Studying Mathematics Thread

[Kernel Sohcahtoa]

My studies in number theory have been interesting so far; however, they got sidetracked last week due to my store inventory (I worked every night of inventory).  The elementary number theory text by Jones, which I'm currently using as a primary text, has the added bonus that it contains fully worked out solutions to the exercises.  As an individual who self-studies mathematics, this is very useful: in the absence of a teacher, should I get stuck on a proof exercise for whatever reason, I can always go to the back and do an analysis of the author's proof and treat this as an exercise in understanding more worked examples; thus, improving my ability to fill in the missing details and self-explain proofs.  In order to ensure that I do not rob myself of the opportunity to work exercises on my own, I'm getting an additional book on number theory (this will give me three number theory books total), which will hopefully provide me with plenty of opportunities to practice proof exercises.   

P.S. For any others here who also self-study mathematics, I am very open to any tips or suggestions that you may have.

[polymath257]

My studies in number theory have been interesting so far; however, they got sidetracked last week due to my store inventory (I worked every night of inventory).  The elementary number theory text by Jones, which I'm currently using as a primary text, has the added bonus that it contains fully worked out solutions to the exercises.  As an individual who self-studies mathematics, this is very useful: in the absence of a teacher, should I get stuck on a proof exercise for whatever reason, I can always go to the back and do an analysis of the author's proof and treat this as an exercise in understanding more worked examples; thus, improving my ability to fill in the missing details and self-explain proofs.  In order to ensure that I do not rob myself of the opportunity to work exercises on my own, I'm getting an additional book on number theory (this will give me three number theory books total), which will hopefully provide me with plenty of opportunities to practice proof exercises.   

P.S. For any others here who also self-study mathematics, I am very open to any tips or suggestions that you may have.

You seem to be doing it right.

[Kernel Sohcahtoa]

I've recently become very interested in abstract algebra again.  As a result, I think that I'll cover congruences via my primary number theory text and then give myself a refresher on abstract algebra (I've actually been doing this already), so that I'll be better prepared to learn the chapters in my number theory text that cover algebraic number theory.

[Kernel Sohcahtoa]

I'm getting close to finishing up my abstract algebra refresher self-study.  I can't stress enough how important it is to be able to take a pen and paper to a proof of a main theorem that is presented in a chapter and then being able to put it into your own words and make sense of it via identifying the various definitions, theorems, corollaries, lemmas, and other mathematical ideas that were involved in successfully writing the proof (this process is referred to as self-explanation).  Hence, IMO, once one gains a general understanding of proofs, then learning a proof-based math subject is a good way to hone one's self-explanation proof skills.

That said, once I'm finished, I'm debating whether to take a break from math or to jump right into teaching myself basic graph theory; IMO, graph theory seems like an interesting topic.

[The Valkyrie]

I'm getting close to finishing up my abstract algebra refresher self-study.  I can't stress enough how important it is to be able to take a pen and paper to a proof of a main theorem that is presented in a chapter and then being able to put it into your own words and make sense of it via identifying the various definitions, theorems, corollaries, lemmas, and other mathematical ideas that were involved in successfully writing the proof (this process is referred to as self-explanation).  Hence, IMO, once one gains a general understanding of proofs, then learning a proof-based math subject is a good way to hone one's self-explanation proof skills.

That said, once I'm finished, I'm debating whether to take a break from math or to jump right into teaching myself basic graph theory; IMO, graph theory seems like an interesting topic.

You lost me at "abstract algebra" but well done.

[Kernel Sohcahtoa]

I'm getting close to finishing up my abstract algebra refresher self-study.  I can't stress enough how important it is to be able to take a pen and paper to a proof of a main theorem that is presented in a chapter and then being able to put it into your own words and make sense of it via identifying the various definitions, theorems, corollaries, lemmas, and other mathematical ideas that were involved in successfully writing the proof (this process is referred to as self-explanation).  Hence, IMO, once one gains a general understanding of proofs, then learning a proof-based math subject is a good way to hone one's self-explanation proof skills.

That said, once I'm finished, I'm debating whether to take a break from math or to jump right into teaching myself basic graph theory; IMO, graph theory seems like an interesting topic.

You lost me at "abstract algebra" but well done.

I couldn't begin to understand your medical expertise/knowledge.  I'm quite certain that I would lose you very quickly.

[Tiberius]

That said, once I'm finished, I'm debating whether to take a break from math or to jump right into teaching myself basic graph theory; IMO, graph theory seems like an interesting topic.

All hail Dijkstra's algorithm.

[Aliza]

This is a really great resource to have here. I'm working on some stuff for this class that I don't really know much about, so having a place where I can just ask a quick question is really helpful. Thanks for this thread.

[polymath257]

I'm getting close to finishing up my abstract algebra refresher self-study.  I can't stress enough how important it is to be able to take a pen and paper to a proof of a main theorem that is presented in a chapter and then being able to put it into your own words and make sense of it via identifying the various definitions, theorems, corollaries, lemmas, and other mathematical ideas that were involved in successfully writing the proof (this process is referred to as self-explanation).  Hence, IMO, once one gains a general understanding of proofs, then learning a proof-based math subject is a good way to hone one's self-explanation proof skills.

That said, once I'm finished, I'm debating whether to take a break from math or to jump right into teaching myself basic graph theory; IMO, graph theory seems like an interesting topic.

How far into the abstract algebra did you get? I always really enjoyed Galois theory. That there is a *proof* that 5th degree polynomials can't be solved via radicals is just *fun*. But you need to do quotient rings and some field theory first.

[Kernel Sohcahtoa]

I'm getting close to finishing up my abstract algebra refresher self-study.  I can't stress enough how important it is to be able to take a pen and paper to a proof of a main theorem that is presented in a chapter and then being able to put it into your own words and make sense of it via identifying the various definitions, theorems, corollaries, lemmas, and other mathematical ideas that were involved in successfully writing the proof (this process is referred to as self-explanation).  Hence, IMO, once one gains a general understanding of proofs, then learning a proof-based math subject is a good way to hone one's self-explanation proof skills.

That said, once I'm finished, I'm debating whether to take a break from math or to jump right into teaching myself basic graph theory; IMO, graph theory seems like an interesting topic.

How far into the abstract algebra did you get? I always really enjoyed Galois theory. That there is a *proof* that 5th degree polynomials can't be solved via radicals is just *fun*. But you need to do quotient rings and some field theory first.

Thanks for your reply, Polymath.  Galois Theory sounds exciting; however, I'm eager to take a break from abstract algebra and explore another topic.  

That said, I've studied the following topics: groups; fundamental theorems of groups; cyclic groups; subgroups; direct products; functions; symmetric groups; equivalence relations and cosets (I really enjoyed this section); counting the elements of a finite group; normal subgroups and quotient groups (this section was neat); homomorphisms (these are cool); homomorphisms and normal subgroups (I enjoyed this section; it covers the isomorphism theorems, which are very neat IMO); Rings (I enjoyed this section).  I'm currently studying subrings, ideals, and quotient rings.  Once I'm finished with this section, I'm either going to pack it in and conclude my self-study or cover the section on ring homomorphisms and then call it quits.