MATH 347 X3H Fundamental mathematics
Time: MWF
12:00-12:50
Location: Altgeld
Hall 447
Instructor: Marius
Junge Course
email
Office hours: tba
Course description
:
Part I: Elementary Concepts
key words: introduction
(here is the text), numbers, sets, functions, relations, groups,
bijection, cardinality
key goals: Discuss what we should do and what we will do,
introduction to mathematical language, first proofs, induction, a feel
for inequalities.
Part II: Properties of numbers
key
words: Congruence relation, some finite rings, Fermat's little Theorem,
rational numbers (the geometric and the formal viewpoint), irrational
numbers (a beginning)
key goals: Understand the arithmetic axioms for Q (R) (also by
comparison to finite rings Zn).
Part III: Discrete
Mathematics (if we find time)
key words: finite probability space, conditional
probability, Bayes formula, some expectations, pigeonhole principle and
applications.
key goals: counting the right way may be trickier than you
think - discrete probability has many down to earth applications.
Part
IV: Continuous
Mathematics
key words: completeness
axioms, least upper bound, archimedean
principle, monotone and convergent sequences, Cauchy sequences,
important
series, limits and continuity
key goals: understanding how to prove and
disprove `for all ... exists...' statements, the art of
'epsilon-delta'-arguments, see the continuum from
the
mathematical point of view.
Books:
John P. d'Angelo
and
Douglas West: problem-solving and proofs
Grading:
Homework
(weekly) and projects (1/3)
Midterms
(2) (1/3)
Final exam (1/3)
Projects should be prepared by individual students or in groups
of two students. Every student has to prepare at least one project.
There are several ways to find projects. 1) Some projects will be
posted in class and I will ask for volunteers. 2) The text book
offers an abounded reservoir of problems, for example four problems at
the beginning of each section, many exercises marked with (!) are very
instructive and we will not find time to discuss most of the
interesting examples. Any of those problems can be used for a project
and you should notify me when finding something which interests you
(reading a little ahead is a good recipe here). Some applications of
the pigeonhole principle are really cool and great for projects. The
solution 3) is a project of my choice (and closer to what I find
interesting ?).
Your creativity and involvement in projects will be pivotal if I have
to decide between grades at the end of the term.
Comments:
This is not an ordinary
course! Here you may learn what mathematics is all about and you may
decide to hate or love it-I hope you will love it. Proofs, ideas, and
concepts will be practiced and discussed here. But this cannot be
done in a content free environment. We have to consider
interesting problems and questions. You will learn that the
question `What are real numbers?' has an answer, but not necessarily an
easy answer.
In this course you will be asked to act and learn by yourself in the
form of solving problems, writing proofs and most importantly
think. I understand (and I do remember) that the first steps towards
own
proofs are time consuming and sometimes confusing. Well,
confusion or milder forms 'that doesn't sound write' are good
first steps towards true understanding - nothing to worry about. To the
contrary, if you have never been puzzled or surprised in a math course
before, then something went wrong.
I am looking forward to share some of the many interesting questions in
mathematics with you this term.
hw1
hw2
hw3
hw4
hw5
3
first projects
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