Solving The Equation

Pell’s Equation is a very simple Diophantine equation that has been known to mathematicians for over 2000 years. Even today research involving this equation continues to be very active, as can be seen by the publication of at least 150 articles related to this equation over the past decade. However, very few modern books have been published on Pell’s Equation, and this will be the first to give a historical development of the equation, as well as to develop the necessary tools for solving the equation. The authors provide a friendly introduction for advanced undergraduates to the delights of algebraic number theory via Pell’s Equation. The only prerequisites are a basic knowledge of elementary number theory and abstract algebra. There are also numerous references and notes for those who wish to follow up on various topics.

In this “powerful personal story woven with a rich analysis of what we all seek” (Sergey Brin, cofounder of Google), Mo Gawdat, Chief Business Officer at Google’s [X], applies his superior logic and problem solving skills to understand how the brain processes joy and sadness—and then he solves for happy. In 2001 Mo Gawdat realized that despite his incredible success, he was desperately unhappy. A lifelong learner, he attacked the problem as an engineer would: examining all the provable facts and scrupulously applying logic. Eventually, his countless hours of research and science proved successful, and he discovered the equation for permanent happiness. Thirteen years later, Mo’s algorithm would be put to the ultimate test. After the sudden death of his son, Ali, Mo and his family turned to his equation—and it saved them from despair. In dealing with the horrible loss, Mo found his mission: he would pull off the type of “moonshot” goal that he and his colleagues were always aiming for—he would share his equation with the world and help as many people as possible become happier. In Solve for Happy Mo questions some of the most fundamental aspects of our existence, shares the underlying reasons for suffering, and plots out a step-by-step process for achieving lifelong happiness and enduring contentment. He shows us how to view life through a clear lens, teaching us how to dispel the illusions that cloud our thinking; overcome the brain’s blind spots; and embrace five ultimate truths. No matter what obstacles we face, what burdens we bear, what trials we’ve experienced, we can all be content with our present situation and optimistic about the future.

Using the latest research in cognitive science and learning theory to craft a multi-sensory learning experience, the book uses a visually rich format designed for the way your brain works, not a text-heavy approach that puts you to sleep.--Publisher's note.

Uses comics to clarify and review the lessons on equations.

The images in this book are in color. For a less-expensive grayscale paperback version, see ISBN 9781680923254. Prealgebra 2e is designed to meet scope and sequence requirements for a one-semester prealgebra course. The text introduces the fundamental concepts of algebra while addressing the needs of students with diverse backgrounds and learning styles. Each topic builds upon previously developed material to demonstrate the cohesiveness and structure of mathematics. Students who are taking basic mathematics and prealgebra classes in college present a unique set of challenges. Many students in these classes have been unsuccessful in their prior math classes. They may think they know some math, but their core knowledge is full of holes. Furthermore, these students need to learn much more than the course content. They need to learn study skills, time management, and how to deal with math anxiety. Some students lack basic reading and arithmetic skills. The organization of Prealgebra makes it easy to adapt the book to suit a variety of course syllabi.

"Prealgebra is designed to meet scope and sequence requirements for a one-semester prealgebra course. The text introduces the fundamental concepts of algebra while addressing the needs of students with diverse backgrounds and learning styles. Each topic builds upon previously developed material to demonstrate the cohesiveness and structure of mathematics. Prealgebra follows a nontraditional approach in its presentation of content. The beginning, in particular, is presented as a sequence of small steps so that students gain confidence in their ability to succeed in the course. The order of topics was carefully planned to emphasize the logical progression throughout the course and to facilitate a thorough understanding of each concept. As new ideas are presented, they are explicitly related to previous topics."--BC Campus website.

This volume considers the computational complexity of determining whether a system of equations over a fixed algebra A has a solution. It examines in detail the two problems this leads to: SysTermSat(A) and SysPolSat(A), in which equations are built out of terms or polynomials, respectively. The book characterizes those algebras for which SysPolSat can be solved in a polynomial time. So far, studies and their outcomes have not covered algebras that generate a variety admitting type 1 in the sense of Tame Congruence Theory. Since unary algebras admit only type 1, this book focuses on these algebras to tackle the main problem. It discusses several aspects of unary algebras and proves that the Constraint Satisfaction Problem for relational structures is polynomially equivalent to SysTermSat over unary algebras. The book’s final chapters discuss partial characterizations, present conclusions, and describe the problems that are still open.