4.1, due on January 31

1. What I struggled with most was the proof of Theorem 4.4 - the existence of polynomials q(x) and r(x). It seems crazy that you can prove something like that with induction! Hopefully we go through it again in class so I can understand it better.

2. I thought this section was so cool! I love working with polynomials - factoring, multiplying, dividing, adding, etc. It seems like all the rules are familiar to what I have learned in the past, only now they are specifically denoted by certain things. Such examples include deg f(x), and R[x].

Response, due on January 28

I spend about 3 hours per homework assignment, so about 8-9 hours per week (not including the reading/blog post, which I spend about a 30-45 min on each). The readings helped prepare me for the lectures, and the lectures prepare me for the homework assignments. I am glad we are forced to read the textbook because otherwise I don’t think I would, and it has been very helpful. Working in groups and coming to your office to ask questions I still had about homework, even after having had lecture and reading the section has helped immensely as well. Probably the most helpful is the homework assignments – when I cannot understand a question, there are ways of getting help from you or others. This extra help has probably been the most helpful to my understanding.

Office hours for me are not possible for me to attend because I have a class. I sometimes come right before or right after class to ask a question or two, even though those technically are not office hours. I have really appreciated your time you have given to help me during these times – I feel when I can understand the homework, I can understand everything more thoroughly, and there is nothing more discouraging than not being able to understand the homework or material. So anyway, I hope it is alright when I stop by with questions. This is my only comment about the class thus far, other than I love it!

3.3, due on January 26

1. I was getting all confused with charts on page 67 and with the roman numeral discussion. I am not sure how we can prove matrices are isomorphic like on page 68 either.

2. What I did like about this section is that it talked about surjective and injective functions. I remember them pretty well from 290 and 341, so it is cool to see them again here in 371. I love it when the mathematics from different classes connect across the board! Life makes so much more sense that way.

3.2, due on January 24

1. I am still struggling with the concept of multiplicative inverses, what a field is, or what a unit is. All I know about mulitiplicative inverses is that if you multiply a number by its multiplicative inverse, you get 1. However, I am not quite sure how to find what that multiplicative inverse is.

2. I thought the first few pages of the section were very logical. It applied directly to basic arithmetic properties, so it was easy to understand. One thing that did not hold for basic arithmetic properties was that (a+b)^2 did not always equal a^2 + 2ab + b^2. I understand this because I understand 3.1 pretty well.

3.1 (second half), due on January 21

1. The second half of this section was rather short so I did not have much that I didn't understand. If I had to choose something though, it would have to be where it introduces subrings - part iv of Theorem 3.2 - where it says we have to show that the solution has to be an element of the set S, which I am not quite sure how to prove. Also, I do not know what a field is.

2. I thought it was interesting that they introduced something called "subrings". I know what sets are; I know what subsets are. I know what rings are; I am learning what a subring is. There are similar correlations between subsets and subrings, and it is interesting to see the comparison.

3.1 (first half), due on January 19

1. Although the section brings up over and over again the word "ring", I am still not quite sure what in the world that is. The textbook says it is a system that shares a minimal number of fundamental properties with the integers and the integers(mod n). I am guessing that is why they started talking about matrices and rings, because matrices are just another system where some of the fundamental properties with the integers is shared with those of the integers(mod n) like multiplication and addition are non-commutative rings with identity. I am still not quite sure what that means though, or what the book means by integral domain or a field. I am hoping that we will talk about it a lot in class - maybe that way I will understand better.

2. I did like it that they started bringing in matrices. I have taken Linear Algebra and liked the class, so naturally I like matrices. So ya, the connection between Linear Algebra and this class is pretty cool. I also think it is very interesting that they are bringing in the complex numbers.

2.3, due on January 14

1. This section was a bit more abstract for me compared with the other sections that I have read so far. I think it might be because there are just so many variables and I am loosing track of what each of their properties were. I am not positive I understand the logic of the proof of Theorem 2.8, it was just really long! I think once I do some of the homework problems this section might be a little easier to understand I hope.

2. Something I did understand and liked very much was the example on page 39. On the last homework assignment, it asked questions like this that we had to solve by guess and check a little bit, but this method is much more understandable and structured. I would like to try some problems like this on the homework assignment.

2.2, due on January 12

1. This section had many properties in it relating integer arithmetic to integer(mod n) arithmetic. I think the most difficult one for me to conceptualize was property 5 for modulo arithmetic: For each [a] in the integers(mod n), the equation [a]+X=[0] has a solution in the integers(mod n). I had to think a bit about what "X" would be, and I finally realized that it would be an equivalence class, more specifically it would be the equivalence class of [-a].

2. The most interesting part of the material was that you could take a large arithmetic problem regarding the integers in modulo n, and you can either compute the modulo 10 arithmetic and then simplify your answer to modulo n, OR you could simplify along the way so you don't end up with a huge number to reduce to modulo n.

Example (in modulo 3): ([4]+[7])*[2] = [22] = [19] = ... = [4] = [1] vs. ([4]+[7])*[2] = ([1]+[1])*[2] = [4] = [1]

2.1, due on January 10

1. The material was fairly simple to understand in this section, however the area that I may not thought about before is Corollary 2.5. It talks about the remainder having the same equivalence class as the dividend. I guess I am not quite sure why they brought in the t-s...

2. I loved the connection with the equals sign and the equivalence class equals sign. In both cases their properties had an identical outcome, and the properties of reflexive, symmetric, and transitive can be applied to modulo. I also find very useful that a=b(mod n) breaks down into n divides a-b which also breaks down to kn=a-b for some integer k. I like this because depending on the context of the proof, you can choose either of the three representations that will be most useful.

1.1-1.3, due on January 7

1. The most difficult part of the material was the material itself. I have seen these theorems numerous times in the past, but I have not been asked to rigorously prove them before. I am finding that it took me like 20min just to try to understand the Division Algorithm. However, I know understanding the material will just take time and practice. I don't think I fully understand any of the theorems... yet. But I will work on it! One thing that I did not understand about the division algorithm, for example, is why a-b(q+1)=r-b....

2. I am actually very excited to learn to prove such theorems rigorously! As I said above, I have seen these theorems (division and euclidean algorithms, and greatest common divisors) in many of my math courses, but I never quite understood the rigorous mathematics behind them. I have always wanted to learn how. Having a solid understanding of the proofs will greatly help me in my future math courses, and additionally as I end up teaching some day. The most interesting part of the material was in the section about greatest common divisors where it talks about the gcd being a linear combination of (a,b). I have never thought about it that way before.

Introduction, due on January 7

I, Sarah Hinton, am a Junior in the Math Education major.

The post-calculus math courses I have taken are Math 290, Differential Equations, and Linear Algebra.

I am taking this class to fulfill a requirement for my major in order to graduate. Additionally, this class will help expand my knowledge and understanding of mathematics.

One teacher that I had was not a very effective teacher in the sense that I probably could have learned more by spending that hour each day of class in the math lab doing my homework than by sitting through his lectures. During class he would write illegibly at lightning speed on the board and regurgitate the theorems and examples from the textbook. It felt like his goal was to get through the material, instead of getting the material through his students. I wish he could have been more personable and interested in finding out the level of understanding amongst his students (via asking if we understood, offering office hours, having a TA for the class, going a little slower).

I am working on memorizing the digits of pi, and I love to ballroom dance, cook, and play tennis. I do not have a cell phone or a facebook page. :)

MWF: 9-10, 10-11, or 12-1