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Math Girls Talk About Integers introduces students to a variety of fun and informative topics in discrete math, including curious features of the prime numbers, tricks for checking for multiples of 3 and 9 (and why those tricks work!), using division remainders to solve some unusual problems, and an in-depth look at proof by mathematical induction. These topics are introduced through conversations between the characters from Math Girls, offering a fun way to learn this serious content. Each chapter comes with review problems and answers, and an appendix gives more challenging, open-ended problems for readers wanting to push the limits of their understanding.
In the early twentieth century, a massive undertaking to rid mathematics of all paradoxes and inconsistencies was underway. Known as Hilbert's program, it sought to provide an unshakable foundation for all of mathematics. Things seemed to be proceeding well until young Kurt Godel stunned the world by proving that Hilbert's goals were unobtainable, that contradiction was part of the warp and weave of any mathematical system. Yet what at the time seemed to be a fatal blow to mathematical consistency now forms the basis of modern logic. Godel's incompleteness theorems are often misunderstood to be a statement of the limits of mathematical reasoning, but in truth they strengthen mathematics, building it up to be more powerful than what had come before. In this third book in the Math Girls series, join Miruka and friends as they tackle the basics of modern logic, learning such topics as the Peano axioms, set theory, and diagonalization, leading up to an in-depth exploration of Godel's famous theorems. Along the way, visit other interesting and important topics such as trigonometry and the epsilon-delta definition of limits, and of course take on challenges from the enigmatic Mr. Muraki. Math Girls 3: Godel's Incompleteness Theorems has something for anyone interested in mathematics, from advanced high school students to college math majors and educators."
From the author of Math Girls comes an exciting new series for learning and reviewing important skills for taking on advanced mathematics! This first volume, Math Girls Talk About Equations and Graphs, develops topics such as using variables in equations, polynomials, setting up systems of equations, proportions and inverse proportions, the relation between equations and their graphs, parabolas, intersections, and tangent lines. These topics are introduced through conversations between the characters from Math Girls, offering a fun way to learn this serious content. Each chapter comes with review problems and answers, and an appendix gives more challenging, open-ended problems for learners wanting to push the limits of their understanding. This book is most suited to middle- or high-school students who have learned basic algebra, or older readers who want to brush up on forgotten math skills. This series came about through requests from readers who enjoyed the excitement of learning aspects of the Math Girls series, but found themselves unprepared to keep up with the mathematical content. We hope that the books in this series will help young mathematicians firm up vital math skills that will allow them to excel in more advanced studies.
Explores a variety of fun and informative topics in trigonometry, from basics like defining the sine and cosine functions, to less frequently seen topics like Lissajous curves and different ways of deriving the value of pi. These topics are introduced through conversations between the characters from the Math Girls series, offering a fun way to learn this serious content. The third in a series aimed at preparing students for advanced mathematics studies.
This fourth entry in the highly acclaimed Math Girls series focuses on the mathematics of computer science and analysis of algorithms. Aimed at anyone interested in mathematics and computer science, from advanced high school students to college students and educators.
This fifth entry in the highly acclaimed Math Girls series focuses on the mathematics of Évariste Galois, the nineteenth-century wunderkind who revolutionized mathematics with work he performed while still a teenager. Mathematicians before him had discovered solutions to general second-, third-, and fourth-degree equations, but a similar "quintic formula" that would allow knowing the solutions to any fifth-degree equation had eluded mathematicians for centuries. Through his ingenious approach of bridging the worlds of groups and fields, young Galois not only showed that such a formula was impossible, he newly developed group theory and the branch of mathematics that today bears his name. Join Miruka and friends to see how Galois developed his theory, along with related topics such as geometric constructions and the angle trisection problem, derivation of the cubic formula, reducible and irreducible polynomials, group theory and field theory, symmetric polynomials, roots of unity, sets and cosets, cyclotomic polynomials, vector spaces, extension fields, and symmetric groups. The book concludes with a tour through Galois's first paper, in which he describes for the first time the necessary and sufficient conditions for a polynomial to be algebraically solved using radicals. Math Girls 5: Galois Theory has something for anyone interested in mathematics, from advanced high school to college students and educators.
Mathematics is beautiful--and it can be fun and exciting as well as practical. Good Math is your guide to some of the most intriguing topics from two thousand years of mathematics: from Egyptian fractions to Turing machines; from the real meaning of numbers to proof trees, group symmetry, and mechanical computation. If you've ever wondered what lay beyond the proofs you struggled to complete in high school geometry, or what limits the capabilities of computer on your desk, this is the book for you. Why do Roman numerals persist? How do we know that some infinities are larger than others? And how can we know for certain a program will ever finish? In this fast-paced tour of modern and not-so-modern math, computer scientist Mark Chu-Carroll explores some of the greatest breakthroughs and disappointments of more than two thousand years of mathematical thought. There is joy and beauty in mathematics, and in more than two dozen essays drawn from his popular "Good Math" blog, you'll find concepts, proofs, and examples that are often surprising, counterintuitive, or just plain weird. Mark begins his journey with the basics of numbers, with an entertaining trip through the integers and the natural, rational, irrational, and transcendental numbers. The voyage continues with a look at some of the oddest numbers in mathematics, including zero, the golden ratio, imaginary numbers, Roman numerals, and Egyptian and continuing fractions. After a deep dive into modern logic, including an introduction to linear logic and the logic-savvy Prolog language, the trip concludes with a tour of modern set theory and the advances and paradoxes of modern mechanical computing. If your high school or college math courses left you grasping for the inner meaning behind the numbers, Mark's book will both entertain and enlighten you.
Developed from celebrated Harvard statistics lectures, Introduction to Probability provides essential language and tools for understanding statistics, randomness, and uncertainty. The book explores a wide variety of applications and examples, ranging from coincidences and paradoxes to Google PageRank and Markov chain Monte Carlo (MCMC). Additional application areas explored include genetics, medicine, computer science, and information theory. The print book version includes a code that provides free access to an eBook version. The authors present the material in an accessible style and motivate concepts using real-world examples. Throughout, they use stories to uncover connections between the fundamental distributions in statistics and conditioning to reduce complicated problems to manageable pieces. The book includes many intuitive explanations, diagrams, and practice problems. Each chapter ends with a section showing how to perform relevant simulations and calculations in R, a free statistical software environment.
A Programmer's Introduction to Mathematics uses your familiarity with ideas from programming and software to teach mathematics. You'll learn about the central objects and theorems of mathematics, including graphs, calculus, linear algebra, eigenvalues, optimization, and more. You'll also be immersed in the often unspoken cultural attitudes of mathematics, learning both how to read and write proofs while understanding why mathematics is the way it is. Between each technical chapter is an essay describing a different aspect of mathematical culture, and discussions of the insights and meta-insights that constitute mathematical intuition. As you learn, we'll use new mathematical ideas to create wondrous programs, from cryptographic schemes to neural networks to hyperbolic tessellations. Each chapter also contains a set of exercises that have you actively explore mathematical topics on your own. In short, this book will teach you to engage with mathematics. A Programmer's Introduction to Mathematics is written by Jeremy Kun, who has been writing about math and programming for 10 years on his blog "Math Intersect Programming." As of 2020, he works in datacenter optimization at Google.The second edition includes revisions to most chapters, some reorganized content and rewritten proofs, and the addition of three appendices.