MAT 420/1  Real Analysis


Karl Weierstrass

    This is the capstone course in your mathematics education at Eastern.  Besides the obvious goal of introducing you to Real Analysis and its methods, I want to immerse you in the culture of mathematics, its rigor, methods, and beauty, to participate in the liberal arts adventure of understanding, appreciating, and even creating ideas in your chosen major.  Prepare yourself for an intellectual journey as we explore the real number system, and see how 19th century mathematicians dealt with the problems of explaining the underlying nature of the Calculus, problems which took the best mathematicians 200 years to solve.  The place of analysis in the History of Mathematics, from Pythagoras, Zeno, and Euclid, to modern times, will be a continuing theme.

We will also explore the underlying nature of mathematics as a construction of the human brain--where does mathematics come from, how does the mind bring mathematics into being, how does mathematical ability fit into the nature vs. nurture argument?   Discussing such questions will require excursions into cognitive science and linguistics, as you will note below from texts required or recommended for this course.

What important rule of logic is Garfield using?

Required Texts:  Analysis with an Introduction to Proof, 3rd ed. by Steven R. Lay, Prentice Hall.
                           The Math Gene by Keith Devlin, Basic Books.  (Devlin is a mathematician with far reaching interests.)

 Library Reserve Text:  Where Mathematics Comes From by George Lakoff and Rafael Nunez, Basic Books.
                                    (Lakoff is a linguist, Nunez a cognitive scientist.)

Tests:  Up to three, including the final exam.

Quizzes:  Most days, the initial five minutes or so of class (1 exam = 10 quizzes; best 10 of each 12 quizzes count).  By the way, the quiz ideas is being continued at the suggestion of previous MAT 420 students.

Attendance:  Your are expected to attend class.  Your participation is important since a cooperative leaning environment will be stressed.  Also, my intention is to discuss the underlying nature of mathematics rather than examine your knowledge solely on tests and quizzes.

Assignments:  These will be published on my web page, along with hints, comments, and maybe the occasional joke.

Topics:  The first semester (MAT 420) will be topics from chapters 2-5, analyzing the nature of the real numbers, sequences, limits, and continuity.  The second semester will build on this, discussing uniform continuity, differentiation, integration, infinite series, and sequences and series of function (chapters 5-9), and other topics as time permits.

Cooperation:  You are encouraged to form study/homework groups.  A seating chart, with phone numbers(optional) will be distributed.  Grading is not done on a curve; it is in your best learning and grade interest to work cooperatively with your classmates.

Problem Sets:  A critical part of the course.  You are encouraged to discuss problems within your study groups, though final work should be your own.  Under no circumstances should you be offering your work to be copied.  Please remember that this is a writing intensive course!  Among these writing assignments, you should not be surprised to see topics related to the the texts by Lakoff and Devlin.

Grading: Based on the exams, quizzes,  and problem sets, but ultimately my evaluation of your mathematical maturity, part of which is demonstrated in the give and take of the classroom.  The exams and quizes sum to 400 points.  A grade in the 'A' range is 90% or better, in the 'B' range 80%-90%, etc.  Problem set work that is significantly different from the exam/quiz grade will modify that grade (typically, the movement is upward).  Finally, this grade estimation can be modified by your class work and attendance.

Help:  A first resource is a careful reading of the text and your notes, then your classmates, especially if you've been clever enough to form a study group!  I am available much more than just my official office hours.  A good method other than coming to my office is to contact me by email.  If my response to your question could be useful to the class, it will be broadcast.  You should check your email daily!


 Electronic Links
     There are many web links which can prove useful to you in this course, as well as other mathematics courses.  Below, I list some of these.  From each, there are connections to others.

Interactive Real Analysis--This is a special online project from Seton Hall University, with many topics directly related to this course.
Mathematics Association of America--The home page, with interesting columns.
History of Mathematics--St. Andrews University site; very extensive!
Mathematics Quotes--Some neat stuff, great to use as an "opener" next time you're trying to impress him or her at a church social or Blarneys.


Waiter:      "Would you like the chicken cordon bleu, Monsieur?"
Descartes: "I think not."  And he disappeared.