
Catalan Numbers with Applications
Thomas Koshy
Like the intriguing Fibonacci and Lucas numbers, Catalan numbers are also ubiquitous. "They have the same delightful propensity for popping up unexpectedly, particularly in combinatorial problems," Martin Gardner wrote in Scientific American. "Indeed, the Catalan sequence is probably the most frequently encountered sequence that is still obscure enough to cause mathematicians lacking access to Sloane's Handbook of Integer Sequences to expend inordinate amounts of energy rediscovering formulas that were worked out long ago," he continued. As Gardner noted, many mathematicians may know the abc's of Catalan sequence, but not many are familiar with the myriad of their unexpected occurrences, applications, and properties; they crop up in chess boards, computer programming, and even train tracks. This book presents a clear and comprehensive introduction to one of the truly fascinating topics in mathematics. Catalan numbers are named after the Belgian mathematician Eugene Charles Catalan (18141894), who "discovered" them in 1838, though he was not the first person to discover them. The great Swiss mathematician Leonhard Euler (17071763) "discovered" them around 1756, but even before then and though his work was not known to the outside world, Chinese mathematician Antu Ming (1692?1763) first discovered Catalan numbers about 1730. Catalan numbers can be used by teachers and professors to generate excitement among students for exploration and intellectual curiosity and to sharpen a variety of mathematical skills and tools, such as pattern recognition, conjecturing, prooftechniques, and problemsolving techniques. This book is not only intended for mathematicians but for a much larger audience, including high school students, math and science teachers, computer scientists, and those amateurs with a modicum of mathematical curiosity. An invaluable resource book, it contains an intriguing array of applications to computer science, abstract algebra, combinatorics, geometry, graph theory, chess, and World Series.

Discrete Mathematics With Applications
Thomas Koshy
This approachable text studies discrete objects and the relationsips that bind them. It helps students understand and apply the power of discrete math to digital computer systems and other modern applications. It provides excellent preparation for courses in linear algebra, number theory, and modern/abstract algebra and for computer science courses in data structures, algorithms, programming languages, compilers, databases, and computation.

Elementary Number Theory with Applications
Thomas Koshy
This second edition updates the wellregarded 2001 publication with new short sections on topics like Catalan numbers and their relationship to Pascal's triangle and Mersenne numbers, Pollard rho factorization method, HoggattHensell identity. Koshy has added a new chapter on continued fractions. The unique features of the first edition like news of recent discoveries, biographical sketches of mathematicians, and applicationslike the use of congruence in scheduling of a roundrobin tournamentare being refreshed with current information. More challenging exercises are included both in the textbook and in the instructor's manual. Elementary Number Theory with Applications 2e is ideally suited for undergraduate students and is especially appropriate for prospective and inservice math teachers at the high school and middle school levels.

Elementary Number Theory with Applications
Thomas Koshy
The advent of modern technology has brought a new dimension to the power of number theory: constant practical use. Once considered the purest of pure mathematics, it is used increasingly now in the rapid development of technology in a number of areas, such as art, coding theory, cryptology, computer science, and other necessities of modern life. Elementary Number Theory with Applications is the fruit of years of dreams and the author's fascination with the subject, encapsulating the beauty, elegance, historical development, and opportunities provided for experimentation and application. This is the only number theory book to show how modular systems can be employed to create beautiful designs, thus linking number theory with both geometry and art. It is also the only number theory book to deal with bar codes, Zip codes, International Standard Book Numbers (ISBN), and European Article Numbers (EAN). Emphasis is on problemsolving strategies (doing experiments, collecting and organizing data, recognizing patterns, and making conjectures). Each section provides a wealth of carefully prepared, wellgraded examples and exercises to enhance the readers' understanding and problemsolving skills

Fibonacci and Lucas Numbers with Applications
Thomas Koshy
The first comprehensive survey of mathematics' most fascinating number sequences Fibonacci and Lucas numbers have intrigued amateur and professional mathematicians for centuries. This volume represents the first attempt to compile a definitive history and authoritative analysis of these famous integer sequences, complete with a wealth of exciting applications, enlightening examples, and fun exercises that offer numerous opportunities for exploration and experimentation. The author has assembled a myriad of fascinating properties of both Fibonacci and Lucas numbersas developed by a wide range of sourcesand catalogued their applications in a multitude of widely varied disciplines such as art, stock market investing, engineering, and neurophysiology. Most of the engaging and delightful material here is easily accessible to college and even high school students, though advanced material is included to challenge more sophisticated Fibonacci enthusiasts. A historical survey of the development of Fibonacci and Lucas numbers, biographical sketches of intriguing personalities involved in developing the subject, and illustrative examples round out this thorough and amusing survey. Most chapters conclude with numeric and theoretical exercises that do not rely on long and tedious proofs of theorems.

Triangular Arrays with Applications
Thomas Koshy
Triangular arrays are a unifying thread throughout various areas of discrete mathematics such as number theory and combinatorics. They can be used to sharpen a variety of mathematical skills and tools, such as pattern recognition, conjecturing, prooftechniques, and problemsolving techniques. While a good deal of research exists concerning triangular arrays and their applications, the information is scattered in various journals and is inaccessible to many mathematicians. This is the first text that will collect and organize the information and present it in a clear and comprehensive introduction to the topic. An invaluable resource book, it gives a historical introduction to Pascal's triangle and covers application topics such as binomial coefficients, figurate numbers, Fibonacci and Lucas numbers, Pell and PellLucas numbers, graph theory, Fibonomial and tribinomial coefficients and Fibonacci and Lucas polynomials, amongst others. The book also features the historical development of triangular arrays, including short biographies of prominent mathematicians, along with the name and affiliation of every discoverer and year of discovery. The book is intended for mathematicians as well as computer scientists, math and science teachers, advanced high school students, and those with mathematical curiosity and maturity.

Common Core Mathematics Standards and Implementing Digital Technologies
Drew Polly
The Advances in Educational Technologies and Instructional Design (AETID) Book Series is a resource where researchers, students, administrators, and educators alike can find the most updated research and theories regarding technology's integration within education and its effect on teaching as a practice.
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