INTRODUCTION TO NUMBER THEORY (3 semester hours)
Basic concepts in problem solving, methods of proof, divisibility, primes, integer sequences, number-theoretic functions, and selected topics.
Not used in major or minor GPA calculations for mathematical sciences majors or minors. It is a required course for the Minor in Mathematics Education.
PREREQUISITE: MATH 229 or consent of department.
TIME and PLACE:
SYLLABUS: The course will cover most of Modules 1-6 of the text.
WITHDRAWAL: The last day for undergraduates to withdraw from a full-session course is Friday, October 21.
GRADING: Grades will be assigned on the basis of 400 points, as follows:
GOALS OF THE COURSE: Since mathematics is a useful tool in everyday life and in many academic fields, it is taught in elementary courses in terms of its applications. Indeed most people think that mathematics is synonymous with applications of mathematics, such as balancing a check book, finding the area of a field, computing the stress on a beam, or determining the orbit of a satellite. These are examples of using mathematics to solve problems in other disciplines. They are not problems in mathematics itself. The view that applied mathematics is all of mathematics is no more a true picture of mathematics than it would be to think that commercial art is all of art. What, then, is mathematics? One of the goals of this course is to provide you with a clearer answer to this question.
A real mathematician is not just a user of mathematics, but is also an inventor of mathematics. This course will give you a glimpse of the work of a mathematician. You will invent your own proofs to the many theorems stated in the notes. You will discover your own counter-examples and solutions to the various questions and problems. This course requires you to actively participate at a level beyond that assumed in an ordinary course. In a standard course, the homework done outside of class reinforces the material presented in lectures or in the text. In this class, however, student generated proofs form the foundation of the course. Just as a music student could not possibly learn how to play an instrument by listening to someone else play, you cannot become a mathematician merely by seeing another person's proofs. Think of this course as like driver's ed in high school, where you were expected to spend your share of time behind the wheel of the training cars. To draw another analogy, this time from sports, the only way to learn to catch a baseball is to put on a glove and play catch. In this course we play ball in a game called number theory, using the rules of logic.
RULES OF THE COURSE: There are five golden rules:
HANDOUTS FROM THE TEXT: The text
EXAM 1. (Oct 15)
EXAM 2. (Dec 5)
THANKSGIVING BREAK: Thanksgiving Break is from November 26 through November 30.
FINAL EXAM: Your final exam will be given in class at the end of the semester, probably on Wednesday, Dec 3, 2012. It is a 100 point exam, covering the material in the second half of the semester. The Final Exam is scheduled for 4:00 - 5:50 p.m., Wednesday, Dec 10, 2014. We will meet during this time to discuss the results of this second test.
About the text: In a way this text will be one of the easiest math books you've ever read. Many of the sections are written in dialogue form, with a minimum of formulas and a lot of space spent on simple explanations. Let's face it, most math students don't read the textbook, except for the exercises and possibly the examples - to be read only when you get stuck on the exercises. Hopefully this book is different. Try reading it and see.
In another way this text will be one of the most challenging math books you might have ever been exposed to. It expects you to think and thinking - especially in mathematics - is hard work. But it can also be fun. That's why the cross-word puzzle is often found near the comics in the newspaper. Thinking isn't so bad if you believe that what you're thinking about is worth the effort. The difference between mathematicians and the rest of the world is that mathematicians actually like thinking about math. It may surprise you to discover that mathematics is not as boring and humdrum as you might believe. Many people who dislike the subject can still get interested in topics such as infinity, chaos theory, space-filling curves, and logical paradoxes. In this course you will discover that not all infinities are the same; that you can have curves with a finite area, but an infinite perimeter; that you can reorder the points on the number line so that they do not take up any length.
One warning: Not all the questions posed in the text are answered for you; not all the theorems and facts are proved for you. You will have to work many of them out yourself - possibly with the help of your instructor or classmates. This is not your typical "how-to" course. Your job, as a student, should not be rote utilization of formulas, but rather reasoning, thinking, and possibly discovering.
Advice about the course: Perhaps the single most important factor in your success in this course is your study habits . Think of learning math as "working out" in the gym. Study at least 3 times per week; do not wait until the day before the exam. Learn mathematics like you would learn a language. Work on the concepts until they make sense. Don't just memorize facts and then forget them a few weeks later. Master as many homework problems as you can - beyond just getting a correct answer. Always come to class! While you're there, listen, think, and ask questions.
Last update: Sept 20, 2012