Spring 2018 -- 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 1/10.
Lecture: MWF 1-2PM, A8
First Lecture is on 1/10.
Teaching Assistants: Jacob Seidman, 1N1, seidj@sas.upenn.edu, and Shané Winner, 4C9, swinner@seas.upenn.edu
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Course Content

(1) representation of numbers, sets, cardinality, 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 and take notes. I will also post notes on Canvas.

Preliminary Schedule

All scheduled events other than exams are subject to change.
1 W 1/10 Introduction, representing numbers, sign-value systems, Roman numerals
2 F 1/12 Positional number systems, decimal number system, binary number system, and other bases
M 1/15 NO CLASS
3 W 1/17 Mayan and Babylonian number systems
4 F 1/19 Sets, elements, ∈, ∉, ellipses, union, ∪, intersection, ∩, subsets, ⊆, natural numbers, integers
5 M 1/22 Sets, complements, set-builder notation
6 W 1/24 Definitions, size of a set, cardinality, 1-to-1 matchings between sets, Cantor's definition of cardinality
7 F 1/26 Cardinality, countable sets, integers and rationals are countable, the Grand Hilbert Hotel
8 M 1/29 Uncountable sets, proof by contradiction, √ 2 is not rational, real numbers are uncountable
9 W 1/31 √ 13 is not rational, introduction to number theory
10 F 2/2 There are infinitely many primes, finding primes, Fermat primes, Mersenne primes
11 M 2/5 Sieve of Erathosthenes, Fundamental Theorem of Arithmetic, divisibility, Euclid's Lemma
12 W 2/7 Existence of prime factorizations, uniqueness of prime factorizations
13 F 2/9 Review of number systems, sets, cardinality, and prime numbers; FAQ
14 M 2/12 FIRST MIDTERM EXAM
15 W 2/14 Gauss' formula, proof by induction
16 F 2/16 Introduction to graph theory, graphs, vertex degrees
17 M 2/19 Degree Sum Formula, connected graphs, regular graphs, complete graphs, bipartite graphs, graph isomorphism
18 W 2/21 Examples of isomorphic and nonisomorphic graphs, planar graphs and faces
19 F 2/23 Euler's formula for planar graphs
20 M 2/26 Non-planarity of the complete graph on 5 vertices; polyhedral graphs
21 W 2/28 Platonic solids
22 F 3/2 Dual polyhedra, from map colorings to graph colorings
SPRING BREAK
23 M 3/12 Greedy coloring algorithm, Every planar graph contains a vertex of degree <6
24 W 3/14 Planar graphs are 6-colorable, Euler paths and Euler cycles
25 F 3/16 Review of graph theory; FAQ
26 M 3/19 SECOND MIDTERM EXAM
27 W 3/21 SNOW DAY
28 F 3/23 Euler paths, Hamiltonian cycles and paths, introduction to secure communication
29 M 3/26 Introduction to secure key exchange, modular arithmetic: addition
30 W 3/28 Modular arithmetic: multiplication, remainders, standard representation
31 F 3/30 Divisibility by 9, modular exponentiation, Fermat's Little Theorem
32 M 4/2 examples of modular exponentiation, Diffie-Helmann key exchange
33 W 4/4 Diffie-Helmann key exchange, public key cryptosystems, digital signature
34 F 4/6 Random numbers, linear congruential generators
35 M 4/9 The birthday paradoxon, Fibonacci numbers
36 W 4/11 Recurrence relations, closed form, characteristic equation
37 F 4/13 Closed form of recurrence relations, golden ratio, amortized loans, rotational symmetry and mirror symmetry
38 M 4/16 Binary operators, identity elements, inverses, groups
39 W 4/18 Uniqueness of identity elements, examples of groups, applications of group theory
40 F 4/20 Review FAQ
41 M 4/23 THIRD MIDTERM EXAM
42 W 4/27 * Special Guest Lecture *