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It literally took me an hour of extreme frustration to write this, and I don't even know if it's correct (probably not):
http://i.imgur.com/tJXpL.png
I mean admittedly, yes, it is my first foray into proof-based mathematics and into number theory, but, WTF, this took me longer than the other supposedly harder exercises--and the result is so trivial, too. (Why do I feel like this was harder than reproving the Rayleigh theorem?)
Does anyone have any suggestions about how I can go about learning proof methods and such and just improving my ability to write proofs in general? Are there books out there that basically teach you problem-solving techniques?
(Comments on the actual proof would be nice, too. I really, really hope it isn't flawed.)
edit fuck my life i only proved the rayleigh theorem in 1 direction FML and i already printed it out
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that looks crazy hard... what level of mathematics is this?
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Is just takes a time, after your first semester or two it becomes second nature. Don't be easy on yourself when you write them
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Oh god I am dreading the time at which I have to start using LaTeX lol.
But yeah, all told, you get used to it after a while.
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Don't worry, proofs aren't supposed to be easy. You get good at them by doing them, like everything else in mathematics. Quick note: some of the notation seems nonstandard. I assume $[x]$ is the floor function -- why not use $\lfloor x \rfloor$? What's $\{x\}$, if not the set containing x -- the non-integer part of x? Maybe that is accepted notation, but none of my number theory courses ever had a use for such things.
On October 01 2012 10:17 Aerisky wrote:
Oh god I am dreading the time at which I have to start using LaTeX lol.
Nooooo tex is awesome. Seriously, once you get used to it, you're never going to want to hand-write mathematics again.
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Yeah, $[x]$ is floor and $\{x\}$ is the fractional part. I'm used to seeing $\lfloor x \rfloor$ but I just decided to stick with the notation that my textbook uses (Elementary Number Theory and Its Applications by Rosen).
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Not going to lie, 3rd and 4th year mathematics based courses were the only subjects I had to scrape by with. I probably could have managed if I'd focused my efforts and studies like a boss, but with a full course load I basically resigned myself to just get a passing grade.
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in general, 1+2+...+x = x(x+1)/2
pretty easy to prove it given that, but idk if your allowed to use it
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Hmm I'm learning proofs too, but my teacher taught us induction. So like prove N=1 true then assume random number k is true then prove the term after (which is k+1) true.
I feel like I'm missing something
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On October 01 2012 10:46 Assault_1 wrote:
in general, 1+2+...+x = x(x+1)/2
pretty easy to prove it given that, but idk if your allowed to use it
How would I use that?
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On October 01 2012 11:00 meguca wrote:Show nested quote +On October 01 2012 10:46 Assault_1 wrote:
in general, 1+2+...+x = x(x+1)/2
pretty easy to prove it given that, but idk if your allowed to use it How would I use that?
Try writing x as [x]+{x}...
To hoot0, you don't need induction for this, and I doubt it would be any easier than showing it directly.
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On October 01 2012 11:04 Iranon wrote:Show nested quote +On October 01 2012 11:00 meguca wrote:On October 01 2012 10:46 Assault_1 wrote:
in general, 1+2+...+x = x(x+1)/2
pretty easy to prove it given that, but idk if your allowed to use it How would I use that? Try writing x as [x]+{x}... To hoot0, you don't need induction for this, and I doubt it would be any easier than showing it directly.
I see.
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Yeah, I avoided Number Theory and Elementary Proofs (double-major dodge). Those kinds of things never clicked for me, and were not as fun as limits and infinite series proofs. I didn't have the brain for a lot of proofs. I envy a lot of math majors haha
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Hi meguca, you should look into the first two chapters of a book called "discrete mathematics and its applications" by rosen.
The first chapter is all about logic and proofs, starting from teaching you basics concepts and boolean logic like AND/OR/XOR into teaching you logical fallacies and how to construct a logical argument. Then, you get into traditional proofs like prove root 2 is irrational. That's where you learn proof by contradiction, proof by induction, etc etc.
Chapter 2 is sets and proofs on sets. The book is not that great to be honest, but it might help you get your foot in the door.
Also, it's good because with some googling you MIGHT be able to find it online. *hint hint*
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99% of high school students won't understand this either. So...
at least you have that working in your favor right?
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Does anyone have any suggestions about how I can go about learning proof methods and such and just improving my ability to write proofs in general? Are there books out there that basically teach you problem-solving techniques?
For the first question, what exactly do you find hard about writing/finding proofs? (In particular, do you find writing proofs or finding proofs hard?) Hard to give a specific answer without knowing what you find hard.
As for problem-solving techniques, the answer is yes:
Polya's "How to Solve It" is a classic text on problem-solving.
Other than that, there are books on math competitions (Competitions using only "elementary math" for high school students, "elementary math" is usually high school algebra, high school geometry, number theory, and combinatorics. Much more of a test of problem-solving ability than deep understanding or knowledge of theory, though those can be useful.) See http://www.artofproblemsolving.com/Store/index.php.
Good luck! (Also, number theory was a nice gate for me into richer and more beautiful areas of math than I had seen, and I hope it will be for you as well. My interests have shifted from number theory since then, but it was a great introduction. glhf :D)
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Don't really have the time for a more thorough response at the moment, but:
Nehsb: My primary issue is in basically concretely writing down logical steps; that is, I often can very easily intuitively understand that something is true, but I find it practically painful to laboriously prove something that seems to be trivial (and it's frustrating, too). I think more familiarity with commonly used proof techniques and 'strategies', so to speak, would help me a lot; I'll definitely take a look at the books you mentioned.
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Just pretend to be Ramanujan. That should give you some mental confidence. Next time you are solving a difficult number theory problem, remember the voice inside your head is none other than Ramanujan.
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proof-based math = math
What is math without proof? Calculus?
And this is not "number theory". This is just a triviality on integers and real numbers, for first year students in the university.
PS : Your proof looks correct.
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EPT
