Other theorems w/ regard to markers (1 Viewer)

[Wacky]

In inequalities if you use AM-GM-HM inequalities - when do you need to prove them first (i.e. can you just quote them in a long question)

What about the rearrangement inequality, the Chauchy-Shwarz (can't spell it) inequality? The .. well all the ones from the Mathematical Toolchest (book) - if they can be used, I'll learn them then.

Also, in other topics stuff like pigeon hole principle, and a lot of the geometry ones from the Toolchest.

Also, is the "e is transcendal, therefore e is irrational" proof inadequate in any way (especially the working... why can't teachers solve linear simultaneous equations with integer coefficients in their head?)

 

[Fosweb]

Wacky: did you ever do any of the USYD/NSW AMOC programs? There was some Pigeon Hole Principle in them (quite a bit), so maybe you should look over that if you have it. They did ask PHP not long ago if i remember rightly (like - something in a q8).

I dont think they'd ever ask a simple php question like: show that there is a min distance of xxx between 2 points in this arrangement... (its not syllabus stuff anyway is it?)

As for the toolchest geometry: they probably 'could' ask a lot of stuff from that. Also: if you said that "This is true because Ptolemy's Theorem says so...", then is the marker going to know what you are talking about? (Is that the point of your question in the first place?)
So my guess is: they will ask you most likely to prove these theorys without actually worrying about names of them.

 

[maniacguy]

Be careful about it. One of our lecturers was telling us about a time 'many moons ago' (read: probably a couple of decades) when they included a moderately complicated geometric theorem as q8 and led up to it by asking students to prove a series of results on the way.

Unfortunately the brighter 4u students were aware of the end result and used it to prove the first couple of parts, which left them quite stuffed when they got to the end. Following that incident (where a number of the brighter 4u students got very low marks for q8!) matters have apparently been revised somewhat.

 

[maniacguy]

As for inequalities - the ones you can use are the basic ones. Simple AM-GM (in a long proof you might be able to quote the extended version), and then the basic (a-b)^2 >= 0, triangle inequality, etc.

Things you can't use without proof::
Harmonic Mean, Cauchy-Schwarz, Rearrangement Inequality, most others from the Toolchest.

(If you strike it lucky and get a lecturer from a university as your marker, though, you might get bonus marks. The chances of that are very very small compared to the chances of getting someone with no idea what you're talking about.)

You can't use a lot of the geometry results in the toolchest without prompting. You'll have covered the ones you can use in class.

Fosweb, the AMOC isn't affiliated with any particular university - rather, it's part of the Australian Mathematics Trust, and academics from a lot of universities help out with the various activities going around.

Incidentally, I haven't seen any sign of the proof drbuchanan is referring to - is it in one of these posts by any chance? (Oh, and on transcendental numbers - you'd probably have to define what they were in order to use them anywhere in a proof. That could take a while in a 4u exam!)

 

[Rahul]

where can i get hold of Mathematical Toolchest? i found it at the mansw site, what if i am looking for it urgently...

 

[maniacguy]

You could try the Australian Mathematics Trust, I suppose (if you want to buy it)
http://www.amt.canberra.edu.au

Otherwise check your library - it may have a copy. I have no idea where else it might be.

You don't really need it as such yet - if you decide you'd like to do maths at university, though, it could come in handy...

 

[Rahul]

oh ok, thanks, i may check the library actually...

it just seemed like some bits of it may help.

can i borrow someones?

 

[turtle_2468]

there's an extremely high chance you don't need it for 4 unit. Roger what maniacguy said though, if you want to do maths at uni... (but then that doesn't explain why you need it urgently). If you want a short list of the relevant results however I'm sure people (including myself) with the book will be more than happy to give you those... most of those are in normal 4U anyway.
BTW, cauchy schwarz won't be needed. I'm guessing most markers don't know it anyway.. ditto rearrangement. That is because the proof of C-S (ha ha) is quite complicated, at very worst they'd walk you through part of it...

pigeon-hole, on the other hand, makes quite a lot of sense. So you could be asked to prove it, or use an easy application of it in a question...

 

[Rahul]

oh ok thanks. i was after the "short list of the relevant results". but since it isnt that important then i might put the ~$30 back in my pocket.

although i would like a "short list", maybe i can make a thread for this, as many people would like to know.

 

[turtle_2468]

By the way in response to maniacguy's post: They really should have read the whole q before they did it, weren't smart enough obviously Just joking!

Okay. Things you might be expected to know. DOn't sue me if I get this wrong, but they would be crazy to expect ppl to know any more than this: (this is stuff not in 4U already, eg AM-GM and alternate segt)
Hmm. I can only think of pigeonhole. And only in a very generalised context. Basically says that if you have n holes, and at least kn+1 objects (k, n positive integers of course) then there is some hole with at least k+1 objects in it. Proof by contradiction, I'll leave as an exercise.

 

[Fosweb]

Well...
With n holes, and kn + 1 objects to go in these holes:
If NO holes have k+1 objects (ie the most number of objects in any hole was k), then you couldnt have anymore than kn objects, which is the contradiction. So then at least one hole muct have at least k+1 objects in it.

Its used in set theory isnt it? And questions like:
Choose any 5 positive integers. Prove that from these 5 integers, there will always be at least one set of 3 integers whose sum is divisible by 3.