Maths Discussion Thread (1 Viewer)

[Sy123]

A thread to discuss maths in general of topics of all ranges (can be school based as well)

Ask questions and discuss certain things.

 

[Lord Iron-Balls]

Excellent thread

 

[Carrotsticks]

Question for discussion:

We know that we can take non-integer powers, as counter intuitive as it may seem initially.

What does it mean to take a non-integer factorial?

Hint: I touched on it briefly in my 4U trial paper.

 

[barbernator]

Question for discussion:

We know that we can take non-integer powers, as counter intuitive as it may seem initially.

What does it mean to take a non-integer factorial?

Hint: I touched on it briefly in my 4U trial paper.

is there a function representation of the factorials? because if you define the function in terms of x you could just read off the function/graph?

 

[deswa1]

Question for discussion:

We know that we can take non-integer powers, as counter intuitive as it may seem initially.

What does it mean to take a non-integer factorial?

Hint: I touched on it briefly in my 4U trial paper.

Oh I've seen this before. I think it can be defined by a function gamma(z) (gamma being the capital letter)- I'll look more into it. That's why if you go to wolfram and plot y=x!, its a smootly defined curve. And I think you can even define it for complex numbers but that just flies WAY over my head haha

 

[barbernator]

Oh I've seen this before. I think it can be defined by a function gamma(z) (gamma being the capital letter)- I'll look more into it. That's why if you go to wolfram and plot y=x!, its a smootly defined curve. And I think you can even define it for complex numbers but that just flies WAY over my head haha

ahh so its the gamma function, hectic!

 

[Carrotsticks]

is there a function representation of the factorials? because if you define the function in terms of x you could just read off the function/graph?

Yep there is, but as far as you know it would be a piece-wise graph... not a smooth continuous function.

Oh I've seen this before. I think it can be defined by a function gamma(z) (gamma being the capital letter)- I'll look more into it. That's why if you go to wolfram and plot y=x!, its a smootly defined curve. And I think you can even define it for complex numbers but that just flies WAY over my head haha

Indeed has something to do with the gamma function, which is also defined for complex values =)

 

[barbernator]

Yep there is, but as far as you know it would be a piece-wise graph... not a smooth continuous function.

I'd just join all the dots with a nice concave up curve. hey presto!

 

[seanieg89]

lols. Another topic that I got reminded of by a pm: Some people will have seen a 'proof' of Euler's formula before that claims to be within the mx2 syllabus. Here is why I think it is wrong.

Originally Posted by bleakarcher
Hey seanie, just a question. Would you consider this a satisfactory proof of Euler's formul? I remember reading somewhere it's like a poor man's proof of it lol.

Let f(x)=cos(x)+isin(x)
Consider f'(x)/f(x)=[-sin(x)+icos(x)]/[cos(x)+isin(x)]=i
Integrating both sides, log(e)[f(x)]=ix+C
When x=0, f(x) => C=0 => f(x)=cos(x)+isin(x)=e^(ix)


K, going to be really careful and really critical here so you see the sort of issues that can happen when you 'prove' things without care.

First line is fine, we have defined a function from the reals to the complex numbers, this isn't really done in mx2 but it is perfectly legit.

The second is fine, it uses the fact that the derivative of functions from R to C behave like the regular derivative, in that (f+g)'=f'+g' and (cf)'=cf' for functions f,g constant c. This should be proven by first principles before taking it as fact, but the proof is pretty much identical to the real valued analogue.

The third line is my first and main issue. Firstly, you are integrating a complex function. This, like differentiation is legit...it turns out we can integrate the real and imaginary parts separately and add them, although this requires proof.
More importantly, you are integrating the LHS using a pattern that is valid for real functions, but how do we know this is the case for complex functions? In fact this line does not even make sense, because we have not defined what log(z) is for complex z! (And f(x) is certainly complex for all real x). In this line we are basically crossing our fingers and hoping that there is a function called log defined on the complex plane (except possibly at 0) which has derivative 1/z everywhere, then we are pretending this function exists!

The next step involves raising e to the power of both sides. But each side is complex? What does it mean to raise a number to a complex power?

We are also crossing our fingers and pretending that our magical function log on the complex plane from before cancels out when e is raised to the power of it (in the sense of our brand new magical way of raising numbers to complex powers.) Because we are using the same letters for e and log as we did for our familiar e and log from the real line to the real line, they must obey all the same properties right?

On the final line we have arrived at the conclusion...but wait a minute, what does the RHS even mean? We have gone from not having complex powers defined, to knowing what a certain number to a complex power is?! That's the most immediate way to tell that this argument is balls, it doesn't 'prove' anything.

Hope this helps . Feel free to ask for clarification on anything.

 

[Nooblet94]

lols. Another topic that I got reminded of by a pm: Some people will have seen a 'proof' of Euler's formula before that claims to be within the mx2 syllabus. Here is why I think it is wrong.

Originally Posted by bleakarcher
Hey seanie, just a question. Would you consider this a satisfactory proof of Euler's formul? I remember reading somewhere it's like a poor man's proof of it lol.

Let f(x)=cos(x)+isin(x)
Consider f'(x)/f(x)=[-sin(x)+icos(x)]/[cos(x)+isin(x)]=i
Integrating both sides, log(e)[f(x)]=ix+C
When x=0, f(x) => C=0 => f(x)=cos(x)+isin(x)=e^(ix)


K, going to be really careful and really critical here so you see the sort of issues that can happen when you 'prove' things without care.

First line is fine, we have defined a function from the reals to the complex numbers, this isn't really done in mx2 but it is perfectly legit.

The second is fine, it uses the fact that the derivative of functions from R to C behave like the regular derivative, in that (f+g)'=f'+g' and (cf)'=cf' for functions f,g constant c. This should be proven by first principles before taking it as fact, but the proof is pretty much identical to the real valued analogue.

The third line is my first and main issue. Firstly, you are integrating a complex function. This, like differentiation is legit...it turns out we can integrate the real and imaginary parts separately and add them, although this requires proof.
More importantly, you are integrating the LHS using a pattern that is valid for real functions, but how do we know this is the case for complex functions? In fact this line does not even make sense, because we have not defined what log(z) is for complex z! (And f(x) is certainly complex for all real x). In this line we are basically crossing our fingers and hoping that there is a function called log defined on the complex plane (except possibly at 1) which has derivative 1/z everywhere, then we are pretending this function exists!

The next step involves raising e to the power of both sides. But each side is complex? What does it mean to raise a number to a complex power?

We are also crossing our fingers and pretending that our magical function log on the complex plane from before cancels out when e is raised to the power of it (in the sense of our brand new magical way of raising numbers to complex powers.) Because we are using the same letters for e and log as we did for our familiar e and log from the real line to the real line, they must obey all the same properties right?

On the final line we have arrived at the conclusion...but wait a minute, what does the RHS even mean? We have gone from not having complex powers defined, to knowing what a certain number to a complex power is?! That's the most immediate way to tell that this argument is balls, it doesn't 'prove' anything.

Hope this helps . Feel free to ask for clarification on anything.

So is there a proof that's within range of the mx2 course? The proof I've seen used Taylor series, but I've only got a vague understanding of how they work.

 

[seanieg89]

Not that I have seen, the theorems statement contains an undefined object...any 'proof' of it must involve some definitions so the meaningless collection of symbols e^{i*pi} is given some meaning. And taylor series are the standard way of defining what e^{something} is. (As well as what sin{something} and cos{something} are.)

 

[Sy123]

What does it mean to take a number to an imaginary/complex power?

like e^ix for instance, is it possible to give it an interpretation like we give to powering by real number systems?

 

[seanieg89]

What does it mean to take a number to an imaginary/complex power?

like e^ix for instance, is it possible to give it an interpretation like we give to powering by real number systems?

Yes although we lose uniqueness, things like i^i have infinitely many values....think of it as a much worse version of the problem we get when considering square roots of positive numbers, there are two! I explained how this works in the p00n thread, within the last two pages or so.

PS. Also this is a nice way of defining real^real. In fact we NEED complex numbers to make sense of things like (-1)^pi.

 

[Sy123]

Yes although we lose uniqueness, things like i^i have infinitely many values....think of it as a much worse version of the problem we get when considering square roots of positive numbers, there are two! I explained how this works in the p00n thread, within the last two pages or so.

PS. Also this is a nice way of defining real^real. In fact we NEED complex numbers to make sense of things like (-1)^pi.

Alright thanks.

I watched this video today about the harmonic series paradox, about how the series:



and



And the paradox where if we geometrically represent the converging series by an infinite series of squares each with 1/i length and width.
Yet when we put the squares all together in the same row, the bottom part is of the harmonic series (the diverging one), yet the squares areas converge.

How does this work? Is it about what infinity means?

 

[bleakarcher]

Btw guys that 'proof' of Euler formula I actually found once in an MX2 trial paper haha, they need to start introducing more rigour into the course..

 

[seanieg89]

Alright thanks.

I watched this video today about the harmonic series paradox, about how the series:



and



And the paradox where if we geometrically represent the converging series by an infinite series of squares each with 1/i length and width.
Yet when we put the squares all together in the same row, the bottom part is of the harmonic series (the diverging one), yet the squares areas converge.

How does this work? Is it about what infinity means?

Its an example of why proving things with pictures can be problematic. There are lots of things like this, eg Gabriel's horn has infinite surface area but finite volume.

 

[Sy123]

Its an example of why proving things with pictures can be problematic. There are lots of things like this, eg Gabriel's horn has infinite surface area but finite volume.

But we still have to consider that, geometrically if we have this shape of infinite length in the base, and a finite area, what can we make of it?

A broader question though is, why do we arrive at these paradoxes because of pictures?

Is it because algebra representation is not as axiomatic as geometry or the other way around?

 

[Trebla]

Hate to be a spoiler but the whole point of the extra-curricular topics forum is so you can discuss topics of interest. Right now this thread seems to have the intention of putting what the forum is for into a single thread. This is not advisable. Please discuss separate topics in separate threads.

 

[Sy123]

Well I was thinking the rest of the forum could be used for posting math problems, discussing math events and specific proofs and stuff like that. I intended this thread to be more general mathematical theory, random tidbits from everything in discussion.

But I guess if you wish then it cant continue (please consider having a thread like this though)

 

[lolcakes52]

I believe this is relevant to your the harmonic question.