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# How to get better at maths. watch

1. (Original post by around)
Counterexamples (off the top of my head, from when I did UG maths):

Hilbert's Basis Theorem
Kantorovich-Rubenstein Theorem
Special Adjoint Functor Theorem (the general one is a doddle, in comparison)
Most of the Sylow theorems

I mean, yeah, most of them have some kind of key idea, but it's still not trivial to derive the full proof from that idea.
There are some magic proofs of Sylow, but Imre Leader did a really well-motivated one. Hilbert Basis has a key idea which works: "do the obvious thing with the ideals consisting of leading coefficients of nth degree polynomials". The proof of QR we were given is, admittedly, magic.

Those are the only ones of your list I've encountered yet.
2. (Original post by Smaug123)
There are some magic proofs of Sylow, but Imre Leader did a really well-motivated one. Hilbert Basis has a key idea which works: "do the obvious thing with the ideals consisting of leading coefficients of nth degree polynomials". The proof of QR we were given is, admittedly, magic.

Those are the only ones of your list I've encountered yet.
Sylow's theorem basically relies on you remember to define the right group actions, and then some messy combinatorics to get the right numbers out the end. The proof on Wikipedia is very similar (probably the same) to the one I remember being taught in GRM, and I just had to memorise that.

I'm just prejudiced against HBT because I hate polynomials.
3. Follow these 3 steps, they really do work!

1) Never miss a single lecture, always attend one even if the topic taught is easy.
2) Before attending a class, always get to know what is being taught, ie. learn a sizable chunk of material prior to the lecture, so that you know what you are going to learn; this boosts up confidence.
3) Get more books and just practice. Practice, practice and practice.

Good luck!
4. (Original post by Smaug123)
There are some magic proofs of Sylow, but Imre Leader did a really well-motivated one. Hilbert Basis has a key idea which works: "do the obvious thing with the ideals consisting of leading coefficients of nth degree polynomials". The proof of QR we were given is, admittedly, magic.
I confess I don't remember any details (which usually means I didn't understand it that well, although 25 years gives some level of excuse), but I think the proof of Sylow I used for part II rested on one or two core ideas.

QR is a bit of a weird one - there are proofs that are really based on one or two key ideas, but the ones that you use in exams tend to be a lot more "magic".

The exam process tends to distort things, because sometimes you need to do it by rote just to get the proof done in a reasonable amount of time.

But I think I'd stand by the idea that "every theorem you can only prove by rote" is a theorem you don't really understand.
5. (Original post by around)
Sylow's theorem basically relies on you remember to define the right group actions, and then some messy combinatorics to get the right numbers out the end. The proof on Wikipedia is very similar (probably the same) to the one I remember being taught in GRM, and I just had to memorise that.
(Original post by DFranklin)
I confess I don't remember any details (which usually means I didn't understand it that well, although 25 years gives some level of excuse), but I think the proof of Sylow I used for part II rested on one or two core ideas.
I wrote a summary a while back. Essentially, it goes " for the normaliser of , and use some sensible interpretations of those quantities, the first as the size of an orbit of P under conjugation and the second as the size of a particular quotient group. The size of an orbit can be determined by the class equation." With that key idea, there's only one slightly magic bit, and all three Sylow theorems drop out.
6. (Original post by MathMeister)
Maths has never been about rote-learning at any level.
Well I've always understood everything easily anyway.
Understood everything for the purpose of exams you mean? Because I can't see how you could mean anything else. I'd actually be impressed if an A-level student knew how to prove that the derivative of x^n is nx^(n-1). That's how dodgy A-level mathematics is.
7. (Original post by MrLowIQ)
Understood everything for the purpose of exams you mean? Because I can't see how you could mean anything else. I'd actually be impressed if an A-level student knew how to prove that the derivative of x^n is nx^(n-1). That's how dodgy A-level mathematics is.
I try and learn those sorts of things yeah.
8. (Original post by MathMeister)
I try and learn those sorts of things yeah.
Well then can you post the proof? Since this is from C1.
9. (Original post by MrLowIQ)
Understood everything for the purpose of exams you mean? Because I can't see how you could mean anything else. I'd actually be impressed if an A-level student knew how to prove that the derivative of x^n is nx^(n-1). That's how dodgy A-level mathematics is.
In fairness, I had an excellent A-level teacher who showed this stuff to us.
10. (Original post by MrLowIQ)
Well then can you post the proof? Since this is from C1.
No.. I've better things to do.
As you said you can find it in C1.
11. (Original post by MathMeister)
No.. I've better things to do.
As you said you can find it in C1.
The proof is approximately three lines long. If you know it, it would nearly have been as fast to write it as it would your refusal.
12. (Original post by Smaug123)
The proof is approximately three lines long. If you know it, it would nearly have been as fast to write it as it would your refusal.
I cannot latex
I do know it please stop annoyin'
13. (Original post by MathMeister)
I cannot latex
I do know it please stop annoyin'
Do it on paper and upload the picture.
14. (Original post by Smaug123)
In fairness, I had an excellent A-level teacher who showed this stuff to us.
You were lucky. My teachers had no interest in showing students such things - they just taught the bare minimum required to do well in the exam. We had an excellent trainee teacher for a few months, but he decided that teaching wasn't for him and went back to university to pursue research in number theory.
15. (Original post by MathMeister)
No.. I've better things to do.
As you said you can find it in C1.
I'm actually curious. What proof have you seen in C1? I was referring to the result.
16. (Original post by MrLowIQ)
Understood everything for the purpose of exams you mean? Because I can't see how you could mean anything else. I'd actually be impressed if an A-level student knew how to prove that the derivative of x^n is nx^(n-1). That's how dodgy A-level mathematics is.
You'd be impressed? I thought it was standard. I was asked this in my York interview.

As for OP, I don't have any experience of uni maths, but I'd guess that solving as many problems as possible would help a ton.
17. (Original post by StarvingAutist)
You'd be impressed? I thought it was standard. I was asked this in my York interview.
They wouldn't ask you to solve a standard question in the interview, surely?
18. (Original post by MrLowIQ)
They wouldn't ask you to solve a standard question in the interview, surely?
Well, you know what I mean
19. (Original post by StarvingAutist)
Well, you know what I mean

I posted this earlier but that post isn't appearing until approved by a moderator somehow.
It's from Bostock and Chandler, considered probably the best A-level text around.
Attached Images

20. (Original post by MrLowIQ)
<rigorous proof>
Oh, um, wow that was sure convincing..
Maybe I'm just deluded then

But why don't they include it? The space taken up by that table & paragraph could easily have been used

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