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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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i am a first year student
doing an exercise in the first section of the first chapter of my beginning number theory textbook
yes, it is trivial, i am aware.
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I don't understand why proofs are considered dreadful or difficult.
To me they're the most intuitive and pleasant thing in all of Mathematics.
I'd much rather prove something that is presented without artifice than solve one of those masturbatory mathematical problems that combine different fields of mathematics and physics in weird ways and give you cumbersome algebra to juggle. God do I hate those for the reason that a lot of people memorize those and consider them "important problems" that they expect you to solve quickly when in fact they aren't even usable in practical situations let alone test any relevant part of your analytic thinking. You either know the solution to the puzzle and solve it really fast or you don't already know it and it takes a long time, depending on which intuition you have at the start.
I guess I'm just dissatisfied with the quality of mathematical problems that I've had in my life. I love learning mathematics but man are the textbooks unnecessarily torturous sometimes. I haven't even talked about equations that go far beyond what can be stored in your brain so you write down everything you think so it's immediately slower and laborious.
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On October 01 2012 23:10 Fyodor wrote:
I don't understand why proofs are considered dreadful or difficult.
To me they're the most intuitive and pleasant thing in all of Mathematics.
I'd much rather prove something that is presented without artifice than solve one of those masturbatory mathematical problems that combine different fields of mathematics and physics in weird ways and give you cumbersome algebra to juggle. God do I hate those for the reason that a lot of people memorize those and consider them "important problems" that they expect you to solve quickly when in fact they aren't even usable in practical situations let alone test any relevant part of your analytic thinking. You either know the solution to the puzzle and solve it really fast or you don't already know it and it takes a long time, depending on which intuition you have at the start.
I guess I'm just dissatisfied with the quality of mathematical problems that I've had in my life. I love learning mathematics but man are the textbooks unnecessarily torturous sometimes. I haven't even talked about equations that go far beyond what can be stored in your brain so you write down everything you think so it's immediately slower and laborious.
Yeah man, fuck derivations.
It's not like they created some of the most important equations of all time or anything.
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On October 02 2012 00:06 ymir233 wrote:Show nested quote +On October 01 2012 23:10 Fyodor wrote:
I don't understand why proofs are considered dreadful or difficult.
To me they're the most intuitive and pleasant thing in all of Mathematics.
I'd much rather prove something that is presented without artifice than solve one of those masturbatory mathematical problems that combine different fields of mathematics and physics in weird ways and give you cumbersome algebra to juggle. God do I hate those for the reason that a lot of people memorize those and consider them "important problems" that they expect you to solve quickly when in fact they aren't even usable in practical situations let alone test any relevant part of your analytic thinking. You either know the solution to the puzzle and solve it really fast or you don't already know it and it takes a long time, depending on which intuition you have at the start.
I guess I'm just dissatisfied with the quality of mathematical problems that I've had in my life. I love learning mathematics but man are the textbooks unnecessarily torturous sometimes. I haven't even talked about equations that go far beyond what can be stored in your brain so you write down everything you think so it's immediately slower and laborious.
Yeah man, fuck derivations. It's not like they created some of the most important equations of all time or anything.
Derivation is like the most straight foward stuff in maths lol.
@OP: induction and win?
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It is a big jump from solving equations into proof based mathematics. I have a degree in mathematics, and I remember making the jump.
Get ready to spend hours in a library with a cup of coffee. As courses get more difficult, the proofs can become more complex.
My best advice to you is to really invest the time it takes to write out a nice proof. For those of us that are really good at math we get lazy. We are used to things in Calculus, and differential equations. Things we know how to solve. It truly is a big jump to go from a course where the professor tells us how to solve each "type" of equation, to where the professor shows us how things build on top of each other, and then asks us if something follows from those building blocks, if so, prove it.
It is almost like a new subject.
But I encourage you to keep at it. You are good at math because you are good at solving problems. Your problem now is learning how to write proofs.
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On October 02 2012 00:18 Boblion wrote:Show nested quote +On October 02 2012 00:06 ymir233 wrote:On October 01 2012 23:10 Fyodor wrote:
I don't understand why proofs are considered dreadful or difficult.
To me they're the most intuitive and pleasant thing in all of Mathematics.
I'd much rather prove something that is presented without artifice than solve one of those masturbatory mathematical problems that combine different fields of mathematics and physics in weird ways and give you cumbersome algebra to juggle. God do I hate those for the reason that a lot of people memorize those and consider them "important problems" that they expect you to solve quickly when in fact they aren't even usable in practical situations let alone test any relevant part of your analytic thinking. You either know the solution to the puzzle and solve it really fast or you don't already know it and it takes a long time, depending on which intuition you have at the start.
I guess I'm just dissatisfied with the quality of mathematical problems that I've had in my life. I love learning mathematics but man are the textbooks unnecessarily torturous sometimes. I haven't even talked about equations that go far beyond what can be stored in your brain so you write down everything you think so it's immediately slower and laborious.
Yeah man, fuck derivations. It's not like they created some of the most important equations of all time or anything. Derivation is like the most straight foward stuff in maths lol. @OP: induction and win?
Regardless of whether they're straightforward or not they're relatively significant in discerning relationships that might not have seemed obvious, like with integral equations...I mean good for you if you could have figured out Maxwell relations before Maxwell but...
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Ah, just keep it up. Everyone feels like that from time to time in math. The only thing you can do is keep it up and concentrate on the task at hand.
Also: LaTeX sucks  It's just sooo f***ing stupid to type and compile and then go to google to find out that you didn't make a mistake, it's just align is that is stupid... 
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Ifucking hate LaTeX so much and one of my courses requires problem sets typed up in LaTeX  (
-e- and your proof looks fine as long as the logic is good. I don't know what the notation [x] is so i can't say for sure
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lol i just recently started using LaTeX and it has changed my life, I no longer submit any homework not written in LaTeX :D
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On October 01 2012 14:11 VaySept wrote:
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.
A bit harsh, but true, unless you wrote maths before Gauss or something. Or you're Ramanujan :p
Wasn't surprised to read France at the top of the post^^
Edit : and yeah, fuck LaTeX, I did not choose maths to type nice looking things on a computer and look at stupid error messages
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ah the good ol days when all i had to worry about were proofs and physics hw
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EPT
