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Introduction to Mathematical Reasoning

Purpose of Course  showclose

The main purpose of this course is to bridge the gap between introductory mathematics courses in algebra, linear algebra, and calculus on one hand and advanced courses like mathematical analysis and abstract algebra, on the other hand, which typically require students to provide proofs of propositions and theorems.  Another purpose is to pose interesting problems that require you to learn how to manipulate the fundamental objects of mathematics: sets, functions, sequences, and relations.  The topics discussed in this course are the following: mathematical puzzles, propositional logic, predicate logic, elementary set theory, elementary number theory, and principles of counting.  The most important aspect of this course is that you will learn what it means to prove a mathematical proposition.  We accomplish this by putting you in an environment with mathematical objects whose structure is rich enough to have interesting propositions.  The environments we use are propositions and predicates, finite sets and relations, integers, fractions and rational numbers, and infinite sets. Each topic in this course is standard except the first one, puzzles.  There are several reasons for including puzzles.  First and foremost, a challenging puzzle can be a microcosm of mathematical development.  A great puzzle is like a laboratory for proving propositions.  The puzzler initially feels the tension that comes from not knowing how to start just as the mathematician feels when first investigating a topic or trying to solve a problem.  The mathematician “plays”  with the topic or problem, developing conjectures which he/she then tests in some special cases. Similarly, the puzzler  “plays” with the puzzle.  Sometimes the conjectures turn out to be provable, but often they do not, and the mathematician goes back to playing.  At some stage, the puzzler (mathematician) develops sufficient sense of the structure and only then can he begin to build the solution (prove the theorem).  This multi-step process is perfectly mirrored in solving the KenKen problems this course presents.  Some aspects of the solutions motivate ideas you will encounter later in the course.  For example, modular congruence is a standard topic in number theory, and it is also useful in solving some KenKen problems. Another reason for including puzzles is to foster creativity.

Optionally, you can read this New York Times article by Benedict Carney, “Tracing the Spark of Creative Problem-Solving”, to get the whole story; note that this may require you to sign up for the New York Times online for free.

Course Information  showclose

Welcome to MA111: Introduction to Mathematical Reasoning!  Below, please find some general information on the course and its requirements.
 
Course Designer: Harold Reiter
 
Primary Resources: This course is comprised of a range of different free, online materials.  However, the course makes primary use of the following resources:
Requirements for Completion: In order to complete this course, you need to work through each unit and all of its assigned materials.  Pay special attention to Units 1 and 2 as these lay the groundwork for understanding the more advanced, exploratory material presented in latter units.  You will also need to complete:
  • Subunit 1.2 Activity
  • Subunit 1.3 Activity
  • Sub-subunit 1.4.1 Activity
  • Sub-subunit 1.4.2 Activity
  • Unit 1 Assessments
  • Sub-subunit 3.4.5.1 Activity
  • Sub-subunit 3.4.5.2 Activity
  • Sub-subunit 3.7.1 Activity
  • The Final Exam 
Note that you will only receive an official grade on your Final Exam.  However, in order to adequately prepare for this exam, you will need to work through the assignments and assessment listed above.
 
In order to “pass” this course, you will need to earn a 70% or higher on the Final Exam.  Your score on the exam will be tabulated as soon as you complete it.  If you do not pass the exam, you may take it again.
 
Time Commitment: This course should take you a total of 112.25 hours to complete.  Each unit includes a time advisory that lists the amount of time you are expected to spend on each subunit.  These should help you plan your time accordingly.  It may be useful to take a look at these time advisories and to determine how much time you have over the next few weeks to complete each unit, and then to set goals for yourself.  For example, Unit 1 should take approximately 31.5 hours to complete.  Perhaps you can sit down with your calendar and decide to complete subunit 1.1 (a total of 4 hours) on Monday night; subunit 1.2, which is optional (a total of 4 hours) on Tuesday night; subunits 1.3 and 1.4 (a total of 4.25 hours) on Wednesday night; etc.
 
Tips/Suggestions: As you read, take careful notes on a separate sheet of paper.  Mark down any important equations, formulas, definitions, and proofs that stand out to you.  These notes will be useful to review as you study for the Final Exam.

Khan Academy  
This course features a number of Khan Academy™ videos. Khan Academy™ has a library of over 3,000 videos covering a range of topics (math, physics, chemistry, finance, history and more), plus over 300 practice exercises. All Khan Academy™ materials are available for free at www.khanacademy.org.

Learning Outcomes  showclose

Upon successful completion of this course, the student will be able to:
  • Read and dissect proofs of elementary propositions related to discrete mathematical objects such as integers, finite sets, graphs and relations, and functions.
  • Translate verbal statements into symbolic ones by using the elements of mathematical logic.
  • Determine when a proposed mathematical argument is logically correct.
  • Determine when a compound sentence is a tautology, a contradiction, or a contingency.
  • Translate riddles and other brainteasers into the language of predicates and propositions.
  • Solve problems related to place value, divisors, and remainders.
  • Use modular arithmetic to solve various equations, including quadratic equations in Z6, Z7, Z11 and Diophantine equations.
  • Prove and use the salient characteristics of the rational, irrational, and real number systems to verify properties of various number systems.
  • Use mathematical induction to construct proofs of propositions about sets of positive integers.
  • Classify relations as being reflexive, symmetric, antisymmetric, transitive, a partial ordering, a total ordering, or an equivalence relation.
  • Determine if a relation is a function, and if so, whether or not it is a bijection.
  • Manipulate finite and infinite sets by using functions and set operations.
  • Determine if a set is finite, countable, or uncountable.
  • Use the properties of countable and uncountable sets in various situations.
  • Recognize some standard countable and uncountable sets.
  • Determine and effectively use an appropriate counting tool to find the number of objects in a finite set. 

Course Requirements  showclose

In order to take this course, you must:

√    Have access to a computer.

√    Have continuous broadband Internet access.

√    Have the ability/permission to install plug-ins or software (e.g. Adobe Reader or Flash).

√    Have the ability to download and save files and documents to a computer.

√    Have the ability to open Microsoft files and documents (.doc, .ppt, .xls, etc.).

√    Have competency in the English language.

√    Have read the Saylor Student Handbook.

Unit Outline show close


Expand All Resources Collapse All Resources
  • Unit 1: Logic  

    In this unit, you will begin by considering various puzzles, including Ken-Ken and Sudoku.  You will learn the importance of tenacity in approaching mathematical problems including puzzles and brain teasers.  You will also learn why giving names to mathematical ideas will enable you to think more effectively about concepts that are built upon several ideas.  Then, you will learn that propositions are (English) sentences whose truth value can be established.  You will see examples of self-referencing sentences which are not propositions.  You will learn how to combine propositions to build compound ones and then how to determine the truth value of a compound proposition in terms of its component propositions.  Then, you will learn about predicates, which are functions from a collection of objects to a collection of propositions, and how to quantify predicates.  Finally, you will study several methods of proof including proof by contradiction, proof by complete enumeration, etc.

    Unit 1 Time Advisory   show close
    Unit 1 Learning Outcomes   show close
  • 1.1 Sudoku  
    • Reading: Wikipedia’s “Sudoku”

      Link: Wikipedia’s “Sudoku” (PDF)

      Instructions: Please click on the link above and read the entire article.  Try not to get sidetracked looking at variations.  Pay special attention to the growth of the number of Latin squares as the size increases.  Note that if you want to look ahead at the type of problem you will be asked to solve, check the file “Logic.pdf” at the end of Unit 2.

      Reading this article should take approximately 30 minutes.

      Terms of Use: The article above is released under a Creative Commons Attribution-Share-Alike License 3.0.  You can find the original Wikipedia version of this article here.

      See a broken link? Please let us know!

    • Reading: Mathematical Circles Topics: Tom Davis’ “Sudoku”

      Link: Mathematical Circles Topics: Tom Davis’ “Sudoku” (PDF)

      Instructions: Please click on the link above.  Scroll down the webpage to “Games,” and select the “Sudoku” link to download the PDF.  Read Tom Davis’ paper, paying special attention to the way he names the cells and to his development of language.  Next, if you have not done Sudoku puzzles before, Web Sudoku and Daily Sudoku and are two popular sites.  Do one or two before moving on to Ken-Ken.  Please note that this reading also covers the topics outlined in sub-subunits 2.1.2 and 2.1.3.

      This will take you about 3 hours if you have not done Sudoku before, and about 2 hours if you have.

      Terms of Use: Please respect the copyright and terms of use on the webpages displayed above.

      See a broken link? Please let us know!

  • 1.1.1 What Is a Latin Square?  
  • 1.1.2 Building a Language around Sudoku  

    Note: This topic is covered by the Davis reading assigned below subunit 1.1.  In particular, please focus on the introduction, especially the bulleted list of terminology on page 2 of the PDF.

  • 1.1.3 X-Wing and Unique Candidates Strategies  

    Note: This topic is covered by the Davis reading assigned below subunit 1.1.  In particular, please focus on Section 6, “X-Wings and Swordfish,” and Section 11, “Unique Solution Constraints.”

  • 1.2 Ken-Ken  
    • Activity: The New York Times’ “Ken-Ken Puzzles”

      Link: The New York Times’  “Ken-Ken Puzzles” (HTML)

      Instructions: Please keep in mind that this activity is optional.  After reading Harold et al.’s paper, click on the link above to access Ken-Ken puzzles, and attempt to complete one of these puzzles.  Note that you can choose the level of difficulty (easier, medium, and harder).  After a few practices, challenge yourself to attempt a Ken-Ken puzzle that is at the next level of difficulty.  Do not allow yourself to get addicted!

      You should dedicate no more than 1 hour to practicing Ken-Ken puzzles.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Reading: Harold Reiter, John Thornton, and Patrick Vennebush’s “Using Ken-Ken to Build Reasoning Skills”

      Link: The Saylor Foundation: Harold Reiter, John Thornton, and Patrick Vennebush’s “Using Ken-Ken to Build Reasoning Skills” (PDF)

      Instructions: This article is optional.  If you have an interest in solving Ken-Ken problems, then you will find this section interesting.  Otherwise, omitting it will not hinder your understanding of subsequent material.  Should you choose to work through this section, please click on the link above, and read the paper by Harold Reiter, et al. for an introduction to Ken-Ken.  Complete the exercises in the PDF.  After you have completed the exercises, check Harold Reiter and John Thornton’s “Solutions to Using Ken-Ken to Build Reasoning Skills.”  Please note that this reading covers the topics outlined in sub-subunits 1.2.1 through 1.2.3.

      Reading this article and completing the exercises should take approximately 3 hours.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 1.2.1 Parity  

    Note: This topic is covered by the reading assigned below subunit 1.2.  Please read Section 3 of “Using Ken-Ken to Build Reasoning Skills” to learn how to use parity in Ken-Ken puzzles. 

  • 1.2.2 Counting  

    Note: This topic is covered by the reading assigned below subunit 1.2.  Please read Section 4 of “Using Ken-Ken to Build Reasoning Skills” to learn how to use counting.

  • 1.2.3 Stacked Cages  

    Note: This topic is covered by the reading assigned below subunit 1.2.  Please read Section 5 of “Using Ken-Ken to Build Reasoning Skills” to learn how to use the idea of stacked cages.

  • 1.2.4 X-Wing Strategy  

    Note: This topic is covered by the reading assigned below subunit 1.2.  Please read Section 6 of “Using Ken-Ken to Build Reasoning Skills” to learn how to use the X-wing strategy.

  • 1.2.5 Pair Analysis  

    Note: This topic is covered by the reading assigned below subunit 1.2.  Please read Section 6 of “Using Ken-Ken to Build Reasoning Skills” to learn how to use the idea of stacked cages.

  • 1.2.6 Parallel and Orthogonal Cages  

    Note: This topic is covered by the reading assigned below subunit 1.2.  Read Section 7 of “Using Ken-Ken to Build Reasoning Skills” to learn how to use parallel and orthogonal cages.

  • 1.2.7 Unique Candidates  

    Note: This topic is covered by the reading assigned below subunit 1.2.  Please read Section 8 of “Using Ken-Ken to Build Reasoning Skills” to learn how to use the unique candidate rule.

  • 1.2.8 Modular Arithmetic  

    Note: This topic is covered by the reading assigned below subunit 1.2.  Please read Section 9 of “Using Ken-Ken to Build Reasoning Skills,” even though you have not yet studied modular arithmetic.  When you get to this part of the course, you will be asked to come back and take another look at this section.

  • 1.3 SET  
    • Activity: Set Enterprises’ “Daily Puzzle”

      Link: Set Enterprises’ “Daily Puzzle” (HTML)

      Instructions: Please note that this activity is optional.  If you choose to work through this activity, please click on the link above, read the game rules by clicking on the “daily puzzle rules” link, and play a bit.

      You should dedicate no more than 1 hour to exploring SET.

      Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.

      The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

      Submit Materials

  • 1.4 Other Brain Teasers  
    • Web Media: Khan Academy’s “Brain Teasers”

      Link: Khan Academy’s “Brain Teasers” (YouTube)

      Instructions: Please click on the link above and pick out a few videos to watch on brain teasers.  The puzzle will be introduced to you at the beginning of the video.  You should pause the video and attempt to solve the puzzle before viewing the solution.  Watch the solutions only if you absolutely cannot solve the puzzle; then, go back and reattempt the problem.

      You should spend approximately 1 hour on this site, watching a few of these videos and attempting to solve the problems.

      Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. It is attributed to the Khan Academy.

      See a broken link? Please let us know!

  • 1.4.1 Truth Tellers and Liars  
    • Activity: University of Chicago’s Math Explorer’s Club: Antonio Montalban and Yannet Interian’s “Module on Puzzles”

      Link: University of Chicago’s Math Explorer’s Club: Antonio Montalban and Yannet Interian’s “Module on Puzzles” (HTML)

      Instructions: Please click on the link above and work on the problems on this webpage: liars and truth-tellers puzzles, the Rubik’s cube, knots and graphs, and arithmetic and geometry.

      Solving these problems should take approximately 2 hours.

      Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.

      The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

      Submit Materials

  • 1.4.2 Coin Weighing Puzzles  
    • Activity: Cut the Knot: Alexander Bogomolny’s “A Fake Among Eight Coins”

      Link: Cut the Knot: Alexander Bogomolny’s “A Fake Among Eight Coins” (HTML)

      Instructions: Problems about finding the counterfeit coin among a large group of otherwise genuine coins are quite abundant.  Please click on the link above and attempt to solve the problem on this webpage.  Solutions appear at the bottom of the webpage.  If this type of logical thinking interests you, attempt to find similar problems to solve with an online search.

      You should spend approximately 15 minutes attempting to solve this problem.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 1.5 Propositional Logic  
  • 1.5.1 Compound Proposition  

    Note: This topic is covered by the lecture and article assigned below subunit 1.1.

  • 1.5.1.1 Logical Connectives, Conjunction, Disjunction, and Negation  
  • 1.5.1.2 Implication and Other Boolean Connectives  

    Note: This topic is covered by the video lecture assigned below subunit 1.1.  In particular, focus on and review the lecture from the 25-minute mark to the 32-minute mark for a discussion on implication and other Boolean connectives.

  • 1.5.1.3 Properties of Connectives  
  • 1.5.2 Truth Tables  
  • 1.5.2.1 What Is a Truth Table?  

    Note: This topic is covered by the Koehler reading assigned below sub-subunit  1.5.2.  Make sure to review the “Logical Operations and Truth Tables” section for an introduction that helps define truth tables.

  • 1.5.2.2 The Boolean Algebra of Propositions  

    Note: This topic is covered by the Koehler reading assigned below sub-subunit 1.5.2.  Make sure to review the section on “Boolean Algebra.”

  • 1.5.2.3 Tautologies, Contingencies, and Contradictions  

    Note: This topic is covered by the Koehler reading assigned below sub-subunit 1.5.2.  Make sure to read the definitions of tautology and contradictions (terms highlighted in bold) in the opening paragraphs of the “Logical Operations and Truth Tables” section.  Please note that a contingency is simply a proposition that is caught between tautology (at the top) and contradiction (at the bottom).  In other words, it is a proposition which is true for some values of its components and false for others.  For example “if it rains today, it will snow tomorrow” is a contingency, because it can be true or false depending on the truth values of the two component propositions. 

  • 1.5.2.4 Logical Equivalence  

    Note: This topic is covered by the Koehler reading assigned below sub-subunit 1.5.2.  In particular, focus on the text after the heading “Equivalence” toward the end of the “Logical Operations and Truth Tables” section.

  • 1.6 Predicate Logic  
  • 1.6.1 Universal and Existential Qualifiers  

    Note: This topic is covered by the video lectures on predicates and quantifiers assigned below subunit 1.6.

  • 1.6.1.1 Negating Existential and Universal Predicates  

    Note: This topic is covered by the video lectures on predicates and quantifiers assigned below subunit 1.6.  Note that the negation of an existentially quantified predicate is a universally quantified one, and vice-versa. 

  • 1.6.1.2 The Algebra of Predicates  

    Note: This topic is covered by the video lectures on predicates and quantifiers assigned below subunit 1.6.  The main idea here is that predicates can be manipulated in much the same way as numbers, sets, or propositions as we have seen already in the course. 

  • 1.6.2 Building Proofs  

    Note: This topic is covered by the lecture assigned below subunit 1.6.

  • 1.6.2.1 Modus Ponens and Modus Tollens  
  • 1.6.2.2 Proofs by Contradiction  
    • Reading: California State University, San Bernardino: Peter Williams’ “Notes on Methods of Proof”

      Link: California State University, San Bernardino: Peter Williams’ “Notes on Methods of Proof” (HTML)

      Instructions: Please click on the link above and read the following sections: “Introduction”, “Definition and Theorems”, “Disproving Statements”, and “Types of Proofs”.  The types of proofs include Direct Proofs, Proof by Contradiction, Existence Proofs, and Uniqueness Proofs.  You may stop the reading here; we will cover the sixth one, Mathematical Induction, later in the course.

      Reading these sections should take approximately 1 hour.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 1.6.2.3 Problem Solving Strategies  
  • 1.6.2.4 Contrapositive and Equivalent Forms  
  • Unit 1 Assessment  
  • Unit 2: Sets, Part I  

    In this unit, you will explore the ideas of what is called ‘naive set theory.’  Contrasted with ‘axiomatic set theory,’ naive set theory assumes that you already have an intuitive understanding of what it means to be a set.  You should mainly be concerned with how two or more given sets can be combined to build other sets and how the number of members (i.e. the cardinality) of such sets is related to the cardinality of the given sets.

    Unit 2 Time Advisory   show close
    Unit 2 Learning Outcomes   show close
  • 2.1 What Is a Set? Set Builder Notation  
  • 2.1.1 The Empty Set, the Universal Set  
  • 2.1.2 The Characteristic Function of a Set (Also Called the Indicator Function)  

    Note: This topic is covered by the Bernstein reading assigned below sub-subunit 2.1.1.  In particular, make sure to focus on “Definition 2: Characteristic Function,” starting on page 10 of the PDF.

  • 2.1.3 Sets with Sets as Members  
  • 2.2 Building New Sets from Given Sets  
  • 2.2.1 Properties of Union, Intersection, and Complementation  
    • Reading: Old Dominion University: Shunichi Toida’s “Properties of Set Operation”

      Link: Old Dominion University: Shunichi Toida’s “Properties of Set Operation” (HMTL)

      Instructions: Please click on the link above and read the entire webpage.  It is important that you become aware that sets combine under union and intersection in very much the same ways that numbers combine under addition and multiplication.  For example, AUB=BUA is a way to say union is commutative in the same way as x + y = y + x says addition is commutative.  One difference, however, is that the properties of addition and multiplication are defined as part of the number system (in our development) whereas the properties of sets under the operations we have defined are provable and hence must be proved.

      Reading this webpage should take approximately 15 minutes.

      Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.

      See a broken link? Please let us know!

    • Reading: Simpson College’s Department of Computer Science: Lydia Sinapova’s “Boolean Algebra”

      Link: Simpson College’s Department of Computer Science: Lydia Sinapova’s “Boolean Algebra” (PDF)

      Instructions: Please click on the link above, scroll down the webpage to week 7, and click on the link for “Boolean Algebra” to download the lecture as a PDF.  Please read this entire lecture, paying special attention to the definition of Boolean Algebra and to the isomorphism between the two systems of propositional logic and that of sets.  Work the three exercises at the bottom of the PDF and then have a look at the solutions at the end of the document.  Note that this reading also covers the topic outlined in sub-subunit 2.2.2.

      Reading this lecture and completing these exercises should take approximately 1 hour.

      Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.

      See a broken link? Please let us know!

  • 2.2.2 The (Boolean) Algebra of Subsets of a Set  
  • 2.2.3 Using Characteristic Functions to Prove Properties of Sets  
  • 2.3 The Cartesian Product of Two or More Sets  
  • 2.3.1 The Disjoint Union and Addition  
  • 2.3.2 The Cartesian Product and Multiplication  
  • 2.4 Counting Finite Sets  
  • 2.4.1 The Cardinality of the Power Set of a Set  
  • 2.4.2 The formula |A| + |B| = |A U B| + |A U B|  
  • Unit 3: Introduction to Number Theory  

    This unit is primarily concerned with the set of natural numbers N = {0, 1, 2, 3, . . .}.  The axiomatic approach to N will be postponed until the unit on recursion and mathematical induction.  This unit will help you understand the multiplication and additive structure of N.  This unit begins with integer representation: place value.  This fundamental idea enables you to completely understand the algorithms we learned in elementary school for addition, subtraction, multiplication, and division of multi-digit integers.  The beautiful idea in the Fusing Dots paper will enable you to develop a much deeper understanding of the representation of integers and other real numbers.  Then, you will learn about the multiplicative building blocks, the prime numbers.  The Fundamental Theorem of Arithmetic guarantees that every positive integer greater than 1 is a prime number or can be written as a product of prime numbers in essentially one way.  The Division Algorithm enables you to associate with each ordered pair of non-zero integers – a unique pair of integers – the quotient and the remainder.  Another important topic is modular arithmetic.  This arithmetic comes from an understanding of how remainders combine with one another under the operations of addition and multiplication.  Finally, the unit discusses the Euclidean Algorithm, which provides a method for solving certain equations over the integers.  Such equations with integer solutions are sometimes called Diophantine Equations.

    Unit 3 Time Advisory   show close
    Unit 3 Learning Outcomes   show close
  • 3.1 Place Value Notation  
    • Reading: The Saylor Foundation: Harold Reiter’s “Fusing Dots”

      Link: The Saylor Foundation: Harold Reiter’s “Fusing Dots” (PDF)

      Instructions: Please click on the link above and read this essay, “Fusing Dots,” paying special attention to the exercises at the end.  Please note that this reading covers all of the subunits assigned below subunit 3.1.  You may find the second half of this reading very difficult.  Try to read through Laurie Jarvis’ “Understanding Place Value” first (sub-subunit 3.1.12) and then come back to this more challenging paper.  You can access the solutions for selected problems here (PDF).  Don’t worry about understanding all of the details your first time through the reading.  Instead, concentrate on the material in the first five sections of the document, and then attempt to generally understand the subsequent sections on Fusing Dots.  The supporting details will become more familiar as you work through the various subunits.

      Reading this essay should take approximately 2 hours, and completing the exercises should take approximately 2 hours.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 3.1.1 Representing Integers in Base b Notation  

    Note: This topic is covered by the reading assigned below subunit 3.1.  In particular, please read Section 2 of “Fusing Dots.”

  • 3.1.1.1 Base 5 Notation: Doing Arithmetic in Base 5  

    Note: This topic is covered by the reading assigned below subunit 3.1.  In particular, work the problem that involves finding the product of two numbers both given in base 5 notation, without translating to decimal notation.

  • 3.1.1.2 Decimal Notation  
  • 3.1.1.3 Using Repeated Subtraction  

    Note: This topic is covered by the reading assigned below subunit 3.1.  In particular, please focus on Section 3 of “Fusing Dots.”

  • 3.1.1.4 Using Repeated Division  

    Note: This topic is covered by the reading assigned below subunit 3.1.  In particular, please focus on Section 4 of “Fusing Dots.”

  • 3.1.2 Representing Rational Numbers in the Unit Interval in Base b Notation  

    Note: This topic is covered by the reading assigned below subunit 3.1.  In particular, please focus on Section 5 of “Fusing Dots.”

  • 3.1.2.1 Using Repeated Subtraction  

    Note: This topic is covered by the reading assigned below subunit 3.1.  In particular, please focus on Section 5 of “Fusing Dots.”

  • 3.1.2.2 Using Repeated Multiplication  

    Note: This topic is covered by the reading assigned below subunit 3.1.  In particular, please focus on Section 5 of “Fusing Dots.”

  • 3.1.2.3 Translating between Representations  

    Note: This topic is covered by the reading assigned below subunit 3.1.  In particular, please focus on Section 5 of “Fusing Dots.”

  • 3.1.2.4 Representing Repeating Base b Numbers as Quotients  

    Note: This topic is covered by the reading assigned below subunit 3.1.  In particular, please focus on Section 5 of “Fusing Dots.”

  • 3.1.3 Other Interesting Methods of Representation  
  • 3.1.3.1 Base phi Notation  

    Note: The Base number here is the irrational number phi.  This topic is covered by the reading assigned below subunit 3.1.  In particular, please focus on Section 6 of “Fusing Dots” and see problem 6 at the end of Section 6. 

  • 3.1.3.2 Fibonacci Representation  

    Note: This topic is covered by the reading assigned below subunit 3.1.  In particular, please focus on Section 6 of “Fusing Dots” and see problem 7 at the end of Section 6. 

  • 3.1.3.3 Cantor’s Representation  

    Note: This topic is covered by the reading assigned below subunit 3.1.  In particular, focus on Section 6 of “Fusing Dots.”  Problems 12 through 15 at the end of Section 6 all deal with Cantor’s representation, also known as factorial notation.  

  • 3.1.3.4 Base Negative 4 Notation  

    Note: This topic is covered by the reading assigned below subunit 3.1.  In particular, please focus on Section 7 of “Fusing Dots.”  Problem 2 at the end of section 10 is devoted to base negative 4 notation and arithmetic. 

  • 3.2 Prime Numbers  
  • 3.2.1 An Infinitude of Primes  
  • 3.2.2 Conjectures about Primes  
  • 3.2.2.1 The Twin Prime Conjecture  
    • Reading: Plus Magazine: “Mathematical Mysteries: Twin Primes”

      Link: Plus Magazine: “Mathematical Mysteries: Twin Primes” (HTML)

      Instructions: Please click on the link above and read this webpage.  Take note of the definition of Brun’s constant.  Also note that this is related to the Intel’s famous $475 million recall of Pentium chips.  Please also feel free to click on the link to “Enumeration to 1e14 of the twin primes and Brun’s constant” link at the end of the webpage to read associated content.

      Reading this webpage should take approximately 30 minutes.

      Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.

      See a broken link? Please let us know!

  • 3.2.2.2 Goldbach’s Conjecture  
  • 3.2.2.3 The Riemann Hypothesis  
  • 3.3 Fundamental Theorem of Arithmetic (FTA)  
  • 3.3.1 A Proof of FTA  

    Note: This topic is covered by the readings assigned below subunit 3.3.  In particular, please focus on Bogomolny’s “Euclid’s Algorithm” reading and section 2 and 3, “Euclid’s Algorithm” and “Alternative Proof”, in the Wikipedia article.

  • 3.3.2 Some Applications of FTA  

    Note: This topic is covered by the readings assigned below subunit 3.3.  In particular, please focus on Section 1 “Applications” of the Wikipedia article.  Nearly all the proofs of irrationality of the square root of a composite non-square number depend on FTA.  Of course, there are also many other applications.

  • 3.3.3 A System that Does Not Have Unique Factorization  
  • 3.4 Modular Arithmetic, the Algebra of Remainders  
  • 3.4.1 Division by 3, 9, and 11  
  • 3.4.2 Chinese Remainder Theorem  

    Note: This topic is covered by the video lectures assigned below subunit 3.4.  In particular, please focus on “Modular Arithmetic 3” and “Modular Arithmetic 4.”

  • 3.4.3 Building the Field Z?  
  • 3.4.4 Fundamental Theorems of Modular Arithmetic  

    Note: This topic is covered by the video lectures assigned below subunit 3.4.  In particular, please focus on “Modular Arithmetic 1” and “Modular Arithmetic 2.”

  • 3.4.5 Square Roots in Modular Arithmetic  
  • 3.4.5.1 The Addition of Remainders  
    • Activity: The Art of Problem Solving’s “2000 AMC 12 Problems”

      Link: The Art of Problem Solving’s “2000 AMC 12 Problems” (HTML)

      Instructions: Please click on the link above and try to solve the problem before checking the solution.  This problem asks: what is the units’ digit of the 2012th Fibonacci number?  See if you can work this using your understanding that remainders work perfectly with respect to addition.  After you have attempted this problem, review the solution on this webpage.

      Completing this assignment should take approximately 15 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 3.4.5.2 The Multiplication of Remainders  
    • Activity: Oracle ThinkQuest: “The Units Digit of Powers of a Number”

      Link: Oracle ThinkQuest: “The Units Digit of Powers of a Number” (HTML)

      Instructions: By now, the following type of problem should be familiar: what is the units’ digit of the expression 7^2012 X 13^2011?  See if you can work this using your understanding that remainders work perfectly with respect to multiplication.  In other words, if you know the remainder when N is divided by d, then you can find the remainder when N^3 is divided by d.

      The solution to this question is mentioned below, but please only check it after you have attempted the problem.  After you have completed this problem, click on the link above, and work to solve the problem on this webpage.  After you have attempted the problem, click on the link to see the solution.

      Solution: The solution to the initial problem mentioned above is that the remainder when N^3 is divided by d is the same as when the r^3 is divided by d, where r is the remainder when N is divided by d.

      Completing this assignment should take approximately 30 minutes.

      Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.

      See a broken link? Please let us know!

  • 3.5 Functions in Number Theory  
  • 3.5.1 The Floor and the Ceiling Functions  
  • 3.5.2 The Greatest Common Divisor (GCD) and Least Common Multiple (LCM) of Two Integers  
  • 3.5.3 The Sigma-Function, Summing Divisors  
  • 3.6 The Lattice of Divisors of an Integer  
  • 3.6.1 Using GCD and LCM to Define a Lattice  

    Note: This topic is covered by the reading, “Just the Factors Ma’am,” assigned below Subunit 3.5.4.  In particular, review Sections 3 and 4.

  • 3.6.2 Properties of the Lattice Dn of Divisors of n  

    Note: This topic is covered by the reading, “Just the Factors Ma’am,” assigned below Subunit 3.5.4.  In particular, review Sections 3 and 4.

  • 3.6.3 Counting the Divisors of a Number  

    Note: This topic is covered by the reading, “Just the Factors Ma’am,” assigned below Subunit 3.5.4.  In particular, review Sections 3 and 4.

  • 3.7 The Euclidean Algorithm  
  • 3.7.1 Another Look at the Division Algorithm  
    • Activity: The University of Western Australia: Greg Gamble’s “Number Theory 1”

      Link: The University of Western Australia: Greg Gamble’s “Number Theory 1” (PDF)

      Instructions: Note that you have already read this essay in sub-subunit 3.5.1.  Please click on the link above, and select the “v.3.0 numberI.pdf” link under “1995 Lectures” to download the PDF.  Review the section on “Division Algorithm” again, and then attempt the 3 sample problems in the lecture.

      Reviewing this section and attempting the sample problems should take approximately 30 minutes.

      Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.

      See a broken link? Please let us know!

  • 3.7.2 Solving Ax + By = C over the Integers  
    • Reading: DavData’s “Solving Ax + By = C”

      Link: DavData’s “Solving Ax + By = C” (HTML)

      Instructions: Please click on the link above and read the brief text on solving Ax + By = C.

      Reading this text should take approximately 15 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage linked above.

      See a broken link? Please let us know!

    • Reading: Carnegie Mellon University: Victor Adamchick’s Concepts of Mathematics: “Integer Divisibility”

      Link: Carnegie Mellon University: Victor Adamchick’s Concepts of Mathematics: “Integer Divisibility” (PDF)

      Instructions: Please click on the link above and then click on “Schedule” on the left side of the webpage.  Scroll down the webpage to the section “Integer Divisibility,” and select the “Linear Diophantine Equations” link to download Lecture 5 as a PDF.  Please read this student-friendly version of the lecture, which discusses solving an integer divisibility type of equation.  You should focus on solving linear Diophantine equations.  In particular, you should be able to find a single solution and then generate all solutions from the one you found.

      Reading this lecture should take approximately 1 hour.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage linked above.

      See a broken link? Please let us know!

  • Unit 4: Rational Numbers  

    In this unit, you will learn to prove some basic properties of rational numbers.  For example, the set of rational numbers is dense in the set of real numbers.  That means that strictly between any two real numbers, you can always find a rational number.  The distinction between a fraction and a rational number will also be discussed.  There is an easy way to tell whether a number given in decimal form is rational: if the digits of the representation regularly repeat in blocks, then the number is rational.  If this is the case, you can find a pair of integers whose quotient is the given decimal.  The unit discusses the mediant of a pair of rational fractions, and why the mediant does not depend on the values of its components, but instead on the way they are represented.

    Unit 4 Time Advisory   show close
    Unit 4 Learning Outcomes   show close
    • Reading: The Saylor Foundation: Harold Reiter’s “Fractions”

      Link: The Saylor Foundation: Harold Reiter’s “Fractions” (PDF)

      Instructions: Please click on the link above and read this article.  Pay special attention to the five problems on rational numbers at the beginning of the paper.  Problem 10 will enable you to appreciate the different between the value of a number and the numeral used to express it.  Pay special attention to Simpson’s Paradox in the paper.  Try the practice problems at the end of the reading.  After you have attempted these problems, please check the solutions against the Saylor Foundation’s “Fractions Solutions” (PDF).

      Please note that this reading covers the topics outlined for subunit 4.1, as well as inclusive sub-subunits 4.1.1 through 4.1.3, and subunit 4.2, as well as sub-subunits 4.2.1 through 4.2.3.

      Reading this article and taking notes should take approximately 3 hours.  You should also spend approximately 3 hours working on the problems provided in the text.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 4.1 Fractions and Rational Numbers Are Not the Same  
  • 4.1.1 Numbers and Their Numerals  

    Note: This topic is covered by the reading assigned below the Unit 4 introduction.  In particular, please focus on first 2 paragraphs after the text “Fraction versus rational number,” especially where numbers and numerals are in bold font as this will help you understand the relationship between the two.

  • 4.1.2 The Mediant of Two Fractions  

    Note: This topic is covered by the reading assigned below the Unit 4 introduction.  Make sure to work on problem 10 in the essay to help you better understand the mediant of two fractions. 

  • 4.1.3 Building New Rational Numbers from Given Ones  

    Note: This topic is covered by the reading assigned below the Unit 4 introduction.  In particular, pay attention to problems 1-3 under “Rational Numbers.”

  • 4.2 Representing Rational Numbers as Decimals  
  • 4.2.1 Every Rational Number Has a Nice Decimal Representation  

    Note: This topic is covered by the reading assigned below the Unit 4 introduction.  In particular, focus on problem 4 to learn more about decimal representation.

  • 4.2.2 Turning Repeating Decimals into Fractions  
  • 4.2.3 Density of Rational Numbers  

    Note: This topic is covered by the reading assigned below the Unit 4 introduction.  An interesting property of the rational numbers is that between any two rational numbers we insert another rational number.  This property is called density.  We say the rational numbers are dense in the real numbers.  The same property holds for irrational numbers.  Try proving these propositions.  Problem 6 in the essay discusses this property.
     
    This topic is also covered in the reading assigned below subunit 4.3.2.  In particular, focus on Section 6, “Density of Rational Numbers.”

  • 4.3 The Existence of Irrational Numbers  
  • 4.3.1 ?2, ?3, and ?6 Are All Irrational Numbers  
  • 4.3.2 Density of Irrational Numbers  
    • Reading: New York University: Lawrence Tsang’s “Real Numbers”

      Link: New York University: Lawrence Tsang’s “Real Numbers” (PDF)

      Instructions: Please click on the link above to access Professor Tsang’s webpage.  Select the link to “HW1” to download the PDF.  Please read pages 7 through 9, from “Density of Rational Numbers” through “Density of Irrational Numbers.”  Please note that this reading also covers the topic of Density of Rational Numbers outlined for sub-subunit 4.2.3.

      Reading this article, taking notes, and reviewing the proofs several times should take approximately 30 minutes.

      Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.

      See a broken link? Please let us know!

  • 4.3.3 Algebraic versus Transcendental Numbers  
  • 4.4 The Field of Rational Numbers  
    • Reading: New York University: Lawrence Tsang’s “Real Numbers”

      Link: New York University: Lawrence Tsang’s “Real Numbers” (PDF)

      Instructions: Please click on the link above to access Professor Tsang’s webpage.  Select the link to “HW1” to download the PDF.  Read pages 1-7 of the text.  The first 6 pages discuss the field and order axioms for real numbers.  The Completeness Axiom on page 6 is what distinguishes the rational numbers from the real numbers – the latter is COMPLETE, while the former is not.  This resource covers the topics for sub-subunits 4.4.1 through 4.4.3.

      Reading this article and taking notes should take approximately 1 hour.

      Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.

      See a broken link? Please let us know!

  • 4.4.1 Axioms for a Field  

    Note: This topic is covered by the reading assigned below subunit 4.4.  In particular, please focus on Section 4.1 “Field Axioms” on pages 2 and 3. 

  • 4.4.2 Proofs of Properties  

    Note: This topic is covered by the reading assigned below subunit 4.4.  For example, look at the proofs of theorems 4.9 and 4.12.

  • 4.4.3 Order and Incompleteness  

    Note: This topic is covered by the reading assigned below subunit 4.4.  In particular, focus on Section 4.2 “Ordering Axiom” and Section 5 “Well-Ordering Principle.”

  • Unit 5: Mathematical Induction  

    In this unit, you will prove propositions about an infinite set of positive integers.  Mathematical induction is a technique used to formulate all such proofs.  The term recursion refers to a method of defining sequences of numbers, functions, and other objects.  The term mathematical induction refers to a method of proving properties of such recursively defined objects.

    Unit 5 Time Advisory   show close
    Unit 5 Learning Outcomes   show close
  • 5.1 Mathematical Induction Is Equivalent to the Well-Ordering Property of N  
  • 5.1.1 The Well Ordering Property of N  
  • 5.1.2 Strong and Weak Induction  
    • Web Media: YouTube: isallaboutmath’s “Mathematical Induction (Part III)”

      Link: YouTube: Is All About Math’s “Mathematical Induction (Part III)” (YouTube)

      Instructions: Please click on the link above and watch the brief video, which provides an informative discussion on the principle of mathematical induction and the well-ordering principle of the natural numbers.  It specifically addresses the notion of strong mathematical induction.

      Watching this video and pausing to take notes should take approximately 30 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Reading: YouTube: The Mathsters’ “Induction 12”

      Link: YouTube: The Mathsters’ “Induction 12” (YouTube)

      Instructions: Please click on the link above and watch the brief video, which discusses why the principle of strong mathematical induction implies the well-ordering principle of the natural numbers.

      Watching this video and pausing to take notes should take approximately 15 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

  • 5.2 Proofs of Summations and Products  
  • 5.2.1 Sums and Products  
  • 5.2.2 Divisibility  
  • 5.2.3 Recursively Defined Functions  
  • Unit 5 Assessment  
  • Unit 6: Relations and Functions  

    In this unit, you will learn about binary relations from a set A to a set B.  Some of these relations are functions from A to B.  Restricting our attention to relations from a set A to the set A, this unit discusses the properties of reflexivity(R), symmetry(S), anti-symmetry(A), and transitivity(T).  Relations that satisfy R, S, and T are called equivalence relations, and those satisfying R, A, and T are called partial orderings.

    Unit 6 Time Advisory   show close
    Unit 6 Learning Outcomes   show close
  • 6.1 Binary Relations on a Set A  
    • Assessment: The Saylor Foundation’s “Relations Homework Set”

      Link: The Saylor Foundation’s “Relations Homework Set” (PDF)

      Instructions: Please click on the link above, and complete the following problems, showing all work.  These problems cover the properties of relations and interrelationships among them, as well as specific examples of relations.  When you are done, check your work against the answers provided in the accompanying solutions file, The Saylor Foundation’s “Relations Homework Set Solutions” (PDF).

      This exercise set should take between 1 and 2 hours to complete, depending on your comfort level with the material.

      See a broken link? Please let us know!

    • Web Media: YouTube: MathDoctorBob’s “Binary Relations”

      Link: YouTube: MathDoctorBob’s “Binary Relations” (YouTube)

      Instructions: Please click on the link above and watch the video.  It may be worth spending some time watching this video twice.  Note that the lecturer spends some time discussing the definitions of the properties below for sub-subunits 6.1.1 through 6.1.5.  The examples he provides exhibit several properties.  These are the defining properties of an equivalence relation (see subunit 6.4) and Partial Ordering (see subunit 6.5).  Note that this resource covers the topics outlined for sub-subunits 6.1.1 through 6.1.6.

      Watching this video twice and taking notes should take approximately 1 hour.

      Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.

      See a broken link? Please let us know!

  • 6.1.1 Symmetry S  

    Note: This topic is covered by the video lecture assigned below subunit 6.1.

  • 6.1.2 Reflexivity R  

    Note: This topic is covered by the video lecture assigned below subunit 6.1.

  • 6.1.3 Anti-symmetry AS  

    Note: This topic is covered by the video lecture assigned below subunit 6.1.

  • 6.1.4 Transitivity T  

    Note: This topic is covered by the video lecture assigned below subunit 6.1.

  • 6.2 Binary Relations from A to B  
  • 6.2.1 Relations that Are Functions  
  • 6.2.2 Injections  
  • 6.2.3 Surjections  

    Note: This topic is covered by the video assigned below subunit 6.2.2.  A surjective function is one for which every element in the codomain is mapped to by an element in the domain.  For such functions, the codomain and range are equal.

  • 6.2.4 Bijections  

    Note: This topic is covered by the video assigned below subunit 6.2.2.  A bijection is a function that is both one-to-one and onto.

  • 6.3 Equivalence Relations  
    • Web Media: YouTube: MathDoctorBob’s “Binary Relations”

      Link: YouTube: MathDoctorBob’s “Binary Relations” (YouTube)

      Instructions: Please click on the link above and watch the last 10 minutes of this video again.  It is especially important that you understand the relationship between an equivalence relation and the partition it induces.

      Reviewing this section of the lecture and pausing to take notes should take approximately 15 minutes.

      Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

      See a broken link? Please let us know!

    • Reading: Old Dominion University: Shunichi Toida’s “Equivalence Relations”

      Link: Old Dominion University: Shunichi Toida’s “Equivalence Relations” (HTML and Java)
       
      Instructions: Please click on the link above, and read the entire webpage on equivalence relations.  Then, answer the four questions at the bottom of the webpage.
       
      Reading and answering the questions should take about 1 hour to complete.
       
      Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.

      See a broken link? Please let us know!

  • 6.4 Partial Orderings  
  • Unit 7: Sets, Part II  

    In this unit, you will study cardinality.  One startling realization is that not all infinite sets are the same size.  In fact, there are many different size infinite sets.  This can be made perfectly understandable to you at this stage of the course.  In Unit 7.4.3, section (d)iii, you learned about bijections from set A to set B.  If two sets A and B have a bijection between them, they are said to be equinumerous.  It turns out that the relation equinumerous is an equivalence relation on the collection of all subsets of the real numbers (in fact on any set of sets).  The equivalence classes (the cells) of this relation are called cardinalities.

    Unit 7 Time Advisory   show close
    Unit 7 Learning Outcomes   show close
  • 7.1 Cantor Diagonalization Theorem: The Existence of Uncountable Sets of Real Numbers  
  • 7.1.1 Proof of the Theorem  
  • 7.1.2 Even the Cantor Set Is Uncountable, the Base-3 Connection with the Cantor Ternary Set  
  • 7.1.3 Other Examples of Uncountable Subsets of R  
  • 7.2 The Rational Numbers Are Countable  
  • 7.2.1 The Proof  
  • 7.2.2 The Algebraic Numbers Are Countable  
  • 7.3 Other Bijections  
  • 7.3.1 Bijections on Sets of Real Numbers  

    Note: This topic is covered in Dr. Kirby’s reading assigned below subunit 7.3. 

  • 7.3.2 The Cube and the Interval Are Equinumerous  
  • Unit 8: Combinatorics  

    In this unit, you will learn to count.  That is, you will learn to classify the objects of a set in such a way that one of several principles applies.

    Unit 8 Time Advisory   show close
    Unit 8 Learning Outcomes   show close
  • 8.1 Counting Problems as Sampling Problems with Conditions on the Structure of the Sample  
  • 8.1.1 Sampling with Replacement, Order Matters  

    Note: This topic is covered by the reading assigned below subunit 8.1.  

  • 8.1.2 Sampling without Replacement, Order Matters  

    Note: This topic is covered by the reading assigned below subunit 8.1.

  • 8.1.3 Sampling with Replacement, Order Does Not Matter  

    Note: This topic is covered by the reading assigned below subunit 8.1.

  • 8.1.4 Sampling without Replacement, Order Does Not Matter  

    Note: This topic is covered by the reading assigned below subunit 8.1.

  • 8.2 The Inclusion-Exclusion Principle  
  • 8.2.1 The Case with Just Two Sets  
  • 8.2.2 The Proof  
  • 8.2.3 Other Examples  
  • 8.3 The Pigeon-Hole Principle (PHP)  
  • 8.3.1 The Standard Principle  

    Note: This topic is covered, in part, by the video assigned below subunit 8.3.

  • 8.3.2 Using the PHP Idea in Other Settings  
  • Final Exam  

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