Proving simple things (elementary number theory)

[Elwin.Martin]

Proving "simple" things (elementary number theory)

Hey, I was wondering if I could ask for some help proving simple things in number theory, like divisibility things etc. The kind of stuff that you stare at and say "Duh, properties of real numbers...next?" but maybe don't know how to start as a beginner.

I can't ask for specifics because the only problems I can think of I was told not to ask for help on, but I am sure it's legal to ask questions to a community about proof.

I don't have time to prove 50 other things to get a feel for proofs, I'm too slow. I do have time to read a whole lot of basic number theory proofs if that would help, I just don't know where to look. I have no resources beyond like, three simple proofs that are only tangentially related to what I need to know in general. If my only hope is in proving a lot of other things, I might just take the hit and worry long term.

In the long term, would books like Polya's be of use? I'm currently a "Find that" person rather than a "Show that" as most people fresh out of high school are.

Thanks for your time, I appreciate any guidance the community has.

 

[HallsofIvy]

I get the feeling, reading your post, that you are telling us that you really don't like mathematics, don't want to spend much time on it, and are looking for a short cut. Did you really mean to say that?

 

[Elwin.Martin]

I get the feeling, reading your post, that you are telling us that you really don't like mathematics, don't want to spend much time on it, and are looking for a short cut. Did you really mean to say that?

No, I was asking for two different things. I was asking for the best method to work on proving things that looks trivial, but aren't in the short term and what I should work on for learning to prove things in the long term.

Impatient? A little bit. I posted that at almost 3 am after working on the same problem set for the entire night and getting everything, but the question about divisibility. (p divides a, p divides a^2+b^2...show that p divides b). I'm still not sure how to show something that seems so simple...I just can't think of anything besides convoluted brute force that lacks rigor. I spent 3 hours on something that will probably end up taking a quarter of a page and it got a bit frustrating. It's not hard to forget about the math when you're worried about your grades, at least it's not for me.

I's not like I don't want to learn. It's too late for the short term that I was asking about anyway, but I would like some advice on proving things in the long term. I've never had to formally prove things before and while I've been told I would just pick it up now that I'm in the more abstract maths, I'm not able to prove everything I've needed to for class and it concerns me. I don't expect it to be easy, but I'm sure there's something I can do to help myself besides bothering my instructor about everything I can't do on my own. Again, does anyone support books like Polya's? I have access to it, but I haven't looked at it since I don't know if I can trust a book with such a seemingly bold title...

Sorry for sounding impatient, but don't judge me so quickly. I was looking for a short term solution to one problem and I'll admit that, but I didn't write off all of mathematics.

 

[disregardthat]

Definitely try a book like Polya's if you want a beginners guide to proofs, and starting early (most people get the hang of it at university level) will give you a great advantage. Don't listen to Halls, just spend a lot of time solving problems and doing proofs from books like this

In my experience starting with the elementary and basics makes you in the long run much more aware of the formal mechanics of more abstract proofs as well as you are used to proving things from scratch, and not just jumping from theorem to theorem which e.g. tend to be the deal in many higher-level courses.

As to your simple problem, remember to use every bit of information you get. If a is divisible by p, then a = a'*p for some integer a'... Go on and plug that in, and do the same for a^2+b^2. See what you get.

 

[Pithikos]

Search for "TCC - Discrete Mathematics". It's a set of videos VERY well explained which will help you a lot. I didn't open a book myself under the course. I just watched these videos and solved simple exercises on my own. Else there is Khan Academy of course.

 

[Elwin.Martin]

Definitely try a book like Polya's if you want a beginners guide to proofs, and starting early (most people get the hang of it at university level) will give you a great advantage. Don't listen to Halls, just spend a lot of time solving problems and doing proofs from books like this

In my experience starting with the elementary and basics makes you in the long run much more aware of the formal mechanics of more abstract proofs as well as you are used to proving things from scratch, and not just jumping from theorem to theorem which e.g. tend to be the deal in many higher-level courses.

As to your simple problem, remember to use every bit of information you get. If a is divisible by p, then a = a'*p for some integer a'... Go on and plug that in, and do the same for a^2+b^2. See what you get.

Great, I'll try to look at it after I finish my notes, I think the University's library has it. Yeah, I'm hoping that getting a strong set of fundamentals from a book like this will help...my Abstract course isn't "hard" yet, but it might get there before the end of the semester.

Thanks, I think being awake helped too.

By the way, do you have any recommendations besides his book? I'll see if I like it, but I'm just seeing if there are other author's who are good. I've heard a lot about the book, but I've never actually looked in it (yet).

 

[Elwin.Martin]

Search for "TCC - Discrete Mathematics". It's a set of videos VERY well explained which will help you a lot. I didn't open a book myself under the course. I just watched these videos and solved simple exercises on my own. Else there is Khan Academy of course.

You mean TTC? Like, this?
http://www.thegreatcourses.com/tgc/courses/course_detail.aspx?cid=1456
"Most recently, the MAA named Professor Benjamin the 2006–2008 George Pólya Lecturer"?

I'd think:
http://www.thegreatcourses.com/tgc/courses/course_detail.aspx?cid=1483
would almost be more appropriate (since I'm also preparing for some math competitions x.x')

 

[Studiot]

You might look at

A First Course in Abstract Algebra

By Fraleigh

This has an introductory chapter on proofs and continues in that vein to quite a reasonably advanced standard, along with plenty of motivational (fun) examples eg Escher's diagrams.

go well

 

[Pithikos]

You mean TTC? Like, this?
http://www.thegreatcourses.com/tgc/courses/course_detail.aspx?cid=1456
"Most recently, the MAA named Professor Benjamin the 2006–2008 George Pólya Lecturer"?

I'd think:
http://www.thegreatcourses.com/tgc/courses/course_detail.aspx?cid=1483
would almost be more appropriate (since I'm also preparing for some math competitions x.x')

Well to be honest.. A lot of the learning experience is absolutely dependent on the professor despite what many say in an effort to defend themselves. Professor Benjamin does explain everything down to the core while smiling all the time, using simple examples so he doesn't get you frustrated or bored. I tried other videos from TTC (Number Theory etc.) but they were boring.. boring professors just standing and talking. Benjamin is plain awesome(and I am not sponsored by him or anything)! He also has a second video course called Joy Of Mathematics. That is also very nice to watch but it's more basic.
The Discrete Mathematics videos have a lot of proofs and Number Theory inside so I think it's very appropriate for your needs.

As about reading books I personally never learned much in that way. At least not in a fast way. Reading a book puts you to have to sit and visualise each step and many times makes you visualise incorrectly. Watching a video gives you the right visualisation and in my opinion is much easier to hold you focused(as long as the explanation is simple enough).

 

[disregardthat]

By the way, do you have any recommendations besides his book?

I personally believe solving elementary but hard mathematical problems can be of great benefit while being a lot of fun as well. A book called "The art and craft of problems solving" by Paul Zeits has helped me a great deal, and will hone your skills in proving things, but also solving problems. I liked it a lot, and would recommend it to anyone who are interested in mathematical problem solving. If you are as you say interested in mathematical competitions, this book will serve you very well. Prepeare to spend several hours on simply stated but hard problems. I also agree with Studiots suggestion, though it is much more particular.

I don't agree with people saying that reading books by yourself isn't of great help. All that is needed is persistence and time, and you will get much more in return compared to watching videos.

 

[Elwin.Martin]

I personally believe solving elementary but hard mathematical problems can be of great benefit while being a lot of fun as well. A book called "The art and craft of problems solving" by Paul Zeits has helped me a great deal, and will hone your skills in proving things, but also solving problems. I liked it a lot, and would recommend it to anyone who are interested in mathematical problem solving. If you are as you say interested in mathematical competitions, this book will serve you very well. Prepeare to spend several hours on simply stated but hard problems. I also agree with Studiots suggestion, though it is much more particular.

:) I have it, I'm going to start it soon if I can (tonight if I finish my other work)...

I don't agree with people saying that reading books by yourself isn't of great help. All that is needed is persistence and time, and you will get much more in return compared to watching videos.

Didn't someone say that learning Mathematics is DOING Mathematics?