From the catalog: An introduction to logical deductive reasoning and mathematical proof through diverse topics in higher mathematics. Specific topics include set and function theory, modular arithmetic, proof by induction, and the cardinality of infinite sets. May also consider additional topics such as graph theory, number theory, and finite state automata.
What that means: At this point in your mathematical career, you have likely taken so much Calculus! Math 2020 is an introduction to what mathematics is outside of Calculus. Along the way you will learn how to make a clear mathematical argument – this is a skill, not an innate ability. You will see that Mathematics has its own language, which you will learn like any other language - slowly! You will become a creative and bold problem solver, and you will think about Mathematics in completely new ways.
Some ideas/topics we will touch on:
Patterns in Mathematics. Do patterns have to continue? Could there be a list of numbers which seems to follow a pattern for 1,000,000 terms and then breaks down? How would you know?
Operations and Number Systems. We use integers and properties of integers without thinking about them. What makes a number system? What properties of the integers are important? Could you have a number system where the product of two non-zero numbers equaled zero? Wouldn’t that be interesting! How do you show that a number is irrational? Why is the square root of 2 irrational? Why is e irrational? It’s actually very hard to show that pi is irrational!
Symmetry. How is symmetry mathematical? Can we use symmetries as a tool to distinguish between different objects? The mathematical term for something which can be used to differentiate between objects is an invariant. Given a set of symmetries, can you find an object which has exactly those symmetries? We will introduce the notion of a mathematical group which will give us the framework to discuss symmetries of different objects.
Cryptography. Why is your communication over the internet secure? How is cryptography based in mathematics? How does one attempt to break an encryption scheme? You might be surprised that the answer includes factoring, and factoring techniques unlike any you ever thought about! This unit is an application of our work on number systems and groups.
Infinity. How do we compare the sizes of infinite sets? Are there more integers or positive integers? More integers or rational numbers? More even integers or odd integers? More rational numbers or irrational numbers? What do these questions even mean for infinite sets? To answer these questions (and more) we will study functions in new non-calculus ways. We will see that certain types of functions form a number system of their own.
You will gain an appreciation for how much mathematics there is past calculus! You will begin to see that there are interesting and important connections between different areas of Mathematics.
Mathematics is a language – it’s just not English! You will learn how to represent both mathematical and non-mathematical statements using symbolic notation. Concurrently, you will learn to take a symbolic expression and understand what statement it represents.
You will learn how to make a rigorous mathematical argument. What needs to be said to justify a statement? What doesn’t need to be said? What logical rules govern mathematical deduction?
You will learn how to read a detailed mathematical argument by isolating the main steps and understanding the justification of each.
You will become a creative problem solver, and you will learn how to be wrong. To be mathematically fearless involves much experimentation, and rarely do we find a correct solution on our first try. You will learn that an incorrect approach often gives you the tools you need to solve a problem correctly. (If it makes you feel better, Henry is wrong all the time - just ask anyone in our class who took Linear Algebra with him.)
Making a precise mathematical argument is a learned skill, not an innate ability. Everyone is starting from zero in this class! We will all struggle at the beginning, and our abilities will grow over the semester. Like any skill, the more you practice the faster you will improve!
It’s also important to know that Henry is still improving at writing mathematical proofs! This is an endeavor that we will begin together in Math 2020 and that you will work on for your entire mathematical career. It will get easier over time, and Math 2020 will provide you with the tools you need to succeed.
We will build these together based on your responses to the Welcome Survey. Some nicely-articulated axioms we may like to strive to uphold are the following, written by Federico Ardila.
Mathematical potential is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries.
Everyone can have joyful, meaningful, and empowering mathematical experiences.
Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.
Every student deserves to be treated with dignity and respect.
The textbook for the class is Introduction to Mathematical Structures and Proofs by Larry J. Gerstein, second edition. It is available for free from the Bowdoin library. You must have the correct edition so that you do the right problems for your homework!
Jan 21- Induction
Jan 26 - Induction
Jan 28 - Induction (HW 1 due Jan 30)
Feb 2 - Tower of Hanoi
Feb 4 - Strong Induction (HW 2 due Feb 6)
Feb 9 - Sets
Feb 11 - Sets, Power Sets (HW 3 due Feb 13)
Feb 16 - Sets, Binary Operations
Feb 18 - Binary Operations (HW 4 due Feb 20)
Feb 23 - Binary Operations, Short Proofs
Feb 25 - In-class Exam 1 (take-home portion Feb 26-29)
Mar 2 - Short Proofs, Distributive Property
Mar 4 - Distributive Property, Divisibility (HW 5 due Mar 6)
Mar 23 - Divisibility, gcd
Mar 25 - Bezout's Identity, Modular Arithmetic (HW 6 due Mar 27)
Mar 30 - Modular Arithmetic
Apr 1 - Modular Arithmetic (HW 7 due Apr 3)
Apr 6 - Modular Arithmetic
Apr 8 - Functions (HW 8 due Apr 10)
Apr 13 - Functions
Apr 15 - In-class Exam 2 (take-home portion Apr 16-19)
Apr 20 - Functions
Apr 22 - Cardinality (HW 9 due Apr 24)
Apr 27 - Cardinality
Apr 29 - Cardinality (HW 10 due May 1)
May 4 - Cardinality
May 6 - Cardinality
There will be times this semester where you will be asked to read a section or watch a short video about a topic related to the course. Learning mathematics is not a spectator sport. Reading mathematics is not like reading a novel; watching mathematics is not like watching an action thriller. Some paragraphs are easy to digest, but you may find yourself looking at one line of text for five or more minutes trying to understand what the author is trying to say. Use the pause button when watching a video. As you read or watch, take notes, just as you do in class. This is crucial! If questions arise, write them down and ask during drop-in hours.
Homework will be due weekly at 8pm on Fridays, to be submitted through Gradescope. Homework will be posted on Gradescope the week before it is due. You are encouraged to write your solutions on your iPad or to scan your work. If you scan your work, please write in dark pen on white paper. Each homework assignment must be turned in with the Math 2020 coversheet.
All homework will be posted on Gradescope.
You will be assigned a homework group of 3-5 students from the class. Homework groups are expected to meet and discuss the problems each week outside of class.
Although you will be working in groups on the homework, all homework must be written up independently, in your own words.
Homework which is not neat and legible will not be graded. Late homework will be graded at the discretion of the grader.
There are two regular exams and one final exam.
There will be a quiz on Thursday, February 19. It will count as a homework assignment for grading purposes. It will be available on Gradescope and you will take 30 minutes to complete it at your own convenience, but you must take it on February 19. The point is for you to get an idea of my expectations before I grade your exams.
First Exam:
Part 1: Closed book, closed notes, Wednesday, February 25, in class.
Part 2: Open book, open notes, take-home exam, available on Gradescope Thursday, February 26 at 8am and due on Sunday, February 29 at 8pm.
Second Exam:
Part 1: Closed book, closed notes, Wednesday, April 15, in class.
Part 2: Open book, open notes, take-home exam, available on Gradescope Thursday, April 16 at 8am and due on Sunday, April 19 at 8pm.
Final Exam: The final exam will be an open book, open notes, take-home exam available after our last class meeting and due one week later. More specific details to come. The final exam is not meant to take you a week, but I assume you will have some other things going on at the time and could use the flexibility.
You are required to go to two math department seminars this semester. After each, you will submit a short response paper. Submission of two responses is necessary to receive any of the "Participation" portion of your grade.
Talk to me if you have a scheduling conflict which prohibits you from attending these seminars. More info to come.
Beyond the two required seminars you will attend, you are expected to be engaged and active in class. You will earn your "Participation" grade by attending the required seminars, by being engaged and active in class, and by completing one of the following:
Share your solutions to an in-class worksheet with the class. You can sign up to share your solutions here, and only the first two people to sign up for a given worksheet will be given the credit. These two people must not have worked on the worksheet together in class. I will update the list as the semester progresses. Solutions should be legible, well-written, and correct. Solutions must be shared by the Friday (at 8pm) of the week the worksheet is given. If you are one of the first two people signed up for a given worksheet, email your materials to the class at math2020b_spring@bowdoin.edu
Write a two-to-three-page expository paper on a topic adjacent to the material we cover, to be shared with the class. If you are interested in earning your "Participation" grade this way, talk to Henry about what topics would be appropriate.
Attend two more seminars than the required two (for a total of four), and write reflections on them.
Your course letter grade will be determined based on the following table.
Participation.................10%
Homework....................30%
Exam 1...........................20%
Exam 2...........................20%
Final Exam.....................20%
There will be notes available for every class, shared following class, so that you will know what has been covered if you must miss a class.
Missing a class due to illness does not give you an automatic extension on work due.
I will be as flexible as possible if you need accommodation due to illness. Please don’t hesitate to reach out to me if difficult circumstances occur. If you are not comfortable speaking with me, I encourage you to talk with your dean, who can contact me if necessary.
If you will miss or need accommodation on more than one homework assignment due to illness, I will need a note from your dean.