content downloaded from 195.251.161.31 on Mon, 27 Feb 2017 22:02:42 UTC All use subject to http://about.jstor.org/terms ARTICLES The Discovery of the Series Formula for 7r by Leibniz, Gregory and Nilakantha RANJAN ROY Beloit College Beloit, WI 53511 1. Introduction The formula for -r mentioned in the title of this article is 4 3 57 . (1) One simple and well-known moderm proof goes as follows: x I arctan x = | 1 +2 dt x3 +5 - +2n + 1 x t2n+2 + -w3 - +(-I)rl2n+1 +(-I)l?lf dt. The last integral tends to zero if Ix < 1, for 'o t+2dt < jt dt - iX2n+3 20 as n oo. -
Useful Relations in Permutations and Combination 1. Useful Relations
Useful Relations in Permutations and Combination 1. Useful Relations - Factorial n! = n.(n-1)! 2. n퐶푟= n푃푟/r! n n-1 3. Pr = n( Pr-1) 4. Useful Relations - Combinations n n 1. Cr = C(n - r) Example 8 8 C6 = C2 = 8×72×1 = 28 n 2. Cn = 1 n 3. C0 = 1 n n n n n 4. C0 + C1 + C2 + ... + Cn = 2 Example 4 4 4 4 4 4 C0 + C1 + C2 + C3+ C4 = (1 + 4 + 6 + 4 + 1) = 16 = 2 n n (n+1) Cr-1 + Cr = Cr (Pascal's Law) n퐶푟 =n/퐶푟−1=n-r+1/r n n If Cx = Cy then either x = y or (n-x) = y. 5. Selection from identical objects: Some Basic Facts The number of selections of r objects out of n identical objects is 1. Total number of selections of zero or more objects from n identical objects is n+1. 6. Permutations of Objects when All Objects are Not Distinct The number of ways in which n things can be arranged taking them all at a time, when st nd 푃1 of the things are exactly alike of 1 type, 푃2 of them are exactly alike of a 2 type, and th 푃푟of them are exactly alike of r type and the rest of all are distinct is n!/ 푃1! 푃2! ... 푃푟! 1 Example: how many ways can you arrange the letters in the word THESE? 5!/2!=120/2=60 Example: how many ways can you arrange the letters in the word REFERENCE? 9!/2!.4!=362880/2*24=7560 7.Circular Permutations: Case 1: when clockwise and anticlockwise arrangements are different Number of circular permutations (arrangements) of n different things is (n-1)! 1. -
1. More Examples Example 1. Find the Slope of the Tangent Line at (3,9)
1. More examples Example 1. Find the slope of the tangent line at (3; 9) to the curve y = x2. P = (3; 9). So Q = (3 + ∆x; 9 + ∆y). 9 + ∆y = (3 + ∆x)2 = (9 + 6∆x + (∆x)2. Thus, we have ∆y = 6 + ∆x. ∆x If we let ∆ go to zero, we get ∆y lim = 6 + 0 = 6. ∆x→0 ∆x Thus, the slope of the tangent line at (3; 9) is 6. Let's consider a more abstract example. Example 2. Find the slope of the tangent line at an arbitrary point P = (x; y) on the curve y = x2. Again, P = (x; y), and Q = (x + ∆x; y + ∆y), where y + ∆y = (x + ∆x)2 = x2 + 2x∆x + (∆x)2. So we get, ∆y = 2x + ∆x. ∆x Letting ∆x go to zero, we get ∆y lim = 2x + 0 = 2x. ∆x→0 ∆x Thus, the slope of the tangent line at an arbitrary point (x; y) is 2x. Example 3. Find the slope of the tangent line at an arbitrary point P = (x; y) on the curve y = ax3, where a is a real number. Q = (x + ∆x; y + ∆y). We get, 1 2 y + ∆y = a(x + ∆x)3 = a(x3 + 3x2∆x + 3x(∆x)2 + (∆x)3). After simplifying we have, ∆y = 3ax2 + 3xa∆x + a(∆x)2. ∆x Letting ∆x go to 0 we conclude, ∆y 2 2 mP = lim = 3ax + 0 + 0 = 3ax . x→0 ∆x 2 Thus, the slope of the tangent line at a point (x; y) is mP = 3ax . Example 4. -
Combinatorics
Combinatorics Problem: How to count without counting. I How do you figure out how many things there are with a certain property without actually enumerating all of them. Sometimes this requires a lot of cleverness and deep mathematical insights. But there are some standard techniques. I That's what we'll be studying. We sometimes use the bijection rule without even realizing it: I count how many people voted are in favor of something by counting the number of hands raised: I I'm hoping that there's a bijection between the people in favor and the hands raised! Bijection Rule The Bijection Rule: If f : A ! B is a bijection, then jAj = jBj. I We used this rule in defining cardinality for infinite sets. I Now we'll focus on finite sets. Bijection Rule The Bijection Rule: If f : A ! B is a bijection, then jAj = jBj. I We used this rule in defining cardinality for infinite sets. I Now we'll focus on finite sets. We sometimes use the bijection rule without even realizing it: I count how many people voted are in favor of something by counting the number of hands raised: I I'm hoping that there's a bijection between the people in favor and the hands raised! Answer: 26 choices for the first letter, 26 for the second, 10 choices for the first number, the second number, and the third number: 262 × 103 = 676; 000 Example 2: A traveling salesman wants to do a tour of all 50 state capitals. How many ways can he do this? Answer: 50 choices for the first place to visit, 49 for the second, . -
Inclusion‒Exclusion Principle
Inclusionexclusion principle 1 Inclusion–exclusion principle In combinatorics, the inclusion–exclusion principle (also known as the sieve principle) is an equation relating the sizes of two sets and their union. It states that if A and B are two (finite) sets, then The meaning of the statement is that the number of elements in the union of the two sets is the sum of the elements in each set, respectively, minus the number of elements that are in both. Similarly, for three sets A, B and C, This can be seen by counting how many times each region in the figure to the right is included in the right hand side. More generally, for finite sets A , ..., A , one has the identity 1 n This can be compactly written as The name comes from the idea that the principle is based on over-generous inclusion, followed by compensating exclusion. When n > 2 the exclusion of the pairwise intersections is (possibly) too severe, and the correct formula is as shown with alternating signs. This formula is attributed to Abraham de Moivre; it is sometimes also named for Daniel da Silva, Joseph Sylvester or Henri Poincaré. Inclusion–exclusion illustrated for three sets For the case of three sets A, B, C the inclusion–exclusion principle is illustrated in the graphic on the right. Proof Let A denote the union of the sets A , ..., A . To prove the 1 n Each term of the inclusion-exclusion formula inclusion–exclusion principle in general, we first have to verify the gradually corrects the count until finally each identity portion of the Venn Diagram is counted exactly once. -
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. -
STRUCTURE ENUMERATION and SAMPLING Chemical Structure Enumeration and Sampling Have Been Studied by Mathematicians, Computer
STRUCTURE ENUMERATION AND SAMPLING MARKUS MERINGER To appear in Handbook of Chemoinformatics Algorithms Chemical structure enumeration and sampling have been studied by 5 mathematicians, computer scientists and chemists for quite a long time. Given a molecular formula plus, optionally, a list of structural con- straints, the typical questions are: (1) How many isomers exist? (2) Which are they? And, especially if (2) cannot be answered completely: (3) How to get a sample? 10 In this chapter we describe algorithms for solving these problems. The techniques are based on the representation of chemical compounds as molecular graphs (see Chapter 2), i.e. they are mainly applied to constitutional isomers. The major problem is that in silico molecular graphs have to be represented as labeled structures, while in chemical 15 compounds, the atoms are not labeled. The mathematical concept for approaching this problem is to consider orbits of labeled molecular graphs under the operation of the symmetric group. We have to solve the so–called isomorphism problem. According to our introductory questions, we distinguish several dis- 20 ciplines: counting, enumerating and sampling isomers. While counting only delivers the number of isomers, the remaining disciplines refer to constructive methods. Enumeration typically encompasses exhaustive and non–redundant methods, while sampling typically lacks these char- acteristics. However, sampling methods are sometimes better suited to 25 solve real–world problems. There is a wide range of applications where counting, enumeration and sampling techniques are helpful or even essential. Some of these applications are closely linked to other chapters of this book. Counting techniques deliver pure chemical information, they can help to estimate 30 or even determine sizes of chemical databases or compound libraries obtained from combinatorial chemistry. -
Counting: Permutations, Combinations
CS 441 Discrete Mathematics for CS Lecture 17 Counting Milos Hauskrecht [email protected] 5329 Sennott Square CS 441 Discrete mathematics for CS M. Hauskrecht Counting • Assume we have a set of objects with certain properties • Counting is used to determine the number of these objects Examples: • Number of available phone numbers with 7 digits in the local calling area • Number of possible match starters (football, basketball) given the number of team members and their positions CS 441 Discrete mathematics for CS M. Hauskrecht 1 Basic counting rules • Counting problems may be hard, and easy solutions are not obvious • Approach: – simplify the solution by decomposing the problem • Two basic decomposition rules: – Product rule • A count decomposes into a sequence of dependent counts (“each element in the first count is associated with all elements of the second count”) – Sum rule • A count decomposes into a set of independent counts (“elements of counts are alternatives”) CS 441 Discrete mathematics for CS M. Hauskrecht Inclusion-Exclusion principle Used in counts where the decomposition yields two count tasks with overlapping elements • If we used the sum rule some elements would be counted twice Inclusion-exclusion principle: uses a sum rule and then corrects for the overlapping elements. We used the principle for the cardinality of the set union. •|A B| = |A| + |B| - |A B| U B A CS 441 Discrete mathematics for CS M. Hauskrecht 2 Inclusion-exclusion principle Example: How many bitstrings of length 8 start either with a bit 1 or end with 00? • Count strings that start with 1: • How many are there? 27 • Count the strings that end with 00. -
The Legacy of Leonhard Euler: a Tricentennial Tribute (419 Pages)
P698.TP.indd 1 9/8/09 5:23:37 PM This page intentionally left blank Lokenath Debnath The University of Texas-Pan American, USA Imperial College Press ICP P698.TP.indd 2 9/8/09 5:23:39 PM Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. THE LEGACY OF LEONHARD EULER A Tricentennial Tribute Copyright © 2010 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13 978-1-84816-525-0 ISBN-10 1-84816-525-0 Printed in Singapore. LaiFun - The Legacy of Leonhard.pmd 1 9/4/2009, 3:04 PM September 4, 2009 14:33 World Scientific Book - 9in x 6in LegacyLeonhard Leonhard Euler (1707–1783) ii September 4, 2009 14:33 World Scientific Book - 9in x 6in LegacyLeonhard To my wife Sadhana, grandson Kirin,and granddaughter Princess Maya, with love and affection.