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MA233: Elementary Number Theory

Unit 5: Continued Fractions   In this, the second of our “sightseeing” units, we visit continued fractions.Continued fractionsare a surprisingly ancient topic that consists of writing numbers as fractions within fractions and notsimplifying them! Continued fractions enjoy several properties that allow us to find concise representations of both rational numbers and even of many irrational numbers.

Unit5 Learning Outcomes
Upon successful completion of this unit, you will be able to: - write numbers in continued fraction representation; and - identify certain classes of numbers from their representation as continued fractions.

5.1 Elementary Properties of Continued Fractions   5.1.1 Rational Numbers have Finite Continued Fractions   - Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Introduction to Chapter 7 and Section 7.1: Basic Notations” Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Introduction to Chapter 7 and Section 7.1: Basic Notations” (PDF)

 Instructions: Read the beginning of “Chapter 7 Introduction to
Continued Fractions” and all of “Section 7.1 Basic Notations” on
pages 155-159.  

 Reading these sections, taking notes, and studying the examples
should take approximately 30 minutes.
  • Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory: Continued Fractions Basic Notation Exercises” Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory: Continued Fractions Basic Notation Exercises” (PDF)

    Instructions: Try to do Exercises 1-5 on pages 159-160.

    After attempting the exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.

    Completing this assessment should take approximately 1 hour.

5.1.2 Convergents   - Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Section 7.2: Main Technical Tool” Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Section 7.2: Main Technical Tool” (PDF)

 Instructions: Read “Section 7.2: Main Technical Tool” on pages
160-164.  

 Reading this section, taking notes, and studying the examples
should take approximately 30 minutes.
  • Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Main Technical Tool Exercises” Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Main Technical Tool Exercises” (PDF)

    Instructions: Try to do Exercises 1-4 on page 164.

    After attempting the exercises assigned above, discuss your solutions in the course discussion forum. Feel free to respond to other students’ postings as well. If you haven’t already done so, you will need to create a free account at the link above to participate in the discussions.

    Completing this assessment should take approximately 1 hour.

5.1.3 Sage Lab: “Continued Fractions”   - Web Media: The Saylor Foundation’s Sage Lab: “Continued Fractions” Link: The Saylor Foundation’s Sage Lab: “Continued Fractions” (Sage)

 Instructions: Download the linked set of labs. Upload the sixth one
(5.1.3-SageWS6.sws) to the Sage website where you created an account
(subunit 1.4.2). Work through the lab carefully.  

 Completing this assignment should take approximately 30 minutes.