Kirjahaku

Fractions are everywhere and yet most of us learn only basic and rather dry facts about fractions in primary school. This book makes fractions come to life in a friendly, lively, and accessible way, detailing the history of fractions and their crucial role in the work of mathematicians from various cultures throughout the ages. The book begins by outlining the importance of rational numbers and links ancient Babylonian mathematics with modern processes for determining their decimal expansions and the period length of repeating decimals, which are worked out in full. This then leads to the study of infinite sums, especially to geometric series and the notions of convergence and divergence. The text goes on to explain the importance of the Fibonacci numbers, as well as the Cantor set and the Sierpinski carpet. Much of elementary number theory is introduced including congruence classes, the Euler phi function, the Euclidean algorithm, and some Diophantine equations. The book also discusses many historical applications of fractions, including Christiaan Huygens's cogwheeled planetarium and Archimedes's approximation to the value of pi, as well as an extensive study of the importance of Egyptian fractions. Finally, it outlines modern applications of fractions, such as the fair apportionment of a cake, variations on slicing a pizza, probability questions involving markings on a stick, and ways to divide a bar of gold in order to pay wages for various numbers of days. Accessible to anyone with a passion for the history of mathematics who wishes to delve deeper into the wonderful world of fractions, this book will also be of special interest to teachers of mathematics and students of all ages.

A colorful tour through the intriguing world of mathematics The world of modern mathematics abounds with fascinating, unusual ideas–ideas and concepts even seasoned mathematicians often wonder about. Mathematical Journeys takes you on a grand tour of the best of modern math–its most elegant solutions, most clever discoveries, most mind-bending propositions, and most impressive personalities. Writing with a light touch while showing the real mathematics, author Peter Schumer introduces you to the history of mathematics, number theory, combinatorics, geometry, graph theory, and "recreational mathematics." Requiring only high school math and a healthy curiosity, Mathematical Journeys helps you explore all those aspects of math that mathematicians themselves find most delightful. You’ll discover brilliant, sometimes quirky and humorous tidbits like how to compute the digits of pi, the Josephus problem, mathematical amusements such as Nim and Wythoff’s game, pizza slicing, and clever twists on rolling dice. For a glimpse of the minds that gave birth to the math, read the profiles of such great thinkers as Paul Erdös and Leonhard Euler. Each chapter of the book focuses on some interesting piece of mathematics, giving the history and requisite math background, the solution of a problem or two, and some indication of natural generalizations and related areas of study. Whether you’re a math novice curious to learn what your calculus class left out or a math lover ready for the green chicken contest (What’s that? Read the book!), Mathematical Journeys will give you a true taste of what mathematicians themselves find most exciting about math.

This is a book for an undergraduate number theory course, senior thesis work, graduate level study, or for those wishing to learn about applications of number theory to data encryption and security. With no abstract algebra background required, it covers congruences, the Euclidean algorithm, linear Diophantine equations, the Chinese Remainder Theorem, Mobius inversion formula, Pythagorean triplets, perfect numbers and amicable pairs, Law of Quadratic Reciprocity, theorems on sums of squares, Farey fractions, periodic continued fractions, best rational approximations, and Pell’s equation. Results are applied to factoring and primality testing including those for Mersenne and Fermat primes, probabilistic primality tests, Pollard’s rho and p-1 factorization algorithms, and others. Also an introduction to cryptology with a full discussion of the RSA algorithm, discrete logarithms, and digital signatures. Chapters on analytic number theory including the Riemann zeta function, average orders of the lattice and divisor functions, Chebyshev’s theorems, and Bertrand’s Postulate. A chapter introduces additive number theory with discussion of Waring’s Problem, the pentagonal number theorem for partitions, and Schnirelmann density.