How to get better at maths.

Counterexamples (off the top of my head, from when I did UG maths):

Hilbert's Basis Theorem
Quadratic reciprocity
Kantorovich-Rubenstein Theorem
Beck's monadicity 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.

Reply 61

around

13

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.

Reply 62

AkashdeepDeb

1

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!

Reply 63

DFranklin

18

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.

Reply 64

Smaug123

15

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 "

\dfrac{|G|}{|P|} = \dfrac{|G|}{|N|} \dfrac{|N|}{|P|}

for

N

the normaliser of

P

, 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.

Reply 65

MrLowIQ

1

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.

Reply 66

MathMeister

10

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.

Reply 67

MrLowIQ

1

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.

Reply 68

Smaug123

15

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.

Reply 69

MathMeister

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.

Reply 70

Smaug123

15

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.

Reply 71

MathMeister

10

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'

(edited 9 years ago)

Reply 72

rayquaza17

17

Original post by MathMeister

I cannot latex
I do know it please stop annoyin'

Do it on paper and upload the picture. :biggrin:

Reply 73

MrLowIQ

1

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.

(edited 9 years ago)

Reply 74

MrLowIQ

1

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. :confused:

Reply 75

StarvingAutist

17

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.