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ÉCLAT MATHEMATICS JOURNAL Lady Shri Ram College for Women ECLAT´ MATHEMATICS JOURNAL Lady Shri Ram College For Women Volume X 2018-2019 HYPATIA This journal is dedicated to Hypatia who was a Greek mathematician, astronomer, philosopher and the last great thinker of ancient Alexandria. The Alexandrian scholar authored numerous mathematical treatises, and has been credited with writing commentaries on Diophantuss Arithmetica, on Apolloniuss Conics and on Ptolemys astronomical work. Hypatia is an inspiration, not only as the first famous female mathematician but most importantly as a symbol of learning and science. PREFACE Derived from the French language, Eclat´ means brilliance. Since it’s inception, this journal has aimed at providing a platform for undergraduate students to showcase their understand- ing of diverse mathematical concepts that interest them. As the journal completes a decade this year, it continues to be instrumental in providing an opportunity for students to make valuable contributions to academic inquiry. The work contained in this journal is not original but contains the review research of students. Each of the nine papers included in this year’s edition have been carefully written and compiled to stimulate the thought process of its readers. The four categories discussed in the journal - History of Mathematics, Rigour in Mathematics, Interdisciplinary Aspects of Mathematics and Extension of Course Content - give a comprehensive idea about the evolution of the subject over the years. We would like to express our sincere thanks to the Faculty Advisors of the department, who have guided us at every step leading to the publication of this volume, and to all the authors who have contributed their articles for this volume. We hope that Eclat´ continues to motivate students to go beyond the prescribed limits of the text and to explore many more avenues in the field of mathematics. We are open to any suggestions, corrections and submissions from our readers. This journal is dedicated to Hypatia who was a Greek mathematician, astronomer, philoso- pher and the last great thinker of ancient Alexandria. The Editorial Team Kushagri Tandon, B. Sc. (Hons.) Mathematics, 3rd Year Namrata Lathi, B. Sc. (Hons.) Mathematics, 3rd Year Khuisangmi Konghay, B. Sc. (Hons.) Mathematics, 2nd Year Lipika Parekh, B. Sc. (Hons.) Mathematics, 2nd Year CONTENTS TOPIC PAGE (1) History of Mathematics (a) Euler’s Identity 1 Nayana Nair (b) Decoding the Enigma: Story of Turing and his Machine 5 Tanvi Vohra (2) Rigour in Mathematics (a) Representation of Integers as Sums of Squares 10 Kushagri Tandon (b) Counting Homomorphisms from Zm into Zn 22 Khuisangmi Konghay (3) Extension of Course Content (a) Symmetry Relations between Crystals 27 Richa Sharma (b) Numerical Integration 33 Manishika Negi (c) Non-Orientable Surfaces 41 Rajlaxmi Adwant, Simran, Lipika Parekh (4) Interdisciplinary Aspects of Mathematics (a) Online Ad Auctions 46 Anam Ali and Prakarti Walia (b) Mathematics in Shoelaces 52 Anvita Jain and Shradha Rajpal (c) Elliptic Curve Cryptography 57 Jaya Sharma History of Mathematics Mathematics is one of the oldest academic discipline involving stimulating and intriguing concepts. It is far beyond the ken of one individual and to make any contribution to the evolution of ideas, and understanding of the motivation behind the ideas is needed. The section covers the genesis of mathematical ideas, the stream of thought that created the problem and what led to its solution. The aim is to acquaint the readers with historically important mathematical vignettes and make them inured in some important ideas of Mathematics. Euler’s Identity Nayana Nair Abstract Euler’s identity is the paradigm of mathematical beauty. This equality con- nects five fundamental mathematical constants through three basic arithmetic operations in an elegant equation that displays the powerful connection between them. This paper discusses the significance of Euler’s identity and the various approaches in explaining the same. 1 Leonard Euler Euler’s strokes of genius laid the groundwork for most of the mathematics we have today. He was born in Basel, Switzerland in 1707 and was tutored by Johann Bernoulli at the University of Basel, who instantly discovered his talent for math- ematics. Euler spent most of his career in St. Petersburg and Berlin. His powers of memory and concentration were legendary. Despite his complete loss of vision in 1771, his research continued unabated and his talent to perform prodigious men- tal computations proved indispensable. Like Beethoven who wrote music he never heard, Euler created mathematics he never saw. Euler’s incredibly voluminous and diverse work fills 74 massive volumes of the Opera omnia. Euler left hardly an area of mathematics untouched, putting his mark on such diverse fields as analysis, num- ber theory, mechanics, hydrodynamics, cartography, topology, weights, measures and even the theory of music, to name a few. 2 Euler’s Elegant Equation eiπ = 1 − This peculiar equation first appears in Euler’s Introductio, published in Lausanne in 1748. A unique fusion of exponential and trigonometric, real and complex, this identity is considered as one of the most remarkable identities in all of mathemat- ics and is filled with cosmic beauty. It was from this that Euler deduced strange consequences such as ii = 1 about which the Harvard mathematician Benjamin √eπ Peirce is reported to have said, Gentlemen, we have not the slightest idea of what this equation means, but we may be certain that it means something very important. One of Euler’s greatest achievements was introducing a connection between exponential functions that shoot off to infinity as x becomes large and trigonometric functions that oscillate between the values -1 and 1 with the help of complex numbers. 1 This equation has had a significant impact on the world of mathematics and other sciences, changing many different areas of work and study, and is considered as one of his greatest and most significant contributions. 2.1 Derivation While Euler was solving the differential equa- tion, y(2) + y = 0, he found four solutions ix ix y = sin x, y = cos x, y = e and y = e− . This strange mix of trigonometric and exponen- tial equations showed that these trigonometric functions must be some combination of e and Figure 1: Leonard Euler the imaginary number i. Many continuous, dif- ferentiable functions can be written as a series called a Taylor series. This series for the expo- nential function ex is still valid if x is a complex number. For sine, cosine and ex the series is as follows: x3 x5 x7 sin x = x + + − 3! 5! − 7! ··· x2 x4 x6 cos x = 1 + + − 2! 4! − 6! ··· x2 x3 x4 x5 x6 x7 ex = 1 + x + + + + + + + 2 3! 4! 5! 6! 7! ··· Replacing x with ix in the expansion of ex we get the following series: (ix)2 (ix)3 (ix)4 (ix)5 (ix)6 (ix)7 eix = 1 + ix + + + + + + + 2 3! 4! 5! 6! 7! ··· x2 x3 x4 x5 x6 x7 = 1 + ix i + + i i + − 2 − 3! 4! 5! − 6! − 7! ··· x2 x4 x6 x3 x5 x7 = 1 + + + i x + + − 2 4! − 6! ··· − 3! 5! − 7! ··· = cos x + i sin x For x = π eiπ = 1 − x x n ix n Since e = limn (1 + ) , we can conclude that the value of limn (1 + ) →∞ n →∞ n tends closer and closer to -1. This is equivalent to n repeated multiplications in the complex plane with the final value coinciding with -1. 2 2.2 Theoretical Uses Euler’s identity has several applications within mathematics. It allows standard trigonometric functions to be expressed in terms of exponential functions as cos x = ix ix ix ix e +e− e e− 2 and sin x = −2i and can be used in verifying the addition and subtrac- tion formulas for sine and cosine. 2.2.1 Logarithms of Negative Numbers Euler’s identity gave a greater understanding of complex numbers. The unusual idea of finding the logarithm of a negative number became a reality as from the identity it follows that ln( 1) = iπ. Any odd multiple of π will give an expression − for ln( 1), i.e ln( 1) = i(2n 1)π where n Z. − − − ∈ 2.2.2 Complex Numbers Raised to Complex Numbers The bizarre yet remarkable consequence of this identity is that it allows us to com- pute complex powers of complex numbers which turn out to be real numbers. For iπ π 2 example substituting x = 2 in the identity we get e = i, now raising this to i we get i π i = e −2 0.2078796 ≈ These repercussions of Euler’s identity allows us to manipulate complex numbers and has profound implications in complex number theory. 3 Modern Applications Euler’s identity has extensive applications in electronics and engineering. It is im- plemented in linear, time invariant input-output boxes also known as LTI boxes which takes values of xn(t) and output the values of yn(t) through multiplication by some constant which could be a complex number. With the help of Euler’s formula, scientists are able to understand planetary retrograde motion which is the apparent motion of a planet in a direction opposite to that of other bodies in a system with respect to a particular point, at complex values of time which earlier would have seemed illogical. 4 Conclusion Euler’s identity brought about a paradigm shift in the imaginary field and its relation with the real world. This identity has a profound impact on nearly every field of 3 theoretical mathematics as well as in real world problem solving. From formulating the basic laws of fluid mechanics to remarkable research in number theory and analytical mechanics, Euler’s indispensable and influential contributions to various branches of mathematics make him a true mathematical pioneer. References [1] Dunham, W., Euler: The Master of Us All, The Mathematical Association of America (1999) [2] Maor, E., e: The Story of a Number, The Princeton University Press (1994) [3] Mallett, T., Christensen, S.
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