Spring 2019 -- Math 170: Ideas in Math



General Info

Instructor: Julia Hartmann, DRL 4N38
(215)898-7847, hartmann@math.upenn.edu
Office Hours MF 2.15-3PM
Even though that day is on a Monday schedule by the university calendar, I will not hold office hours on 1/16.
Lecture: MWF 1-2PM, A8
First Lecture is on 1/16.
Teaching Assistants: Christopher Bailey, 4C19, bailey89@sas.upenn.edu,
Anschel Schaffer-Cohen, 3N2B, anschel@math.upenn.edu, office hrs F 9-11
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Course Content

(1) representation of numbers, sets, cardinality, basic logic, proofs, irrational and prime numbers; (2) graph theory, Platonic solids; (3) modular arithmetic, cryptography, random number generators, symmetry, and groups; other topics depending on time and interest.

Grading Policy

Homework will be assigned most weeks (11 assignments total), the assignments will be posted on Canvas. The due date is posted with each respective assigment. No late homework will be accepted but the lowest score will be dropped.


There will be three noncumulative in-class exams on the following dates:
Your final grade will be determined based on the following:

10 Assignments40 points each
3 Exams200 points each
Total1000 points
There will be a few in-class quizzes for extra credit. These will not be announced in advance. There is no need to study for the quizzes.

Expectations

Please sign up for Canvas notifications as I will make announcements there.
There is no textbook for the course. It is important that you attend the lecture. I will post notes on Canvas.

Preliminary Schedule

All scheduled events other than exams are subject to change.
1 W 1/16 Introduction, representing numbers, sign-value systems, Roman numerals, place-value systems, decimal numbers
2 F 1/18 Positional number systems, decimal number system, binary number system, hexadecimal system, Mayan and Babylonian number systems
M 1/21 NO CLASS
3 W 1/23 Sets, elements, ∈, ∉, ellipses, union, ∪, intersection, ∩, subsets, ⊆
4 F 1/25 natural numbrs, integers, complements, set-builder notation
5 M 1/28 Definitions, size of a set, cardinality, 1-to-1 matchings between sets, Cantor's definition of cardinality
6 W 1/30 Cardinality, countable sets, integers and rationals are countable, the Grand Hilbert Hotel
7 F 2/1 Introduction to mathematical arguments and logic: statements, negation, conjunction (and), disjunction (or)
8 M 2/4 Quantifiers, proof by contradiction, √ 2 is not rational
9 W 2/6 Uncountability of the real numbers, introduction to Number Theory
10 F 2/8 Prime numbers, there are infinitely many primes, finding primes, Fermat primes, Mersenne primes
11 M 2/11 Sieve of Erathostenes, Fundamental Theorem of Arithmetic, divisibility, Euclid's Lemma
12 W 2/13 Gauss' formula, proof by induction
13 F 2/15 Review
14 M 2/18 FIRST MIDTERM EXAM
15 W 2/20 SNOW DAY
16 F 2/22 Induction; Introduction to graph theory, graphs, edges, vertices, degree of a vertex
17 M 2/25 Degree Sum Formula, connected graphs, regular graphs, complete graphs, bipartite graphs, graph isomorphism
18 W 2/27 Examples of isomorphic and nonisomorphic graphs, planar graphs and faces, Euler's formula for connected planar graphs
19 F 3/1 Proof of Euler's formula, degree of a face, degree sum formula for faces
SPRING BREAK
20 M 3/11 Non-planarity of the complete graph on 5 vertices; polyhedral graphs, platonic solids
21 W 3/13 Platonic solids, dual graphs and dual polyhedra, map colorings, greedy coloring algorithm
22 F 3/15 every planar graph contains a vertex of degree at most 5; every planar graph is 6-colorable
23 M 3/18 Euler cycles and Euler paths, criterion for Euler cycles
24 W 3/20 Criterion for Euler paths, Fleury's algorithm, Eulerization, Hamiltonian cycles and paths
25 F 3/22 Traveling salesman problem, weighted graphs, nearest neighbor algorithm, sorted edges algorithm
26 M 3/25 minimal spanning trees, Kruskal's algorithm
27 W 3/27 Review
28 F 3/29 SECOND MIDTERM EXAM
29 M 4/1 Introduction to cryptography, rules of modular arithmetic
30 W 4/3 remainders, standard representation, divisibility by 9
31 F 4/5 modular exponentiation, Fermat's Little Theorem
32 M 4/8 Examples of modular exponentiation, Diffie-Helman key exchange
33 W 4/10 Public key cryptosystems, digital signature, random numbers
34 F 4/12 Linear congruential generators, randomness testing, the game of set
35 M 4/15 The birthday problem, Fibonacci numbers
36 W 4/17
37 F 4/19
38 M 4/22
39 W 4/24
40 F 4/26
41 M 4/28 Review
42 W 5/1 THIRD MIDTERM EXAM