HSC 2015 MX2 Marathon (archive) (2 Viewers)

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glittergal96

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Re: HSC 2015 4U Marathon

We will prove existence and uniqueness separately.

Existence we do inductively.

It is trivial for n=1.

Now suppose that there exist positive integers such that

Then



The coefficients clearly being positive integers.

Uniqueness follows from the fact that we cannot write the same real number in more than one way in this form.

For if we have and (if b=d then we can subtract the surdic terms and also get a=b), then , which contradicts the well-known fact that is irrational. (I can prove this if you like, but I think most people have seen it before.)

Proving that is done inductively, as



An interesting extension question (more suitable for the advanced thread) is to show that:

a) There are always positive integer solutions to the equation if D is a nonsquare positive integer. (Hint: use the pigeonhole principle and consider the problem of approximating an irrational number well by a rational one.)

b) The solutions are precisely the pairs where () and is the solution pair with smallest (Hint: again consider multiplying numbers of the form together.)

c) Find a closed form (non-recursive) expression for the solutions.


(If a is too hard, then just try b and c, as they are a bit easier I reckon.)
 
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glittergal96

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Re: HSC 2015 4U Marathon

Also, an application for the above extension problem in geometry:

Find all triangles with sides lengths that are consecutive positive integers that also have their area as an integer. You may find it helpful to first find an expression for a triangles area purely in terms of side lengths.
 

porcupinetree

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Re: HSC 2015 4U Marathon

So why can you differentiate/integrate over the complex field?
Good question, but not one I know the answer to. What I would say, in this situation, is that i is a constant and doesn't affect rates of change (why should it?). Which is not a very good justification. Again, I don't really know
 

seanieg89

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Re: HSC 2015 4U Marathon

You would have to include a definition of the exponential function applied to complex numbers or what it means to raise things to complex powers, otherwise the LHS is a meaningless string of characters.

Eg/ Try a logically similar question. Prove that Sean(x)=cot(7x).
 

seanieg89

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Re: HSC 2015 4U Marathon

For this to make sense you would need to:

a) Define what you mean by "log(z)", and "e^z". The former is subtle as it needs what is called a branch cut.
b) Properly define differentiation and integration of functions from the real line to the complex plane and from the complex plane to itself, and get some analogue of the fundamental theorem of calculus. (Think about what your x integral even means, as x is a complex number!)
c) Prove that the primitive of 1/z is log(z) for your complex definition of log(z).

But as I stated in my previous post, the question itself is nonsensical without some definition of the complex exponential in the first place.
 

porcupinetree

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Re: HSC 2015 4U Marathon

Lets get some inequalities up in here:

 
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