Okay . Thank you !

Oh, thanks for your clarification.
But, again, does this proof is constructive since √2^√3, and that the results is irrational?

I don't quite understand this bit. Mind rephrasing the question?

I'm sorry, my bad. What I want to point out was that, do we only need to show just one example/proof to imply that the statement is true or false?
Like in the case of a^b, and that both a and b is irrational(suppose that both a and b is √2). And that if we multiply (a^b)^√2, we will get 2, which is rational.
But on the other hand, if a=√2, and that b=√3, the results will surely be irrational. Now, doesn't both examples contradict each other? So, how does one tell or pick one of these examples that will proof that the statement is true?

This post has been edited by RED-HAIR-SHANKS: Dec 13 2013, 11:22 PM

I'm back! Wow, what a fruitful discussion...

In logic, accordingly, the statement “There exist irrational numbers a and b such that a^b is rational” means “At least one, but not all cases, where two irrational numbers a and b will produce a rational value of a^b.”

This post has been edited by Critical_Fallacy: Dec 13 2013, 11:39 PM

Ok, now I got it, forgive me for my total ignorance. I would like to thank v1n0d too for helping me to delve into the question.

Appreciate v1n0d's efforts in guiding both of you.

The whole idea is to develop your HOTS, even without the knowledge of Gelfond-Schneider Theorem.

The simplest irrational number we know is √2. So, in the simplest form, we have .

However, we do not know whether is rational or irrational. Thus, we have two cases here.

CASE 1: If is rational, then it is self-evident that the theorem is correct because both a and b are irrational from the beginning.

CASE 2: If is irrational, then we can reassign and raise it to the power of , so that , that is obviously rational, which in turn proves the theorem is correct.

CONCLUSION: Therefore, whichever the case is, we can deduce that the theorem is readily in a position to be validated, even when a ≠ b, or having two distinct irrational numbers, e.g. .

P.S.: Gelfond-Schneider Theorem only tells you that is irrational.

This post has been edited by Critical_Fallacy: Dec 14 2013, 01:16 AM

The most compact equation in all of mathematics is surely Euler’s Identity, or .

In this equation, the five fundamental constants coming from four major branches of classical mathematics – arithmetic (0, 1), algebra (i), geometry (π) , and analysis (e) , – are connected by the three most important mathematics operations – addition, multiplication, and exponentiation – into two non-vanishing terms.

Some of my Tutorial followers (in Complex numbers) are probably aware that Euler’s Identity is but one of the consequences of the miraculous Euler formula , because when , , , it follows that .

This post has been edited by Critical_Fallacy: Dec 14 2013, 01:39 AM

Sorry, I slept off last night.
I see that Critical_Fallacy has already answered your question.

Hi! I have a question under partial derivatives, hope you can provide some clue.



I have no problem with part (a), but I face great difficulty with part (b) despite the hint provided by the suggested answer. I have no idea how to find .

Can you shed some light on part (b)? Thank you.

I'm too lazy to solve it till the end but here's a kickstart for you:


Use product rule to solve for ∂/∂x(∂z/∂r) ..

Product rule:



P.s: Just realised I made a silly mistake there.. notice that ∂/∂x(∂z/∂y) = ∂^2z/∂x∂y
Sorreh.

This post has been edited by VengenZ: Dec 15 2013, 09:51 PM

Posting after a long hiatus:


p/s: How's everyone doing?

Done with SPM . Starting on more 'advanced' concepts for now to get used to them before PreU . How are the lecturers at USM by the way ? How 'rigorous' is the curriculum for Mech Eng ?

USM has brought in plenty of good lecturers due to their apex status. If you're gunning for Mech Eng, I'll suggest UTM as that's their premier course. Most of the other public unis in Malaysia have adopted their syllabus and local reference material was written by lecturers there. They also have strong ties with the industry. If I remember correctly, they signed an MoU so that their Pure Mech degree is recognized by MIT.

This post has been edited by v1n0d: Dec 15 2013, 09:33 PM

I'm not into Mechanical Engineering , I'm more geared towards Physics at the moment . This might change , that's why I'm keeping my options open , planning to do A Levels first ( which means IPTA would be out of the picture for me ) , and I've to really dig deep for scholarships after A Levels .

When it comes to Physics, my heart is with UM. I spent a couple of days there in the IPO selection camp, it still ranks as one of my top 3 academic experiences.

I have a secret admirer.



Jokes aside, that is good to know. How were the papers this year? Any "special" things?

The lecturers are okay, I guess. But then I have never walked into lectures by lecturers at other universities, unless you count MIT OCW in. The curriculum, by my estimate, would be slightly above the average IPTA standards. There are some courses that are more intense than others, and then there are those that nobody cares about. And let me tell you one fact about engineering: we are very software intensive. So far I've been exposed to 7 different softwares, and there are more to come.

By the way, USM accepts A-levels, AUSMAT, SAM and such, other than the usual ones, so IPTA is not entirely out of your reach. (This bit of info may be outdated, so don't quote me on this.)

Do you know how I can be involved with the camp ? I've heard of the IMO one but information about IPO is scarce .

How do you think I should prepare for it ? IPO involves a lot of college level Physics , not sure if I can manage to self-study everything .

I remember you asking about USM , so yeah , that's how I got to know which course and where you're studying .

Yes . A lot . If you've followed the SPM Thread , you will see a lot of rants . The biggest one is Moral , where LPM had to come up with an official reply for many angry parents/students . The sudden change in format , where 'nilai' is no longer needed in essay questions , and 80% comprises of KBKK questions . Sejarah was like this too .

I see . I'm still a little bit undecided . I mean I still have a passion for Physics , but interests change , so I'm trying to be exposed to as many things as possible . I'm reading up on Logic , does engineering involve stuff like Mathematical proofs , number theory , etc or does it require only calculus ?

Honestly, I have no idea. My F6 physics teacher picked my buddy and I to take the statewide exam and we scored. Next thing you know, we were part of 40-student pool for national team selections.

Most of the material covered in the selection camp was college level, so we were pretty lost. I do remember an experiment were asked to approximate the value of g without using a stopwatch.