Cambridge Maths Students and Applicants

Original post by newblood

I doubt senior wrangler every year, or the top few are always pure. I cant see why it wouldnt be even among pure/applied, probably changes a lot so can't go by anecdote. Otherwise, the system is flawed to persuade the top to do pure over their subject interests just to get the highest marks...

I suspect the chances of being senior wrangler and doing majority applied courses is significantly less than someone of equal ability doing majority pure courses. I say this because, having discussed it with others, the fastest applied questions generally take significantly longer than their pure counterparts due to the fact they often involve a significant computational element which eats up time. I would say that my fastest question (applied) takes around 25 mins - this basically just writing time, almost no thinking - with the slowest taking around 30. Obviously this is just anecdotal, although I did well enough that I feel it has some weight to it, and I'm sure that people can write faster than I can but even so you'd have to do something special to get the time taken consistently down to a level where you can get 8/9+ questions out on each paper.

(edited 7 years ago)

Reply 6003

Original post by JosephML

I suspect the chances of being senior wrangler and doing majority applied courses is significantly less than someone of equal ability doing majority pure courses. I say this because, having discussed it with others, the fastest applied questions generally take significantly longer than their pure counterparts due to the fact they often involve a significant computational element which eats up time. I would say that my fastest question (applied) takes around 25 mins - this basically just writing time, almost no thinking - with the slowest taking around 30. Obviously this is just anecdotal, although I did well enough that I feel it has some weight to it, and I'm sure that people can write faster than I can but even so you'd have to do something special to get the time taken consistently down to a level where you can get 8/9+ questions out on each paper.

If your slowest is only 30 minutes then there are plenty of us who'd like to know your secrets :tongue:

There's many a question I've banged my head against for far longer (though thankfully not in the exam!).

From my experience what you're saying would make sense. My experience of pure questions is they are conditionally faster if you know what you're doing. I was getting alphas in the Graph Theory questions in 15-20 minutes this year thanks to some quite nice questions set, and I am a reasonably slow leftie who rewrites every lecture because I cannot write them to a sufficiently neat standard sufficiently quickly in real time. Obviously some Pure takes longer than others (There was a Linear Analysis Q this year taking a good 30 minutes just to write out despite being conceptually pretty simple) but there you go.

There's also the question of question quality - whilst it is natural to try for 20/20 on every question I have heard of Senior Wranglers instead aiming to get enough for the alpha and then move swiftly on to try get more due to the sheer number of bonus points they provide.

(edited 7 years ago)

Reply 6004

JosephML

Original post by DJMayes

If your slowest is only 30 minutes then there are plenty of us who'd like to know your secrets :tongue:

There's many a question I've banged my head against for far longer (though thankfully not in the exam!).

Haha, that was a slight exaggeration - I was a bit drunk last night - but it probably isn't far off my absolute peak just before and during the exams (obviously discounting Qs that I don't finish because that would ruin the average :wink:

).

Reply 6005

tridianprime

Are there any prerequisites for Part II Graph Theory? (other courses in IA or IB that need to be taken first)

Thanks.

Reply 6006

newblood

19

Original post by tridianprime

Are there any prerequisites for Part II Graph Theory? (other courses in IA or IB that need to be taken first)

Thanks.

no

Reply 6007

Gregorius

14

Original post by tridianprime

Are there any prerequisites for Part II Graph Theory? (other courses in IA or IB that need to be taken first)

Thanks.

This is a good guide to prerequisites for part II; the advice given for graph theory seems very sound to me. Wilson's little book is a very nice introduction, if you want to see what's in store!

Reply 6008

Original post by tridianprime

Are there any prerequisites for Part II Graph Theory? (other courses in IA or IB that need to be taken first)

Thanks.

Nah, it's accessible to any Part II student. I personally wouldn't bother with books for it; it's usually lectured by Leader who is more than sufficient for the course and who doesn't rely rate any of the books for it except Bollobas' anyway.

Reply 6009

Original post by cloisters

Hi, question regarding Burkill's Analysis (first course): I understand up to where he says he will take the positive integers for granted as well as those operations on them needed for extending the number system. However, while most of the assumptions he subsequently states are in fact used for this purpose (positive integers, the arithmetic operations, the fact that the positive integers exhibit order), I don't see where the fact that 1.a=a.1=a is used. Since he said he will only state things that will be used for extending the number system, I don't see why he should have stated this fact if he didn't then use it somewhere for the purpose of extending the number system. Addition was used in setting up the equation a+x=b which was then used for defining 0 and negative numbers; division was used for solving bx=a; the fact that the positive integers exhibit order is assumed when stating that a+x=b isn't soluble in positive integers for a>=b; nowhere is the fact 1.a=a.1=a used, however.

You use it for a lot of stuff. Say for example in proving that ab = ac means b=c. You multiply both sides by a^(-1) to get (a . a^(-1)) b = (a . a^(-1)) c, and so you get 1.b = 1.c but by the rule, this means 1.b=b and 1.c = c so b=c.

Another example is ab= 0 means a =0 or b=0. To prove, assume a =/=0 then from ab=0, multiply both sides by a^(-1) to get 1.b = 0. But this means b=0 by the rule.

etc... tons of places you need to use it, just places that you think seem very trivial.

Secondly, why is it that when extending the number system to include 0, negative numbers and rational numbers, he didn't show how to construct them, whereas for irrational numbers he went through the pain of doing so via Dedekind cuts. Is it because the real numbers are more "important" in analysis than negative and rational numbers (therefore we must study them in more detail, which includes constructing them)? Or is it because unlike negative and rational numbers, which are each contained in one compact form (-a and a/b respectively), the irrational numbers are scattered about (some are square roots, other cube roots, still others - such as pi - effectively just symbols, not even the solution of any equation with integer coefficients) and so we must define them in a way that gathers them under one compact concept?

Any light you can shed on these two matters, would be appreciated.

Thanks

You do need to construct the integers and rationals from the naturals, he just doesn't bother because he presupposes it's either elementary or too pedantic/trivial/formal to do in a "first course" of real analysis. Anywho, the integers are easy enough, just take the negative of the naturals and union it with the naturals. (although you need to do some work) and rationals aren't too hard either, just quotients of integers where you can define a/b to be the ordereed part (a,b) belonging to a certain equivalence class. So to do all this construction, he'd have to teach things like equivalence classes and the likes that he must have felt just weren't worth the trouble to construct something as as fundamental as the intengers in a first course book.

Reply 6010

Original post by cloisters

Yes but he doesn't actually need those proof does he, since he states them as axioms.

Nooo, he doesn't at all!

As you can see, he's proving that ab = ac => b=c. It's not an axiom, and in that proof, he makes uses the rule a * 1/a * c= 1 * c= c.

So I take it constructing them just isn't as important for analysis purposes (at least in a first course) as constructing the irrational numbers?

Thanks for your replies by the way :smile:

Yep, pretty much!

No worries, are you coming to Cambridge this October? :smile:

Reply 6011

Original post by cloisters

Technically though he isn't using it for extending the number system (the proof you cite is not for one of the 11 axioms he states by the way) which by the time he gives that proof has already been extended to encompass the real numbers, if you see what I mean. Maybe I'm just being pedantic :P

Ah, I see what you mean now! I guess the best you're going to get from me is that it extends the number system by turning the reals into a ordered field, it wouldn't be an ordered field without a multiplicative identity. So it's not quite extending it in the sense of "integers -> rationals" or "rationals -> reals" but in the sense of "set of numbers -> ordered field".

Reply 6012

Original post by cloisters

"The best I'm going to get from you" is exactly what I was after - thanks!

No worries, keep your eyes peeled, you'll probably get someone who (actually, unlike me) knows something about analysis offering you a few tidbits of wisdom. :smile:

Reply 6013

Original post by cloisters

Humble (but only if you aren't aware of your humbleness :wink:

)

No, I just actually haven't ever done any analysis. :lol:

Reply 6014

Original post by cloisters

Hi, question regarding Burkill's Analysis (first course): I understand up to where he says he will take the positive integers for granted as well as those operations on them needed for extending the number system. However, while most of the assumptions he subsequently states are in fact used for this purpose (positive integers, the arithmetic operations, the fact that the positive integers exhibit order), I don't see where the fact that 1.a=a.1=a is used. Since he said he will only state things that will be used for extending the number system, I don't see why he should have stated this fact if he didn't then use it somewhere for the purpose of extending the number system. Addition was used in setting up the equation a+x=b which was then used for defining 0 and negative numbers; division was used for solving bx=a; the fact that the positive integers exhibit order is assumed when stating that a+x=b isn't soluble in positive integers for a>=b; nowhere is the fact 1.a=a.1=a used, however.

Secondly, why is it that when extending the number system to include 0, negative numbers and rational numbers, he didn't show how to construct them, whereas for irrational numbers he went through the pain of doing so via Dedekind cuts. Is it because the real numbers are more "important" in analysis than negative and rational numbers (therefore we must study them in more detail, which includes constructing them)? Or is it because unlike negative and rational numbers, which are each contained in one compact form (-a and a/b respectively), the irrational numbers are scattered about (some are square roots, other cube roots, still others - such as pi - effectively just symbols, not even the solution of any equation with integer coefficients) and so we must define them in a way that gathers them under one compact concept?

Any light you can shed on these two matters, would be appreciated.

Thanks

I'm a bit late to this but it's a combination of a few things:

1) The rational numbers is something he presumably didn't show because it is a simple construction and he didn't want to waste time.

2) The real numbers most certainly are more important to analysis in the sense that you can't really do analysis without them! As you said, it's also much less obvious that real numbers exist and that you can extend the field of rationals to the field of reals in a single well defined way.

However, I was always introduced via the idea of least upper bounds - a single axiom that extends the entire field.

The axiom is very simple - that every non empty set of numbers with an upper bound has a smallest upper bound.

As an example of its use, we prove the square root of 2 exists. Consider the set of numbers whose square is less than 2 - this is obviously non empty and bounded above, so has a least upper bound, say X.

If X^2 is less than 2, then we can find a constant C>0 sufficiently small that (X+C)^2 is less than 2; then X was not an upper bound.

If X^2 > 2, then similarly to above we can find a smaller upper bound, so X was not the least.

Thus X^2 = 2, so X is our square root of 2.

Burkills book may teach this and it's the first thing you do in the Cambridge IA Analysis course but if not I wanted to talk about it because it really is a neat thing and much nicer than the other constructions of the reals in my opinion.

Reply 6015

JosephML

Original post by DJMayes

I'm a bit late to this but it's a combination of a few things:

1) The rational numbers is something he presumably didn't show because it is a simple construction and he didn't want to waste time.

2) The real numbers most certainly are more important to analysis in the sense that you can't really do analysis without them! As you said, it's also much less obvious that real numbers exist and that you can extend the field of rationals to the field of reals in a single well defined way.

However, I was always introduced via the idea of least upper bounds - a single axiom that extends the entire field.

The axiom is very simple - that every non empty set of numbers with an upper bound has a smallest upper bound.

As an example of its use, we prove the square root of 2 exists. Consider the set of numbers whose square is less than 2 - this is obviously non empty and bounded above, so has a least upper bound, say X.

If X^2 is less than 2, then we can find a constant C>0 sufficiently small that (X+C)^2 is less than 2; then X was not an upper bound.

If X^2 > 2, then similarly to above we can find a smaller upper bound, so X was not the least.

Thus X^2 = 2, so X is our square root of 2.

Burkills book may teach this and it's the first thing you do in the Cambridge IA Analysis course but if not I wanted to talk about it because it really is a neat thing and much nicer than the other constructions of the reals in my opinion.

Is this not just a specific example of a Dedekind cut? Haven't really done any analysis since IA but I would imagine that all constructions of the reals are in fact the same but written out slightly differently.. ?

(edited 7 years ago)

Reply 6016

Original post by JosephML

Is this not just a specific example of a Dedekind cut? Haven't really done any analysis since IA but I would imagine that all constructions of the reals are in fact the same but written out slightly differently.. ?

Effectively yes, but I think it's the nicest way of writing it out :tongue:

These two methods are effectively isomorphic when constructing a specific real number but the least upper bound axiom does lend itself nicely to proving the basic results of analysis on the reals.

Some constructions are certainly a bit different in flavour though, even if they result in the same thing. My example for this would be one I pulled from Wikipedia - that is, defining

\mathbb{R}

as the completion of

\mathbb{Q}

with respect to the standard (Euclidean) metric.

This much more analytic construction does give you what you want but I'd argue it's much less effective. That it's ordered isn't as inherently obvious by definition, for example. The proofs directly from this axiom require you to explicitly construct Cauchy sequences as well, which is fine and instructive in its own right but does give them a decidedly different flavour.

Beyond all this it presupposes analytic ideas such as completeness which makes it inherently less suitable for a beginners construction when introducing the topic.

(edited 7 years ago)

Reply 6017

Original post by cloisters

Hi, I've read the section on least upper bounds now and what I'm getting from Burkill's proof of the (existence of) least upper bounds theorem, is that the theorem is itself a specific case of Dedekind's axiom: the case of the dividing number being at the upper extreme of a bounded-above set.

Also, please correct me if I'm wrong, but how is DJMayes's proof a construction of root 2? I can see that it proves the existence of the number, but I don't see how showing that the positive number satisfying X^2=2 exists and is unique is the same as constructing it.

What's the issue? I feel you may have attached some extra significance to the word "construct". A construction is simply something that gives you what youre looking fot explicitly - in this case I constructed the square root of 2 because I have explicitly told you what it is - the Least upper bound of a given set.

If you want an example of a non-constructive proof that it exists, it would assume the square root of 2 doesn't exist and use that to derive a contradiction, most likely to the Fundamental Theorem of Algebra using the polynomial x^2-2. Such an argument would tell you that it exists but gives no hint about how to build it at all.

Reply 6018