Section 4.2

Difficult: The thing that I think will be most difficult will be actually doing the problems. The one in the example in the book seems quite messy, and I am worried about doing the problems myself because it will be easy to make mistakes and get completely wrong answers. Other than that, the reading material was relatively simple because the math basically the same as in the integers.

Reflective: This material is almost exactly the same as in Z, and I am excited to learn how to do problems and find the gcd of two polynomials. It's helpful that the Euclidean Algorithm can be applied to F(x).

January 27 Questions

The homework assignments for this class take me anywhere between 1 and 2 hours. The reading usually helps, and the lecture are really helpful for clearing things up that I didn't understand in the reading. Doing the homework has definitely contributed the most to my learning, the exercises in the book are really helpful for me because that is how I am able to internalize the concepts that I learn from the reading and lecture. I usually find that I don't fully understand the reading until I have done the problems, and I usually also feel confident that if there is something I don't understand about the reading, then I will be able to understand it better when I do the homework.

Section 4.1

Difficult: I found it hard to follow the proof of Theorem 4.4, the Division Algorithm in F(x). It was helpful to have the examples, but I think that if I was able to follow the proof as it was being done I would be able to follow it and understand it better.

Reflective: I thought it was really interesting to learn about how the Division Algorithm can be applied to functions. Also, I think it's cool how there is a rule that can be applied to addition and multiplication of functions. I am looking forward to working with functions while doing problems, although I think it might be a little messy.

Section 3.3

Difficult: It was slightly difficult for me to understand what the book meant when it said that a property is preserved by isomorphism. The examples clarified it a little bit for me, but I still have trouble with an arbitrary property that can be preserved through an isomorphism, and which properties those are.

Reflective: It was interesting to learn about isomorphisms. I learned about them in linear algebra, although I don't remember the third property that this book identifies as being one of the properties of an isomorphism that we learned about in linear algebra. Additionally, I remember learning about bijections in math 290, so it's nice to have some background about this subject to rely on.

Section 3.2

Difficult: I found it a little difficult to understand why subtraction can be used to prove that a subset of a ring is a subring. It's really interesting that part one of Theorem 3.6 accounts for 3 parts of Theorem 3.2, and this was difficult for me to understand at first because I had trouble seeing the connection between the 3 parts of Thm 3.2 and subtraction.

Reflective: I think it's interesting that rings have so many properties in common with the integers. I am looking forward to learning more about fields because I think that they have a lot of interesting properties, and seeing how those properties relate to rings in general.

Section 3.1 Part 2

Difficult: I found it hard to understand the example and theorem about Cartesian products. I've never really been able to understood what they were and how they work.

Reflective: Once again, I think that the idea of a subring is similar to the idea of subspaces of vector spaces in linear algebra. I think it's really interesting that the subset inherits some of the axioms from the ring, such as commutative addition.

Section 3.1

Difficult: The concepts of rings and fields are slightly difficult to understand at first because I'm not sure how they will be used and applied. I want to know what their relationship is to the integers. I also don't know what they mean by "abstracting" algebra.

Reflective: I think that these rings are somewhat similar to vector spaces that we learned about in linear algebra. They are both sets of arbitrary elements that have a certain number of axioms to satisfy. I hope this will come in handy in the future as I am trying to understand what rings and fields are.

Section 2.3

Difficult: I understood all the proofs in this section, but I am having a hard time applying them to examples and seeing how they will come in useful. For example, I understand how the proof of Corollary 2.9 works, but I wish there was an example in the book that would tell me how to use it.

Reflective: I think it's so interesting that the prime numbers create a special case and have special properties. Also, the fact that a and n being relatively prime makes a difference is also really cool. I've never known that prime numbers have so many special properties, and I am excited to learn more about them.

Section 2.2

Difficult: It was a little difficult at first for me to understand how exactly addition and multiplication worked in congruence classes. I was confused by the use of 4 integers a, b, c, and d in Theorem 2.6 and its proof, and then only 2 integers in subsequent examples. However, I think I was probably just overthinking it and I understand how they both work now.

Reflective: I think it is really interesting that Zn is so closely related to Z. Also, it's interesting that there is a way to add and multiply congruence classes, and I hope that we will be able to do more work with these in the future so that we can find out how to apply congruence class addition and multiplication.

Section 2.1

Difficult: It was a little hard at first to wrap my mind around the concept of congruence, even though I've studied them before. I have to think hard to be able to understand what the examples mean and what congruence modulo n means, but I think this will get easier after practice.

Reflective: I remember learning about congruence in Math 290. I think it's really interesting that congruence modulo n has similar properties to the relation of equality. Also, I think Corollary 2.5 Part 1 is interesting, that [a] = [r], with r being the remainder when a is divided by n.

Sections 1.1-1.3

Difficult: The most difficult part of the material was understand the proofs given in the text, particularly the proof of Theorem 1.1 in Section 1.1. There were parts of the proofs that I had a hard time understanding because I couldn't see the connection between two steps. This has always been difficult for me.

Reflective: I think that the Euclidean Algorithm is very interesting. Last semester I took a math education class and we learned about how to teach the greatest common divisor, and I love this way of doing it. I think it's so interesting that the numbers are all so closely connected, and the the common factors of two numbers can be expressed as a linear combination of those two numbers.

Introduction

I am a sophomore and my major is mathematics education.
I have taken Math 290, Fundamentals of Mathematics, and Math 313, Linear Algebra
I am taking this class because I have to for my major and because I want to become better at proving mathematical concepts and theories. I am not very good at proofs right now, so I hope this course will help me.
I have had many outstanding math teachers, but my 290 professor here at BYU stands out. I wasn't very good at the work we had to do for the class, but his door was almost always open and I could come ask him about the homework. This was very helpful since there wasn't a TA for the class, and I was often in his office asking him for help.
Something unique about me is that I am studying Chinese and I have a little brother who is adopted from China.
Your office hours work well for me, as do the TA's office hours.