Transition to Advanced Mathematics A Survey Course

ISBN-10: 0195310764
ISBN-13: 9780195310764
Edition: 2009
List price: $89.00 Buy it from $38.29
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Description: A Transition to Advanced Mathematics: A Survey Course promotes the goals of a ``bridge'' course in mathematics, helping to lead students from courses in the calculus sequence (and other courses where they solve problems that involve mathematical  More...

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Book details

List price: $89.00
Copyright year: 2009
Publisher: Oxford University Press, Incorporated
Publication date: 7/27/2009
Binding: Hardcover
Pages: 768
Size: 7.25" wide x 9.75" long x 1.25" tall
Weight: 3.190
Language: English

A Transition to Advanced Mathematics: A Survey Course promotes the goals of a ``bridge'' course in mathematics, helping to lead students from courses in the calculus sequence (and other courses where they solve problems that involve mathematical calculations) to theoretical upper-level mathematics courses (where they will have to prove theorems and grapple with mathematical abstractions). The text simultaneously promotes the goals of a ``survey'' course, describing theintriguing questions and insights fundamental to many diverse areas of mathematics, including Logic, Abstract Algebra, Number Theory, Real Analysis, Statistics, Graph Theory, and Complex Analysis. The main objective of A Transition to Advanced Mathematics is "to bring about a deep change in the mathematical character of students -- how they think and their fundamental perspectives on the world of mathematics." This text promotes three major mathematical traits in a meaningful, transformative way: to develop an ability to communicate with precise language, to use mathematically sound reasoning, and to ask probing questions about mathematics. In short, we hope that workingthrough A Transition to Advanced Mathematics encourages students to become mathematicians in the fullest sense of the word. A Transition to Advanced Mathematics has a number of distinctive features that enable this transformational experience. Embedded Questions and Reading Questions illustrate and explain fundamental concepts, allowing students to test their understanding of ideas independent of the exercise sets. The text has extensive, diverse Exercises Sets; with an average of 70 exercises at the end of section, A Transition to Advanced Mathematics has almost 3000 distinct exercises. Inaddition, every chapter includes a section that explores an application of the theoretical ideas being studied. We have also interwoven embedded reflections on the history, culture, and philosophy of mathematics throughout the text.

Mathematical Logic
The Formal Language of Sentential Logic
Truth and Sentential Logic
An Algebra for Sentential Logic
Application: Designing Computer Circuits
Natural Deductive Reasoning
The Formal Language of Predicate Logic
Fundamentals of Mathematical Proofs
Notes
Abstract Algebra
The Algebra of Sets
The Division Algorithm and Modular Addition
Modular Multiplication and Equivalence Relations
An Introduction to Groups
Dihedral Groups
Application: Check Digit Schemes
Notes
Number Theory
Prime Numbers
Application: Introduction to Coding Theory and Cryptography
From the Pythagorean Theorem to Fermat's Last Theorem
Irrational Numbers and Fields
Polynomials and Transcendental Numbers
Mathematical Induction
Notes
Real Analysis
Analytic Geometry
Functions and Inverse Functions
Limits and Continuity
The Derivative
Understanding Infinity
The Riemann Integral
The Fundamental Theorem of Calculus
Application: Differential Equations
Notes
Probability and Statistics
Combinatorics
Pascal's Triangle and the Binomial Theorem
Basic Probability Theory
Application: Statistical Inference and Hypothesis Testing
Least Squares Regression
Notes
Graph Theory
An Introduction to Graph Theory
The Explorer and the Traveling Salesman
Shortest Paths and Spanning Trees
Application: Weighted Graphs
Notes
Complex Analysis
Complex Numbers and Complex Functions
Analytic Functions and the Cauchy-Riemann Equations
Power Series Representations of Analytic Functions
Harmonic Functions
Application: Streamlines and Equipotentials
Notes
Answers to Questions
Answers to Odd-Numbered Exercises
Online Resources
References
Index

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