Math 347. Fundamental Mathematics (formerly Math 247)
Syllabus for Instructors using Liebeck "A Concise Introduction to Pure Mathematics"
Goals of the course
Math 247 introduces students to rigorous mathematics, improving their ability to absorb and communicate it via practice in problem-solving and in writing proofs. The course demands effort from the students; they must read the text and work many exercises. They should be writing arguments all the time and must receive a lot of criticism of their work. The standards expected of students should be kept high but reasonable. Math 247 aims to help students with the difficult transition from lower level to upper level courses. Like upper level courses, this course requires independent thinking and understanding - students cannot expect all questions to be instances of procedures listed in class.
Topics that prepare for upperclass courses include logical reasoning, induction, equivalence relations, prime factorization, and limits.
Detailed Syllabus
Instructors are asked to follow and to complete the syllabus below.
M. Liebeck, A Concise Introduction to Pure Mathematics
- (2 hours) Chapter 1 - Sets and proofs
- (2 hours) Chapter 2 - Number systems
- (2 hours) Chapter 3 - Decimals
- (1.5 hours) Chapter 4 - Inequalities
- (0.5 hours) Chapter 5 - nth Roots and Rational Powers
- (1.5 hours) Chapter 6 - Complex Numbers
- (1.5 hours) Chapter 7 - Polynomial Equations
- (4 hours) Chapter 8 - Induction
- (0 hours) Chapter 9 - Euler's Formula and Platonic Solids
- (8 hours) Chapter 10 - Introduction to Analysis [see notes below]
- (3 hours) Chapter 11 - The Integers
- (2 hours) Chapter 12 - Prime Factorization
- (1 hours) Chapter 13 - More on Prime Numbers
- (3 hours) Chapter 14 - Congruence of Integers
- (2 hours) Chapter 15 - Counting and Choosing
- (1 hours) Chapters 16-18 - More on Sets, Equivalence Relations, Functions
- (2 hours) Chapter 19 - Infinity
- (6 hours) Exams and leeway
- (43 hours) Total
Notes on the material - Chapter 1 - Augment with additional material on logic and truth tables, especially for implications.
- Chapter 2 - This chapter uses the Archimedean Property (that for each real number there exists a larger natural number). The Archimedean Property should be stated in class, and if desired, can be proved later by using the Least Upper Bound Axiom for the real numbers.
- Chapter 3 - One could define the limits of sequences and series, here.
- Chapter 3 - Watch out for Liebeck's unusual meaning of decimal expansion for negative numbers.
- Chapter 4 - Perhaps start by defining "a>b if a-b is positive". Then Rules 4.1(2,3,5) follow easily. Rules 4.1(1,4) are readily believed.
Some important inequalities should be added to the material or exercises in this chapter: the triangle inequalities (both upper and lower), and the arithmetic-geometric mean inequality. See for example pages 4-5 of D'Angelo & West "Mathematical Thinking". Some cautionary exercises involving inequalities and quotients could also be assigned. - Chapter 5 - Proof of Proposition 5.2 is optional.
- Chapter 6 - This chapter uses the sine and cosine functions without defining them, and this should be remarked upon. Also, point out that Euler's formula can be justified by the Taylor series of the expontial, sine and cosine functions. (Students should know those series from calculus, but establishing them rigorously lies outside the scope of Math 247.)
Note: Chapter 6 is used in Chapter 7, but not elsewhere in the text. - Chapter 7 - Omit material on solution of cubic equations.
- Chapter 9 - Omit.
- Chapter 10 - Liebeck's book has insufficient material on the Real Numbers and Sequences and Series. It does not even define the limit of a sequence of real numbers. Chapter 10 must be supplemented, or replaced, in order to complete the syllabus. The needed material is contained, for example, in Chapter 13 (3.5 hours) and Chapter 14 (4.5 hours) of D'Angelo & West. An instructor following these chapters can omit Chapter 10 of Liebeck. Notes: in D'Angelo & West, the proofs of the convergence tests in Chapter 14 need only be treated lightly. The material on continuity in Chapter 15 (3 hours) could also be covered, at the instructor's discretion.
- Warning: Chapter 3 shows that every real number is represented by a decimal, but does not show that every infinite decimal expansion converges to a real number, which requires completeness and is implicitly used in Chapter 10 when proving the least upper bound theorem. So if covering Chapter 10 of Liebeck, one must explicitly assume the completeness axiom for the reals (in one of its forms). Note: Math 247 takes an axiomatic approach, and does not construct the reals.
- Chapter 15 - Omit multinomial coefficients.
- Chapters 16, 17, 18: Concentrate on material needed for Chapter 19. Motivate equivalence relations in Chapter 17 by recalling congruence relations, from Chapter 14.
Notes on the exercises
Liebeck's book contains a relatively small number of exercises. Many problems are tricky, and students will require hints.
Ward Henson and Anand Pillay have generated further exercises. Another source is D'Angelo & West "Mathematical Thinking", which has many relevant examples and exercises. - Chapter 13 #4 - beware! Changing the problem to 4k-1 makes it easier.
- Ward Henson, Richard Laugesen, Anand Pillay Last modified July 2, 2002