Unit 4: Numbers that Lack Integrity This unit focuses primarily on objects that solve mathematical problems, but are neither integers, arithmetic combinations of integers, nor even algebraic combinations of integers. The first topic we owe to the Pythagoreans; their well-known theorem of right triangles led them to a discovery they actually found repugnant: the existence of so-called irrational numbers.
As you know, some numbers are not merely irrational; they are imaginary. Our last set of interesting numbers consists of the Gaussian integers, complex numbers whose real and imaginary parts are integers. They resemble the integers in many ways – ring properties, a way to divide with remainder – but they differ in one very important way: the notion of a prime number. We don't go into much depth on that here, but merely introduce the problem, then refine it as the course progresses.
By the 19^{th} century, mathematicians’ attitudes toward numbers that didn't fit a certain scheme were somewhat more accepting, so when they encountered numbers that weren't algebraic, they apparently felt that these numbers in some sense rose above the others, and called them transcendental. Any spiritual or religious connotation is purely coincidental.
Unit4 Learning Outcomes
Upon successful completion of this unit, you will be able to:
- define rational numbers, and explain how they satisfy the properties
of a ring;
- distinguish between rational and irrational numbers;
- explain why some specific numbers are irrational, while others are
algebraic;
- identify the Gaussian integers, perform arithmetic with them, and
explain how they satisfy the properties of a ring, yet their notion
of primality diverges from that of “ordinary” integers;
- define algebraic numbers, and explain how they satisfy the
properties of a ring; and
- distinguish between algebraic and transcendental numbers.
4.1 Rational Numbers, Irrational Numbers, and Their Existence 4.1.1 Field Properties of the Rational Numbers - Assessment: Show that the set **Q of rational numbers forms a field under their ordinary addition and multiplication.** Instructions: Show that the set Q of rational numbers forms a field under their ordinary addition and multiplication.
Completing this assessment should take approximately 15 minutes.
4.1.2 The Pythagoreans’ Unpleasant Surprise: Irrational Numbers - Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Section 3.2: Irrational Numbers” Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Section 3.2: Irrational Numbers” (PDF)
Instructions: Read “Section 3.2: Irrational Numbers” on pages
59-61.
Reading this section, taking notes, and studying the examples
should take approximately 15 minutes.
Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Irrational Numbers Exercises” Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Irrational Numbers Exercises” (PDF)
Instructions: Try to do Exercises 1 and 2 on page 61.
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 30 minutes.
4.2 Gaussian Integers 4.2.1 “Ring Properties” of the Gaussian Integers - Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Introduction to Section 3.3: Gaussian Integers and Section 3.3.1: Ring Properties of Z[i]” Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Introduction to Section 3.3: Gaussian Integers and Section 3.3.1: Ring Properties of Z[i]” (PDF)
Instructions: Read the introduction to “Section 3.3: Gaussian
Integers” and all of “Section 3.3.1: Ring Properties of Z[i]” on
pages 61-63.
Reading this section, taking notes, and studying the examples
should take approximately 15 minutes.
Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Ring Properties of Z[i] Exercise” Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Ring Properties of Z[i] Exercise” (PDF)
Instructions: Try to do Exercise 1 on page 63.
After attempting the exercise assigned above, discuss your solution 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 30 minutes.
4.2.2 Sage Lab: “Arithmetic of Gaussian Integers: Geometry and Algebra” - Web Media: The Saylor Foundation’s Sage Lab: “Arithmetic of Gaussian Integers: Geometry and Algebra” Link: The Saylor Foundation’s Sage Lab: “Arithmetic of Gaussian Integers: Geometry and Algebra” (PDF)
Instructions: Download the linked set of labs. Upload the fifth one
(4.4.2-SageWS5.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.
4.2.3 Division of Gaussian Integers: A Precise Algorithm - Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Section 3.3.2: Division” Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Section 3.3.2: Division” (PDF)
Instructions: Read “Section 3.3.2: Division” on pages 63–68. It
looks long, but has lots of pictures.
Reading this section, taking notes, and studying the examples
should take approximately 25 minutes.
Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Gaussian Division Exercise” Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Gaussian Division Exercise” (PDF)
Instructions: Try to do Exercise 1 on page 68.
After attempting the exercise assigned above, discuss your solution 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 30 minutes.
4.2.4 Primality - Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Section 3.3.3 Primality” Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Section 3.3.3 Primality” (PDF)
Instructions: Read “Section 3.3.3: Primality” on pages 68-69.
Reading this section, taking notes, and studying the examples
should take approximately 10 minutes.
Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Gaussian Primality Exercises” Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Gaussian Primality Exercises” (PDF)
Instructions: Try to do Exercises 1-3 and Exercise 6 on page 69.
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.
4.3 Algebraic and Transcendental Numbers 4.3.1 “Ring Properties” of the Algebraic Numbers - Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Introduction to Section 3.4: Algebraic and Transcendental Numbers and Section 3.4.1: The Algebraic Numbers form a Ring” Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Introduction to Section 3.4: Algebraic and Transcendental Numbers and Section 3.4.1: The Algebraic Numbers form a Ring” (PDF)
Instructions: Read the introduction to “Section 3.4: Algebraic and
Transcendental Numbers” and all of “Section 3.4.1: The Algebraic
Numbers form a Ring” on pages 70-72.
Reading this section, taking notes, and studying the examples
should take approximately 15 minutes.
Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Algebraic Rings Exercises” Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Algebraic Rings Exercises” (PDF)
Instructions: Try to do Exercises 1-3 on page 74.
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.
4.3.2 Liouville’s Number - Reading: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Section 3.4.2: Liouville’s Number” Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Section 3.4.2: Liouville’s Number” (PDF)
Instructions: Read “Section 3.4.2: Liouville’s Number” on pages
72-74.
Reading this section, taking notes, and studying the examples
should take approximately 15 minutes.
Assessment: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Liouville’s Number Exercise” Link: Wissam Raji’s “An Introductory Course in Elementary Number Theory - Liouville’s Number Exercise” (PDF)
Instructions: Try to do Exercise 4 on page 75.
After attempting the exercise assigned above, discuss your solution 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 15 minutes.