MAT214 Numbers, Equations, and Proofs

Schedule: (usually) Tuesdays and Thursdays, 11AM-12:20PM in the Fall semester only. Includes optional evening review sessions run by undergraduate course assistants on a schedule that is set up after classes begin. 

Brief Course Description: An introduction to classical number theory. Topics include Pythagorean triples and sums of squares, unique factorization, Chinese remainder theorem, arithmetic of Gaussian integers, finite fields and cryptography, arithmetic functions and quadratic reciprocity. Usually includes at least one topic, chosen by the instructor, from more advanced or more applied number theory: possibilities include p-adic numbers, cryptography, and Fermat’s Last Theorem. Equal emphasis is given to learning new mathematics and to learning how to construct a mathematical argument by dividing a question into logical steps where each step is explained and justified carefully, giving references if necessary. For most students this will be a completely new and very challenging way to do mathematics, far removed from the process of memorizing algorithms and working through concrete calculations familiar from high school.

Who takes this course? This course is suitable both for prospective mathematics majors and also for non-majors interested in learning how to understand and construct formal mathematical arguments. It is an especially good choice for prospective computer science majors or philosophy majors.

Why take this course?  This course gives an introduction to rigorous proofs and formal mathematical argument as well as an introduction to easily accessible and beautiful topics in elementary number theory. It is ideal for students with a strong mathematical background who want to learn more about proofs in a more algebraic setting.

Prerequisites and Placement Information:  A very strong aptitude for mathematics and genuine mathematical curiosity are essential. Do you like mathematics, especially the part where you try to understand why a method works or why a definition gives a useful point of view? Do you want to make your own conjectures and figure out for yourself whether a mathematical statement is true or false? Would you like to learn how to construct a clear and convincing, even iron-clad, argument to justify a mathematical claim? Would you like to develop an appreciation for the intrinsic value and power of mathematical thinking, separate from considerations of real-world applications or utility?

Placement Information:  While calculus is not a strictly required in this course, students will need the mathematical maturity and independence that are usually acquired in courses like BC calculus. Because the course is more algebraic and abstract compared to calculus, it is quite difficult to judge how much time you will need to master the more abstract aspects of this kind of course.  For many students, this is a big adjustment, unlike the math courses they have taken before, so expect to invest a lot of time early on.

FAQ: 

• You want to be a math major and you are very curious about mathematics beyond calculus. Should you take 214 or 215? If you are pretty sure you want to be a math major, then taking 215 in the fall may be a better choice.  The analysis topics covered in 215 and in 216-217 are more broadly useful in the major. Most math majors will learn more about algebra and number theory in upper-division courses. If you take 214, you will still need to complete 215-217 or 203-204 to prepare for 300-level math courses.

• You are really not sure about a major, but you like math and you are good at it. You have never taken a proof-based class before. MAT214 will not really help you get ready for other majors like engineering or physics, so there may be better choices (like 215 or 216 or maybe 202 or 203, depending on your background and interests) that satisfy prerequisites for other majors. On the other hand, if you will likely do philosophy or computer science or mathematics, 214 can be a good choice. And it is OK to take a class just because it appeals to you. If you are very curious about math beyond calculus, this is a good choice.

• How hard should I expect to work in this class?  It can vary quite a bit for individual students, but we expect that you will need to spend substantial time outside of class, reading the textbook and reviewing your notes in order to understand the proofs and the definitions from the textbook and your class notes well enough to be able to adapt them to create new proofs and construct counterexamples. Following up in office hours or at the undergraduate-led problem sessions or with your study group will likely be important for understanding the new ideas. The problems require both creativity and knowledge, and so you will need to spend unpredictable amounts of time in active contemplation while you wait for inspiration or the right insight. All in all, you should be ready to spend at least 10 hours per week working outside of class. 

• You have read a few number theory books already and maybe you already know what is in 214. How can you tell? Try the Sample Problems. The first problem requires no special background beyond the definition of a prime number, so see if you can solve it. The others concern topics you will learn about if you take this course.

• You have never had a course with rigorous proofs and you are worried it might be too hard. It will probably be a challenge, but the course is designed to be accessible to students who are seeing proofs for the first time. If after a few classes you still find the subject appealing, you should persevere, even if you do find the problem sets to be (overly) challenging. Go to office hours and take advantage of the help available in the evening problem sessions where enthusiastic math majors will help you learn how to think like a mathematician. The course is graded on a generous curve to encourage interested students who want to give this kind of thinking a serious try without undue academic risk. Consult your instructor for advice after the first few weeks if you are still worried about how things are going. Placement adjustments can be made even after the end of drop/add, but the sooner you get to the right course, the better!