1. What was the most difficult part of the reading for you?
Well, unfortunately, since I still don't understand this character malarkey, the beginning was the most difficult for me again. Seriously, what in the world are these L-series things? I'm lost beyond lost. I'm mostly praying that these character things are just here for this short visit, then they'll leave me alone for the rest of my life, because reading this section feels like I'm reading in a different language.
2. What was the most interesting part of the reading for you?
Okay, so, if I stop complaining for a minute and just accept the gibberish that I've read at the beginning of the section, I actually found myself following the Lemmas pretty easily. Again, I had to just accept the L-series and character stuff as they've described them, but then I was able to follow (using my 113 knowledge from 6 years ago) Lemmas 3.3.5-3.3.7 without too much trauma. So, I guess that's interesting... to me...
Sunday, October 30, 2011
Thursday, October 27, 2011
Section 3.6 Due October 28, 2011
1. What was the most difficult part of the reading for you?
First of all, let me say that this class is just a rollercoaster. I mean, continued fractions felt like a breeze, then came Dirichlet characters which have dropped me back into the pit of h-e-double hockey sticks, and now we have this section, which wasn't half bad. So... okay. But, in terms of what was difficult for me, something that is common for me (especially approaching the a midterm) is that everything seems to bleed together. It isn't until I start writing up my study guide for the test that I'm able to really go back, section by section, definition by definition, and remember/distinguish things. This was my problem with this section. Things were a little mushy with the several different arithmetic functions proven to be multiplicative. Truthfully, I hardly remembered any of those functions, but they seem like a good guideline for getting ready for the upcoming test. The other thing that was difficult for me (since my brain already felt mushy) was the Mobius inversion formula. I really struggled to follow this one. Period. It was pretty darn complicated.
2. What was the most interesting part of the reading for you?
I found the Mobius function, and the proof that it is multiplicative, to be pretty interesting. And, believe it or not, I think I understood everything that was said in those two half pages. I almost felt like I might survive this semester without having to take this class again next semester (please, pass me). But then, just when I was feeling good/strong/smart, the Mobius inversion formula came along and ruined everything. I guess what I found to be so interesting about these pages was the way in which th proof of Theorem 3.6.3 really helped me understand exactly what the multiplicitivity stuff was trying to say. (I mostly understood it, but because I had forgotten about the arithmetic functions it was referring to, having the function and the multiplicitivity side by side was nice.) And that's how it felt for me.
First of all, let me say that this class is just a rollercoaster. I mean, continued fractions felt like a breeze, then came Dirichlet characters which have dropped me back into the pit of h-e-double hockey sticks, and now we have this section, which wasn't half bad. So... okay. But, in terms of what was difficult for me, something that is common for me (especially approaching the a midterm) is that everything seems to bleed together. It isn't until I start writing up my study guide for the test that I'm able to really go back, section by section, definition by definition, and remember/distinguish things. This was my problem with this section. Things were a little mushy with the several different arithmetic functions proven to be multiplicative. Truthfully, I hardly remembered any of those functions, but they seem like a good guideline for getting ready for the upcoming test. The other thing that was difficult for me (since my brain already felt mushy) was the Mobius inversion formula. I really struggled to follow this one. Period. It was pretty darn complicated.
2. What was the most interesting part of the reading for you?
I found the Mobius function, and the proof that it is multiplicative, to be pretty interesting. And, believe it or not, I think I understood everything that was said in those two half pages. I almost felt like I might survive this semester without having to take this class again next semester (please, pass me). But then, just when I was feeling good/strong/smart, the Mobius inversion formula came along and ruined everything. I guess what I found to be so interesting about these pages was the way in which th proof of Theorem 3.6.3 really helped me understand exactly what the multiplicitivity stuff was trying to say. (I mostly understood it, but because I had forgotten about the arithmetic functions it was referring to, having the function and the multiplicitivity side by side was nice.) And that's how it felt for me.
Tuesday, October 25, 2011
Section 3.3 - Pg 110, Due October 26, 2011
1. What was the most difficult part of the reading for you?
Geez louize. I should have known. Yesterday was such a nice/easy day. I understood the section and the homework was a breeze. I should have known something crappy was coming. I struggled with this reading from the beginning. I'm not sure why that is, but my guess is that it's because something completely new to me is being introduced, and it has a weird name. I mean "character"? Really? What does that even mean? To me, it's just a symbol. Obviously it's denoted by a symbol here, but I'm really struggling to understand exactly what it does. From the definition, it is a function (okay...) that maps integers to complex numbers. But, if it's a function, how does it belong to a modular residue class? I'm probably not reading that right, but that's how I'm reading the definition. That's where I get lost. Ignoring that, I can follow the rest of the definition of this function....
2. What was the most interesting part of the reading for you?
Um... good question. Mostly I feel confused, and that's not interesting at all, since it's so common these days. I guess it's interesting because these 'characters' ([disgruntled] whatever) follow so many of the rules and theorems we've established over the last several years of math. Like, x(1)><0, x(a1a2)=x(a1)x(a2) and so on. I mean, since I haven't quite figured out what a character is, it's not that "interesting" but it gives me hope that I might be able to catch up on this idea after a good day in class.
Geez louize. I should have known. Yesterday was such a nice/easy day. I understood the section and the homework was a breeze. I should have known something crappy was coming. I struggled with this reading from the beginning. I'm not sure why that is, but my guess is that it's because something completely new to me is being introduced, and it has a weird name. I mean "character"? Really? What does that even mean? To me, it's just a symbol. Obviously it's denoted by a symbol here, but I'm really struggling to understand exactly what it does. From the definition, it is a function (okay...) that maps integers to complex numbers. But, if it's a function, how does it belong to a modular residue class? I'm probably not reading that right, but that's how I'm reading the definition. That's where I get lost. Ignoring that, I can follow the rest of the definition of this function....
2. What was the most interesting part of the reading for you?
Um... good question. Mostly I feel confused, and that's not interesting at all, since it's so common these days. I guess it's interesting because these 'characters' ([disgruntled] whatever) follow so many of the rules and theorems we've established over the last several years of math. Like, x(1)><0, x(a1a2)=x(a1)x(a2) and so on. I mean, since I haven't quite figured out what a character is, it's not that "interesting" but it gives me hope that I might be able to catch up on this idea after a good day in class.
Monday, October 24, 2011
Section 3.2.5 Due on October 24, 2011
1. What was the most difficult part of the reading for you?
Well, this was a pretty short reading (thanks, by the way) so it embarrasses me to say that I struggled to understand Theorem 3.2.5.2 (including Lemma 3.2.5.1 (together, these make up more than half of the section). I guess, for starters, what are [NZM] AND [P]? I'm guessing these are like... appendices? Other books? I really just don't know what in the world these are. But, to get points for this amazing blog, let me attempt to pinpoint where exactly I got lost. (also, I get lost on where the proof for the theorem and the proof for the lemma end....). Really, I'm confused about which proof goes with which theorem. Like, on the top of page 104 it begins a proof.... and it says "Proof (the sequence of primes is infinite)", so is that starting a proof of its own theorem? Is it proving something specifically mentioned/named above? And, I guess I got lost somewhere around the 10th line. I'm not sure how (maybe it's from something earlier that I'm forgetting) they made this leap about q^2+1 needing to be a power of two. I realize that's only the surface of my confusion, but hey, it's a start.
2. What was the most interesting part of the reading for you?
I thought that looking at this crazy fraction thing as being periodic in nature was kind of interesting. I mean, I was sad that it just lists a special case, rather than a proof (as far as I can see), but that it makes sense that if a_0,a_1,....a_n has some sort of periodic pattern that it could mean something (that I also don't understand.) I guess you can be excited that I'll be in class eager to learn what the heck is being taught here, because I am just not following the connections that they are trying to make. Shocker.
Well, this was a pretty short reading (thanks, by the way) so it embarrasses me to say that I struggled to understand Theorem 3.2.5.2 (including Lemma 3.2.5.1 (together, these make up more than half of the section). I guess, for starters, what are [NZM] AND [P]? I'm guessing these are like... appendices? Other books? I really just don't know what in the world these are. But, to get points for this amazing blog, let me attempt to pinpoint where exactly I got lost. (also, I get lost on where the proof for the theorem and the proof for the lemma end....). Really, I'm confused about which proof goes with which theorem. Like, on the top of page 104 it begins a proof.... and it says "Proof (the sequence of primes is infinite)", so is that starting a proof of its own theorem? Is it proving something specifically mentioned/named above? And, I guess I got lost somewhere around the 10th line. I'm not sure how (maybe it's from something earlier that I'm forgetting) they made this leap about q^2+1 needing to be a power of two. I realize that's only the surface of my confusion, but hey, it's a start.
2. What was the most interesting part of the reading for you?
I thought that looking at this crazy fraction thing as being periodic in nature was kind of interesting. I mean, I was sad that it just lists a special case, rather than a proof (as far as I can see), but that it makes sense that if a_0,a_1,....a_n has some sort of periodic pattern that it could mean something (that I also don't understand.) I guess you can be excited that I'll be in class eager to learn what the heck is being taught here, because I am just not following the connections that they are trying to make. Shocker.
Wednesday, October 19, 2011
Section 3.2.2 Due on October 19, 2011
1. What was the most difficult part of the reading for you?
This time, I will not blame the book. I think it's because of the late hour or because it used a bunch of other proofs from earlier in the section that we haven't discussed in class yet, but I had a hard time following Theorem 3.2.2.2 (Fermat's two square theorem), which makes me nervous because it has a name, so it's probably important. I think I just got confused following the logic on the top of page 92 where everything seems to meld together.
2. What was the most interesting part of the reading for you?
Again, maybe it's the late hour, but I really kind of enjoyed reading Lemma 3.2.2.1. I'm not sure if that's just because I could actually follow it (a treat for me these days) or what it was exactly.... maybe I just felt impressed with the way in which a bunch of theorems we talked about (what seems like forever ago) earlier really came into play in a nice and cohesive way during this proof. I don't think I could ever replicate it (so don't ask me to) but I could actually follow it. Three pats on the back for me.
This time, I will not blame the book. I think it's because of the late hour or because it used a bunch of other proofs from earlier in the section that we haven't discussed in class yet, but I had a hard time following Theorem 3.2.2.2 (Fermat's two square theorem), which makes me nervous because it has a name, so it's probably important. I think I just got confused following the logic on the top of page 92 where everything seems to meld together.
2. What was the most interesting part of the reading for you?
Again, maybe it's the late hour, but I really kind of enjoyed reading Lemma 3.2.2.1. I'm not sure if that's just because I could actually follow it (a treat for me these days) or what it was exactly.... maybe I just felt impressed with the way in which a bunch of theorems we talked about (what seems like forever ago) earlier really came into play in a nice and cohesive way during this proof. I don't think I could ever replicate it (so don't ask me to) but I could actually follow it. Three pats on the back for me.
Monday, October 17, 2011
Section 3.2-3.2.1 Due October 17
1. What was the most difficult part of the reading for you?
I had an okay time following the Pythagorean triples proof, which I'll discuss later, but I must say that I did not have such an okay time following the proof about the equation of x^4+y^4=z^2. It started off simple enough, but then I got lost in somewhere around page 89...
2. What was the most interesting part of the reading for you?
I remember doing a proof for Pythagorean triples a few years ago in "History of Math" that seemed more straightforward and easier to understand than his.... I cannot remember what it is, but believe you me, I am motivated to find my notes from that class, since I think it will help me remember/understand this proof a little better. Like I said before, there is something about this book that makes it so difficult for me to follow proofs (even if I've experienced them before).
I had an okay time following the Pythagorean triples proof, which I'll discuss later, but I must say that I did not have such an okay time following the proof about the equation of x^4+y^4=z^2. It started off simple enough, but then I got lost in somewhere around page 89...
2. What was the most interesting part of the reading for you?
I remember doing a proof for Pythagorean triples a few years ago in "History of Math" that seemed more straightforward and easier to understand than his.... I cannot remember what it is, but believe you me, I am motivated to find my notes from that class, since I think it will help me remember/understand this proof a little better. Like I said before, there is something about this book that makes it so difficult for me to follow proofs (even if I've experienced them before).
Tuesday, October 11, 2011
Section 3.1.4b Due on October 12, 2011
1. What was the most difficult part of the reading for you?
Well, I'm not sure if this is the MOST difficult part, but something that seems really basic and is probably important that I didn't understand was really close to the beginning of the reading. So, on page 72, at the beginning of the proof of the Binet formula, it begins by stating x^2-x-1=0, which we know about the golden ratio from deriving it. BUT, right after that, it says "It follows that..." and gives two equations that supposedly "follow" from what was said above, and I do not get it. Where did these come from? HOw did they come from what was said above? It is frustrating for me because I don't understand how these came to be, and I feel like they're pretty important to the rest of what is being proven. Boo.
2. What was the most interesting part of the reading for you?
I guess that for me the most interesting part of the reading was Corollary 3.1.4.1 where we look at the limit of one Fibonacci number over the previous one, as the limit goes to infinity, and how that equals the golden ratio. Also, I thought that the form in which the golden ratio is written (1+1/1+1/....) was pretty interesting.
Well, I'm not sure if this is the MOST difficult part, but something that seems really basic and is probably important that I didn't understand was really close to the beginning of the reading. So, on page 72, at the beginning of the proof of the Binet formula, it begins by stating x^2-x-1=0, which we know about the golden ratio from deriving it. BUT, right after that, it says "It follows that..." and gives two equations that supposedly "follow" from what was said above, and I do not get it. Where did these come from? HOw did they come from what was said above? It is frustrating for me because I don't understand how these came to be, and I feel like they're pretty important to the rest of what is being proven. Boo.
2. What was the most interesting part of the reading for you?
I guess that for me the most interesting part of the reading was Corollary 3.1.4.1 where we look at the limit of one Fibonacci number over the previous one, as the limit goes to infinity, and how that equals the golden ratio. Also, I thought that the form in which the golden ratio is written (1+1/1+1/....) was pretty interesting.
Thursday, October 6, 2011
Section 3.1.3 Due on October 7, 2011
1. What was the most difficult part of the reading for you?
I'm not sure that there was anything terribly 'difficult' about the reading, but what is making me nervous is the thought of keeping these different types of numbers and formulas memorized and separated in my mind. I mean, maybe it'll get better with time, especially if we discuss their differences in class, but for now I couldn't really tell you the difference between Mersenne numbers or Fermat Numbers. I know that I understand how perfect numbers from the other two, but remembering that crazy formula (when compared with the other two) makes me antsy.
2. What was the most interesting part of the reading for you?
Um... it was interesting to me that there are Mersenne/Fermat Numbers and Primes. I also thought that finding the GCD of two Mersenne numbers was interesting. Also, I'm interested (because I don't feel that the book really gave any insight in this section) in what the Fermat and Mersenne numbers have to do with anything. I mean, they even said in the book that they don't know if there are infinite primes in these sequences... so why are we even talking about them? Obviously there will be an answer now... but there isn't one now, and that makes me sad.
I'm not sure that there was anything terribly 'difficult' about the reading, but what is making me nervous is the thought of keeping these different types of numbers and formulas memorized and separated in my mind. I mean, maybe it'll get better with time, especially if we discuss their differences in class, but for now I couldn't really tell you the difference between Mersenne numbers or Fermat Numbers. I know that I understand how perfect numbers from the other two, but remembering that crazy formula (when compared with the other two) makes me antsy.
2. What was the most interesting part of the reading for you?
Um... it was interesting to me that there are Mersenne/Fermat Numbers and Primes. I also thought that finding the GCD of two Mersenne numbers was interesting. Also, I'm interested (because I don't feel that the book really gave any insight in this section) in what the Fermat and Mersenne numbers have to do with anything. I mean, they even said in the book that they don't know if there are infinite primes in these sequences... so why are we even talking about them? Obviously there will be an answer now... but there isn't one now, and that makes me sad.
Tuesday, October 4, 2011
Section 3.1.2 Due on October 5, 2011
1. What was the most difficult part of the reading for you?
I had a rough week last week, but I have taken the test, and I'm re-committed to this section.
Now, having said that, this section really lost me.... again. Is it me? Is it this book? I've mentioned before how grateful I am that I didn't have to buy a book, but I read these proofs and I hardly understand anything. Seriously. What is my problem.
But, let me see if I can get specific. That might help. Perhaps my first problem is totally my fault. I'm not sure that I remember the exact meaning behind "Diverges" or "Converges." Because, I remember I remember that Sum(1/n) as n goes to infinity converges... but doesn't 1/n diverge? I just don't remember what this stuff means. Both go close to zero, so why does one converge and one diverge? I know this is probably elementary (It's been a few years since I've taken a class that talks about this) but this is one of my big hang ups. (Also, my teacher sucked... so I learned tests and procedures, but I don't know meaning behind them.)
I guess that was the most difficult thing for me. Since both of these big proofs listed here seem to rely on those definitions, I felt lost from the begining.
2. What was the most interesting part of the reading for you?
I thought that it was interesting that primes are denser than the sequence of squares. Thinking about it, it's obvious, but I hadn't thought of comparing the two before. But those squares sure do get large rather quickly, while the primes don't "get large" as much as they get more spacy. So, I thought that was interesting, even though that fact relies on the "Converges/Diverges" dilemma I mentioned above.
I had a rough week last week, but I have taken the test, and I'm re-committed to this section.
Now, having said that, this section really lost me.... again. Is it me? Is it this book? I've mentioned before how grateful I am that I didn't have to buy a book, but I read these proofs and I hardly understand anything. Seriously. What is my problem.
But, let me see if I can get specific. That might help. Perhaps my first problem is totally my fault. I'm not sure that I remember the exact meaning behind "Diverges" or "Converges." Because, I remember I remember that Sum(1/n) as n goes to infinity converges... but doesn't 1/n diverge? I just don't remember what this stuff means. Both go close to zero, so why does one converge and one diverge? I know this is probably elementary (It's been a few years since I've taken a class that talks about this) but this is one of my big hang ups. (Also, my teacher sucked... so I learned tests and procedures, but I don't know meaning behind them.)
I guess that was the most difficult thing for me. Since both of these big proofs listed here seem to rely on those definitions, I felt lost from the begining.
2. What was the most interesting part of the reading for you?
I thought that it was interesting that primes are denser than the sequence of squares. Thinking about it, it's obvious, but I hadn't thought of comparing the two before. But those squares sure do get large rather quickly, while the primes don't "get large" as much as they get more spacy. So, I thought that was interesting, even though that fact relies on the "Converges/Diverges" dilemma I mentioned above.
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