Difficult- I think I am understanding more! I followed the the first couple parts of the section... and they made sense! I am very excited about this. However, towards the middle of the section (incorporating the fields) I got confused. Theorem 19.11 is the one I find the hardest to follow. I think it is one of the ones I need to hear explained out loud and hopefully I'll get that. The rest seems not too bad. But I haven't done any exercises yet so hopefully I'm not speaking too soon!
Reflective- I thought the Venn diagram was very useful. It helped me put a picture in my mind of how the different types of rings relate to one another. I like looking at integral domains and different rings that can and cannot be integral domains. The idea and the fact that I get it probably contribute to why I like them. For the same reasoning, I like the divisors of 0 and cancellation laws.
Tuesday, September 21, 2010
Wednesday, September 15, 2010
#10, questions. Due September 16th
One of the most important theorems is theorem 1.2.7 because it is very useful when solving proofs in multiple ways. Theorem 2.1.5 is also an important one that helps in several proofs because it connects modular world to integers. Although we have only seen this theorem for a couple of days I believe theorem 2.3.5 is important because its 2 conditions will help when solving proofs.
I expect to see a lot of questions that take the definitions one step further so that we show we actually understand the definitions. We will have to solve problems involving these definitions to show understanding and how to use the definition. I also expect proofs that tie together multiple theorems so we demonstrate understanding of those and how we can use them together to solve problems.
I expect to see a lot of questions that take the definitions one step further so that we show we actually understand the definitions. We will have to solve problems involving these definitions to show understanding and how to use the definition. I also expect proofs that tie together multiple theorems so we demonstrate understanding of those and how we can use them together to solve problems.
Tuesday, September 14, 2010
#9: Lecture 9 (and 10?), due September 14th
Difficult- Well I was reading through the first page of ring stuff (in the lecture) and I think I understand the axioms and the different types of rings. But then when I got to the examples I had no idea if it they were rings or not. I found the book to be equally challenging. I am starting to get really nervous for this test now with this stuff included also. I think I will need to see examples in class and I will hopefully be able to figure out rings and then the fields.
Reflective- Well the ring stuff seems pretty cool, what I get of it at least. When I figure it out for sure I think I will like it even more. I liked the history section in the book.I think it's great that a woman mathematician helped to develop rings and was able to become a professor! So if she can do it, then hopefully I can too!
Reflective- Well the ring stuff seems pretty cool, what I get of it at least. When I figure it out for sure I think I will like it even more. I liked the history section in the book.I think it's great that a woman mathematician helped to develop rings and was able to become a professor! So if she can do it, then hopefully I can too!
Saturday, September 11, 2010
#8: Lecture 8, due September 12th
Difficult- I think I am getting confused between the equations and the solutions. Okay so I think that if I'm following this correctly, in Zn if n is prime there is a unique solution to the equation ax=b, if a doesn't equal 0. But if n is not prime then only some equations have solutions? In that equation am I supposed to assume that a and b are classes since there are no bars? Because then in the next corollary the equation is back but with bars. Do they mean the same thing? Okay actually I think the second one might be different because it says a and n must be relatively prime.
Reflective- I found it interesting that in Zn if n is prime then there are special properties. I really think the Zn world is cool! When we were doing the tables in class I liked those a lot. Now it is interesting to see when n changes how properties and equations are formed, such as when n is prime. I also liked bringing the Euclidean algorithm in because I understand that so I can follow what that part is saying.
Reflective- I found it interesting that in Zn if n is prime then there are special properties. I really think the Zn world is cool! When we were doing the tables in class I liked those a lot. Now it is interesting to see when n changes how properties and equations are formed, such as when n is prime. I also liked bringing the Euclidean algorithm in because I understand that so I can follow what that part is saying.
Wednesday, September 8, 2010
#7: Lecture 7, due September 9th
Difficult: Well I am hoping this lesson is not too difficult for me because I somewhat remember the scalar addition and multiplication and the properties from Linear Algebra. The tables for Z5 and Z6 I find confusing however. I should be able to figure them out I think because I do recall some of this stuff, but I'm not following where the 0s come into play for addition. I see the pattern in the table but that's obviously not the only way to come up with the numbers. I might get confused when proving theorem 2.2.3 (the properties) but hopefully I can get it down.
Reflective: I don't feel as confused after reading this lecture as I did the last one! I like the modular arithmetic in the beginning of the lecture because that makes sense to me. I can clearly see how a number in one set added to a number in another set equals a number in another set.And it makes sense why the addition would need to be well-defined. I like seeing things this way. Because when it comes to proofs I get lost.
Answers to additional questions:
1. I spend probably about 30 minutes- 1 hour on the homework. The lecture prepared me for them mostly yes. The homework is getting harder though so I'm starting to struggle though and take longer. You should expect a visit from me in the next week.
2. I hate proofs. And that's what this class is. I am super worried. I follow them when you do them, but most I don't know how to formulate and think of the way to do them on my own. I also dislike that everyone has had analysis so I feel like they're at an advantage. Examples contribute most to my learning.
3. I think the class is great, minus the material. That's something you can't change though so I just have to try my best to understand it all.
Reflective: I don't feel as confused after reading this lecture as I did the last one! I like the modular arithmetic in the beginning of the lecture because that makes sense to me. I can clearly see how a number in one set added to a number in another set equals a number in another set.And it makes sense why the addition would need to be well-defined. I like seeing things this way. Because when it comes to proofs I get lost.
Answers to additional questions:
1. I spend probably about 30 minutes- 1 hour on the homework. The lecture prepared me for them mostly yes. The homework is getting harder though so I'm starting to struggle though and take longer. You should expect a visit from me in the next week.
2. I hate proofs. And that's what this class is. I am super worried. I follow them when you do them, but most I don't know how to formulate and think of the way to do them on my own. I also dislike that everyone has had analysis so I feel like they're at an advantage. Examples contribute most to my learning.
3. I think the class is great, minus the material. That's something you can't change though so I just have to try my best to understand it all.
Monday, September 6, 2010
#6':Lecture 6', due September 7th
(Difficult) I am not quite sure what a modulo is. How does a modulo relate with a congruence class? Does the equals sign but with another straight line over it mean congruency? I was confused after reading the first set of notes but when I got to the second set it cleared some things up. Now I don't know if I get Theorem 1.3.5. And I think I'm missing something big here... how are primes related to congruence classes/modulos? (Reflective) I don't think I understand a lot about primes. Because I was unaware that every integer except 0, -1 and 1 is a product of prime numbers and how to do the prime factorization (Theorem 1.3.4&5). I also did not know that p is prime if and only of -p is prime. However I did know that there are infinitely many primes so there's something! But how to prove that... hmmm I don't know. I guess I will learn more about primes in class!
#6:Lecture 6, due September 7th
Difficult- Well I think the most difficult part for me was keeping track of the steps during the euclidean algorithm with large numbers to find u and v. I think during the lecture I was a couple of seconds behind because it was taking me a little a little longer to follow each step. The other hard part for me was the proof of theorem 2.7. There were also several steps in this proof and once again it would take me a couple of seconds to figure out what exactly we were doing in the proof. I worry that if I had to prove something this big on a test I will struggle because there was so much to the proof.
Reflective- I really like the division algorithm. And even though the large euclidean algorithm took me a bit, I really like it doing it also. I like the computation and I like how it works out every time. It also is pretty easy so that's a reason to like it even more. I actually want to do this homework also, so this is a very good thing.
Reflective- I really like the division algorithm. And even though the large euclidean algorithm took me a bit, I really like it doing it also. I like the computation and I like how it works out every time. It also is pretty easy so that's a reason to like it even more. I actually want to do this homework also, so this is a very good thing.
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