Pure Mathematics

Godel, Escher, Bach: An Eternal Golden Braid

Twenty years after it topped the bestseller charts, Douglas R. Hofstadter's Gödel, Escher, Bach: An Eternal Golden Braid is still something of a marvel. Besides being a profound and entertaining meditation on human thought and creativity, this book looks at the surprising points of contact between the music of Bach, the artwork of Escher, and the mathematics of Gödel. It also looks at the prospects for computers and artificial intelligence (AI) for mimicking human thought. For the general reader and the computer techie alike, this book still sets a standard for thinking about the future of computers and their relation to the way we think.

Hofstadter's great achievement in Gödel, Escher, Bach was making abstruse mathematical topics (like undecidability, recursion, and 'strange loops') accessible and remarkably entertaining. Borrowing a page from Lewis Carroll (who might well have been a fan of this book), each chapter presents dialogue between the Tortoise and Achilles, as well as other characters who dramatize concepts discussed later in more detail. Allusions to Bach's music (centering on his Musical Offering) and Escher's continually paradoxical artwork are plentiful here. This more approachable material lets the author delve into serious number theory (concentrating on the ramifications of Gödel's Theorem of Incompleteness) while stopping along the way to ponder the work of a host of other mathematicians, artists, and thinkers.

The world has moved on since 1979, of course. The book predicted that computers probably won't ever beat humans in chess, though Deep Blue beat Garry Kasparov in 1997. And the vinyl record, which serves for some of Hofstadter's best analogies, is now left to collectors. Sections on recursion and the graphs of certain functions from physics look tantalizing, like the fractals of recent chaos theory. And AI has moved on, of course, with mixed results. Yet Gödel, Escher, Bach remains a remarkable achievement. Its intellectual range and ability to let us visualize difficult mathematical concepts help make it one of this century's best for anyone who's interested in computers and their potential for real intelligence. --Richard Dragan

Topics Covered: J.S. Bach, M.C. Escher, Kurt Gödel: biographical information and work, artificial intelligence (AI) history and theories, strange loops and tangled hierarchies, formal and informal systems, number theory, form in mathematics, figure and ground, consistency, completeness, Euclidean and non-Euclidean geometry, recursive structures, theories of meaning, propositional calculus, typographical number theory, Zen and mathematics, levels of description and computers; theory of mind: neurons, minds and thoughts; undecidability; self-reference and self-representation; Turing test for machine intelligence.

Calculus for Dummies

The mere thought of having to take a required calculus course is enough to make legions of students break out in a cold sweat. Others who have no intention of ever studying the subject have this notion that calculus is impossibly difficult unless you happen to be a direct descendant of Einstein.

Well, the good news is that you can master calculus. It's not nearly as tough as its mystique would lead you to think. Much of calculus is really just very advanced algebra, geometry, and trig. It builds upon and is a logical extension of those subjects. If you can do algebra, geometry, and trig, you can do calculus.

Calculus For Dummies is intended for three groups of readers:

• Students taking their first calculus course – If you're enrolled in a calculus course and you find your textbook less than crystal clear, this is the book for you. It covers the most important topics in the first year of calculus: differentiation, integration, and infinite series.
• Students who need to brush up on their calculus to prepare for other studies – If you've had elementary calculus, but it's been a couple of years and you want to review the concepts to prepare for, say, some graduate program, Calculus For Dummies will give you a thorough, no-nonsense refresher course.
• Adults of all ages who'd like a good introduction to the subject – Non-student readers will find the book's exposition clear and accessible. Calculus For Dummies takes calculus out of the ivory tower and brings it down to earth.

This is a user-friendly math book. Whenever possible, the author explains the calculus concepts by showing you connections between the calculus ideas and easier ideas from algebra and geometry. Then, you'll see how the calculus concepts work in concrete examples. All explanations are in plain English, not math-speak. Calculus For Dummies covers the following topics and more:

• Real-world examples of calculus
• The two big ideas of calculus: differentiation and integration
• Why calculus works
• Pre-algebra and algebra review
• Common functions and their graphs
• Limits and continuity
• Integration and approximating area
• Sequences and series

Don't buy the misconception. Sure calculus is difficult – but it's manageable, doable. You made it through algebra, geometry, and trigonometry. Well, calculus just picks up where they leave off – it's simply the next step in a logical progression.

Algebra Workbook For Dummies

Problem topics can include improper numbers, grouping symbols, using proportions, rationalizing fractions, multiplying and factoring expressions, solving linear and quadratic equations, working with inequalities, and using formulas in story problems.

Practical Algebra: A Self-Teaching Guide, 2nd Edition

Practical Algebra If you studied algebra years ago and now need a refresher course in order to use algebraic principles on the job, or if you're a student who needs an introduction to the subject, here's the perfect book for you. Practical Algebra is an easy and fun-to-use workout program that quickly puts you in command of all the basic concepts and tools of algebra. With the aid of practical, real-life examples and applications, you'll learn:

• The basic approach and application of algebra to problem solving
• The number system (in a much broader way than you have known it from arithmetic)
• Monomials and polynomials; factoring algebraic expressions; how to handle algebraic fractions; exponents, roots, and radicals; linear and fractional equations
• Functions and graphs; quadratic equations; inequalities; ratio, proportion, and variation; how to solve word problems, and more

Algebra and Trigonometry: Structure and Method Book 2

Algebra: Structure and Method Book 1

Algebra for Dummies

One of the most commonly asked questions in a mathematics classroom is, "Will I ever use this stuff in real life?" Some teachers can give a good, convincing answer; others hem and haw and stare at the floor. The real response to the question should be, "Yes, you will, because algebra gives you power" – the power to help your children with their math homework, the power to manage your finances, the power to be successful in your career (especially if you have to manage the company budget). The list goes on.

Algebra is a system of mathematical symbols and rules that are universally understood, no matter what the spoken language. Algebra provides a clear, methodical process that can be followed from beginning to end to solve complex problems. There's no doubt that algebra can be easy to some while extremely challenging to others. For those of you who are challenged by working with numbers, Algebra For Dummies can provide the help you need.

This easy-to-understand reference not only explains algebra in terms you can understand, but it also gives you the necessary tools to solve complex problems. But rest assured, this book is not about memorizing a bunch of meaningless steps; you find out the whys behind algebra to increase your understanding of how algebra works.

In Algebra For Dummies, you'll discover the following topics and more:

• All about numbers – rational and irrational, variables, and positive and negative
• Figuring out fractions and decimals
• Explaining exponents and radicals
• Solving linear and quadratic equations
• Understanding formulas and solving story problems
• Having fun with graphs
• Top Ten lists on common algebraic errors, factoring tips, and divisibility rules.

No matter if you're 16 years old or 60 years old; no matter if you're learning algebra for the first time or need a quick refresher course; no matter if you're cramming for an algebra test, helping your kid with his or her homework, or coming up with next year's company budget, Algebra For Dummies can give you the tools you need to succeed.

The Annotated Turing: A Guided Tour Through Alan Turing's Historic Paper on Computability and the Turing Machine

Programming Legend Charles Petzold unlocks the secrets of the extraordinary and prescient 1936 paper by Alan M. Turing

Mathematician Alan Turing invented an imaginary computer known as the Turing Machine; in an age before computers, he explored the concept of what it meant to be computable, creating the field of computability theory in the process, a foundation of present-day computer programming.

The book expands Turing’s original 36-page paper with additional background chapters and extensive annotations; the author elaborates on and clarifies many of Turing’s statements, making the original difficult-to-read document accessible to present day programmers, computer science majors, math geeks, and others.

Interwoven into the narrative are the highlights of Turing’s own life: his years at Cambridge and Princeton, his secret work in cryptanalysis during World War II, his involvement in seminal computer projects, his speculations about artificial intelligence, his arrest and prosecution for the crime of "gross indecency," and his early death by apparent suicide at the age of 41.

Basic Math & Pre-Algebra For Dummies (For Dummies (Math & Science))

Tips for simplifying tricky operations

Get the skills you need to solve problems and equations and be ready for algebra class

Whether you're a student preparing to take algebra or a parent who wants to brush up on basic math, this fun, friendly guide has the tools you need to get in gear. From positive, negative, and whole numbers to fractions, decimals, and percents, you'll build necessary skills to tackle more advanced topics, such as imaginary numbers, variables, and algebraic equations.

• Understand fractions, decimals, and percents

• Unravel algebra word problems

• Grasp prime numbers, factors, and multiples

• Work with graphs and measures

• Solve single and multiple variable equations

Pre-Calculus Demystified

Packed with practical examples, graphs, and Q&As, this complete self-teaching guide from the best-selling author of Algebra Demystified covers all the essential topics, including: absolute value, nonlinear inequalities, functions and their graphs, inverses, proportion and ratio, and much more.