Department of Mathematical Sciences,
Northern Illinois University

MATH 303 Fall 2019
DuSable Hall 302
Monday, Wednesday 6 - 7:15
Richard Blecksmith

| Catalogue description | Prerequisite | Course Objectives | Syllabus | Withdrawal | Grading | Room | Instructor | Goals | Rules | Text Handouts | Student Solutions | Study Guides | Thanksgiving Break | Final Exam | Some advice |

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:

  • 4:00-5:15 Mon Wed DU 302

    INSTRUCTOR:

  • Richard Blecksmith
    Office: Watson Hall 364
    email: richard@math.niu.edu
    Phone: 753-6762
    Office Hours: Mon, Wed 4-6

    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:

    2 hour exams worth 100 points each
    Homework, attendance, writing assignments, inclass presentations: 200 points total
    Total: 400 points

    ABSCENCES: Everyone is allowed to miss one class without penalty. After that, every unexcused absence will result in 10 points deducted from you total 400 points.

    GOALS OF THE COURSE: Scattered throughout these notes are 18 activities, consisting of games such as "Nim" and "Krypto," brainteasers such as "Mental Math" and "Hat Problems," and routine calculations needed to find "Ruth-Aaron Numbers" or to solve the "Postage Stamp Problem." Typically I split a class into groups of 3 or 4 students, asking each group to work on that day's activity, by talking among themselves, brainstorming ideas, and dividing up routine calculations (busy work). Working alone, however, a motivated student, with perserverance and energy, should be able to complete most of these tasks.

    Let's face it, many college mathematics texts go unread, except for homework and selected examples, because they confuse and frustrate their readers. I wanted this text to be something a student could actually read and understand. The sections are split into small paragraph--sized subsections, each containing a predigested idea, ready for consumption (like feeding a baby bird a worm.) Sections do not end with a list of dozens or more boring and routine exercises. Despite this dearth of repetitive exercises, there should be plenty of things for you to think about. Ideas, strange examples, questions, analogies, and some humor (mostly bad) abound on these pages.

    Some topics require a certain level of mathematical rigor, a good thing for students to be exposed to. Mathematics differs from all other subjects in that we can prove what we claim to be true. If you add two odd numbers, the sum will be even. Every time. And there is a proof that explains why this is so. I felt it important to give precise proofs of such important material as the Fundamental Theorem of Arithmetic and the Infinitude of Primes. Many proofs, with hints, are left to the reader.

    As the title of the book suggests, our goal is to go "beyond the standards" set for the K-12 public school curriculum. You will be introduced to topics such as encryption and primality--testing, not normally seen in an elementary course. To be sure, you won't become an expert in cryptology by mastering the simple 31--code in Module I, any more that you will be ready for a career in Stand--up by decoding the Jokes in Activity 6. But you should get a feeling for how number theory plays a role in modern technology and secure communications.

    I hope that you will like thinking about some of the topics presented here. After all, "thinking" is a good thing and "liking math" is something you will pass on to your own students.

    TEXT:

    Beyond the Standards: Exploring the World of Numbers
    by Richard Blecksmith
    (available online)
    NOTE: Each module is in PDF format, so you will need Adobe's Acrobat Reader which is a free and useful download. Click on the above link to get the latest version of the Acrobat Reader.

    ACTIVITIES:

    1. Multiplying with your Fingers (1-5 handed out in class)
    2. Nim
    3. Krypto
    4. The 3 Pails Problem
    5. The Postage Stamp Problem
    6. Jokes
    7. The Sieve of Eratosthenes
    8. The Ruth - Aaron Numbers
    9. Math Magic Trick
    10. Guess the Number
    11. Primality Testing
    12. Fibonacci Numbers
    13. Pythagorean Triples
    14. Fooled Again
    15. Factoring Primes
    16. The ABC Problem
    MODULE I: Computers, Codes, and Pails of Water
    Introduction
    Section 1. Binary Numbers
    Section 2. Error Correcting Codes
    Section 3. Pails of Water
    Section 4. Clock Arithmetic
    Section 5. Secret Codes
    MODULE II: An Elementary Approach to Primes
    Introduction
    Section 1. The Fundamental Theorem of Arithmetic
    Section 2. GCD's and Katie's Theorem
    Section 2(b). Reducing Fractions
    Section 3. Exccursion into Word Problems
    Section 4. The Distribution of Primes
    MODULE III: Exploring Fractions
    Introduction
    Section 1. The Mediant and Victoria's Theorem
    Section 2. Contiguous Fractions
    Section 3. The Farey Sequence
    MODULE IV: Mathematical Heresy
    Section 1. Square root of 2 and Fermat's Descent Method
    Section 2. To Infinity and Beyond
    Section 3. Counting Fractions
    Section 4. The Irrationals - The Dark Matter of the Real Number Line
    Section 5. Third Deadly Sin: the square root of -1

    STUDY GUIDES:
    EXAM 1. Monday, Oct 21
    EXAM 2. Monday, Dec 9 (6 - 7:50 pm)

    Last update: Sept 20, 2012