4.1, due January 31

This section was pretty interesting.  It's a cool idea of writing polynomials as rings but I'm sure it's making complete sense right now.  Is x just a single value like pi or is it a set of possible values? I'm not sure this is right but I thoguht of the Division Algorithm as long division of polynomials.  That part didn't seem too hard until I got to the proof, then I got a little lost.  I think that's one of those proofs that will be easier once we have a discussion about it in class.

Course Reflection, due January 28

There are a few axioms and definitions that apply in several of the topics we have discussed.  Some of these are the Division Algorithm and the 8 Axioms that define a ring.  I can also see how the definition of homomorphism and isomorphism would be very important for throughout the course.  The fact that the class lectures follow so closely with the book really helps me make sense of this material.  The things that I have a hard time understanding from the book become much clearer during lecture.  I also like how the homework is not overly complex.  It requires a higher level of thinking but it is not overly frustrating.  I think that I can personally be better about studying the things in the book that I don't understand the first time through and even reread some of the tough points after lectures.

3.3, due January 26

I've seen the term 'isomorphism" before in some of my other math and math ed classes, but the term "homomorphism" was new to me.  I was a little surprised to see that being isomorphic depended on a bijective function.   It makes sense that it does, but my understanding of isomorphisms was minimal and I  thought two things were said to be isomorphic if they were similar properties like how triangles are similar in angle size.  A bijective function requires this to be true; it's a stronger more sophisticated way of putting it. One question I had while reading was is an isomorphism a function or something like a set or a ring?  THe book uses noth terms but I wasn't srue how to think about it.My first thought about homomorphisms was that they would be different than an isomorphism, but they aren't.  Every isomorphism is a homomorphism, but just a square is a rectangle and a rectangle is not a square, not every homomorphism is not an isomorphism.  Conviently when proving that something is an isomorphism, you prove that it is an homomorphism.

3.2, due January 24

All this talk about rings reminds me of a song: "If you liked it then you shoulda put a ring on it..." Ok now that that is out of my system, let's get back to Abstract Algebra.  This section seemed to follow quite nicley with the things we talked about in class on. The one thing that was not the easiest to understand was Theorem 3.6.  I thought that for S to be a subring of the ring R, it had to satisfy addition and multiplication under R( at least that's the way I understood what Dr. Forcade said), but in Theorem 3.6, S has to be closed under subtraction and mulitplication in S.  The rest of the section follows from simple logic.  Well I guess it's more intuition but you have to prove it first before you can actually use it.  As I read the example of (a+b)^2, I instinctively thought a^2 + 2ab + b^2, but as I read I remembered that R is not necessarily a commutative ring. That'll be a tough one to remember.

pg 48 - end of 3.1, due January 21

So, this last little bit of Seciton 3.1 wasn't all that bad.  The first example was a little confusing when I read throguh it the first time because I forgot what a Cartesian product was.  For some reason it came into my mind that a + a' was adding a congruence class and an integer together and I was baffled.  Then the lightbulb went off and I remembered what a Cartesian product was.  Everything else made sense after that.  I noitced how in our lecture on Wednesday we used Theorem 3.2 when proving or disproving that certain subsets of the integers were rings.  Reading the theorem made that process a little more clear.  In class I thought we were only allowed to do that in special cases, but here it says that we can do it for all cases where S is a subset of the ring R. That's wonderful news!

3.1 - pg 48, due January 19

This section wasn't too bad, although I was slightly confused trying to understand what the elements of a ring are. Are they congruence classes? Are they integers? Does it depend on the psecific ring you are working with?  It might seem silly but when I think "ring", I think "cirlce". Are the elements if a ring related to each other in some circular pattern?  In the book they had a lot of examples using matrices. So is a ring a matrix or is a matrix a ring? I think after a class discussion and finishing the section these things will be clearer.  I do think that it's nice that a ring is defined by the same axioms that exist in the integers and in Z_n. That was a relief. I understand those.

2.3, due January 14

This section was...interesting. I had a time following the proofs the first go around so I had to reread them, much more carefully the second go around.  I thought the prrof of Theorem 2.8 was intersting how they did (1) --> (2), then (2) --> (3), and finally (3) --> (1).  I did a proof like this in one of my MTHED classes and it was one of my favorite proofs from the course.  Using one part of a theorem to prove another part seems like a much simpler and more comprhensible of proving the theorem as a whole.  I think it's pretty cool that when n is prime you can find specific answers to modular arithmetic problems in Z_n.  When n is not prime though there could be multiple solutions, which is interesting, but I much prefer solving problems with one solution and knowing that it is right.

2.2, due January 12

Modular Arithmetic? Hooray! I thought this was one of the coolest parts of MATH290 and was so excited to use it again. My FHE brother mocked my excitement, but he just doesn't understand how cool it really is.  It's so cryptic and yet simple.  The only tricky thing is finding the equivalent congruence class.  Sometimes I forget that [0]=[n] so I'll create an extra congruence class for [n], but this can be easily solved if you create congruence classes based off of the remainder term in the Division Algorithm.  Modular arithmetic seems like it would be very useful in creating numerical codes for security or personal information because you only have n congruence classes but these classes contain every posssible integer.  Modular arithmetic is so fascinating!

"Packing Primes" by Solomon Friedberg

Dr. Friedberg's lecture was quite interesting. He showed the humorous and human side of mathematicians.  Some of the proofs he went through, especially the last two proving that there is an infinite number of primes, went over my head and were hard to follow.  As a whole however, I found the seminar to be very intriguing and intellectually fascinating.  I loved it when he shared the story about the Houston Oilers and his mathematician friend thought he meant the Houston Eulers.  It let his listeners see his true colors and helped those of us that are more dedicated to following sports than the most recent mathematical discoveries identify and connect with.  This recaptivated my interest in his presentation and I consequently got much more out of the lecture. Well done, Dr. Friedberg!

2.1, due January 10

In this reading the most challenging thing was to train my brain to think about congruence in the right way.  I kept trying to read 17=5(mod6) as 5 divides 17-6.  Once I was able to get the true meaning of that expression down the rest came fairly easily.  A small hiccup came when I got to congruence classes, but after reviewing the definition and several examples that too became clear.  When I started reading this section, I was a little excited because I remembered liking our discussion of modular arithmetic in MATH290.  I was a little rusty after a year's absence but it's slowly starting to come back.  It might sound weird but I like congruence classes.  I find them fascinating!

1.1 - 1.3, due on January 7

The hardest part of this material was following the logic that was only written in words.  I am a very visual, hands-on learner, so when the author would walk through a proof and only describe the steps he was taking instead of showing them I had a harder time following the math.  I found the many applications to division absolutley fascinating! I never realized that that was how division really worked. I knew division was repeated subtraction, but some of the things the author did just blew my mind! I really enjoy learning things like that.  I like to be able to apply the new things I am learning to things I already know. That to me is what learning math is all about.

Introduction, due on January 7

I am a junior majoring in Math Education.  I took AP Calculus in High School, so when I came to BYU I jumped right into the upper level math classes.  So far I have taken Linear Algebra, Multivariable Calculus, Differential Equations, MATH 290 and Theory of Analysis 1. I am taking this class because it is part of the requirements to receive a degree in Math Education. The best math teachers I have had are the teachers that truly care that their students learn, not just memorize, the math contained in their course.  They also structured their class in a way that allowed the students to have part in class discussions.  Something unique about me is that I love to laugh. I find at least one thing everyday that makes me laugh, most of the time it's my own clutziness or something silly that I have done. Even on my worst of days, I try to laugh about something.  I have a class during the scheduled office hours.  The hour before or after our class would work best for me. So either 10-10:50 or 12-12:50 on MWF.