stupid proofs ----> e (1 Viewer)

[dark_angel]

How do u prove the irrationality of "e" ?


Ah i think this question came out in some test and i have the solution for this but its like uni stuff i cant even understand the first line lol.

can anyone provide a proof which i could use in exams

(btw how do u prove irrationallity of pie, is there a method for all proofs)

 

[Euler]

It all depends on what you know, and what you need it for.

For exams, they would not ask you to simply prove the irrationality of e. They would guide you through it. This has been done before in previous 4-unit HSC exams, usually in question 7 or 8.

There is no 'standard way' of proving the irrationality of a number. It very much depends on the properties and definition of the number.

I think that the irrationality of pi is harder to prove than the irrationality of e, though I could be wrong.

but for numbers like sqrt(2), you start by assuming sqrt(2)=a/b where a and b are integers that share no common factors...

 

[Xayma]

It was 1993 that had the irrationality of e I think, try searching for answers to that paper.

 

[Calculon]

go to buchanan's site fourunitmaths.cjb.net

 

[dark_angel]

yeah the proof on buchanans site is the one that is phscyco that i couldnt understand

do u have any different methods like the one Euler mentioned?

 

[Euler]

dark_angel, it will do you no harm if you work through the question from the 1993 paper. Then buchanan's pdf file might make more sense.

just by looking at buchanan's pdf file, it's hard to appreciate the simplicity and the ingenuity of the approach. It doesn't explain why one would consider the sum s_n in the first place.

the reason for considering the sum s_n can be found by looking at the Taylor series of e^x at the point 0, and then substituting x=1. The sum s_n is then the truncated (after n terms) evaluated taylor series.

hope this helps.

 

[turtle_2468]

And, just in case anyone asks, you DON'T need to know taylor series for HSC
Although, it MIGHT be given to you... so freak out ppls! (really. Don't worry about it)

 

[Euler]

oh crap! have they taken it out? or was it never in the syllabus, and me just getting delusional again? I thought I remembered doing Taylor series in class....

didn't mean to scare you...

anyway, then...since Taylors series is an alien concept...

In short, e can be written down as an infinite sum, namely,

\sum_{r=0}^\infty 1/(r!)

or, 1 + 1/1! + 1/2! + 1/3! + 1/4! + ...

And the sum s_n is simply just cutting off the infinite sum and stopping at 1/n!, so all you really are doing is approximating e with s_n. The bigger n is, the more closer to e you are.

 

[CM_Tutor]

If Taylor Series have been in the syllabus, it hasn't been for more than 20 years.

 

[KeypadSDM]

What's a Taylor?

 

[Euler]

Originally posted by CM_Tutor
If Taylor Series have been in the syllabus, it hasn't been for more than 20 years.

Damn. I'm being delusional again!

Keypad: Taylor was a guy. Taylor series are approximations to a function using polynomials. The coefficient of $x^n$ is given by the nth derivative evaluated at 0, divided by n!...technically, what i just described was a Maclaurin series, which is a specialised Taylor series. More to come in MATH1901/MATH1001

 

[Grey Council]

keypad, must you? seriously

:-\

 

[dark_angel]

do u guys reckon this question was just a one-off thing that u had to know to get the marks or is it in the syllabus relating to some distant notion. are there any other questions like this that i need to be aware of?

 

[dark_angel]

thanks drbuchanan that helped alot


Wait hold on is their any way i could get my hands on the arnold/arnold/cambridge solutions?!?!?

i know there have beeen posts on it but has anyone like downloaded them before drbuchanan had to take it off his site i.e do the still EXIST?!?!

THANKYOU

 

[Calculon]

Originally posted by dark_angel
thanks drbuchanan that helped alot


Wait hold on is their any way i could get my hands on the arnold/arnold/cambridge solutions?!?!?

i know there have beeen posts on it but has anyone like downloaded them before drbuchanan had to take it off his site i.e do the still EXIST?!?!

THANKYOU

This has been asked too many times...

 

[Estel]

And never satisfactorily answered.

 

[Calculon]

Originally posted by Estel
And never satisfactorily answered.

Yes it has, but then the post was deleted *stares at drbuchanan*

 

[Constip8edSkunk]

sami el hosri's solutions were posted by drbuchanon for quite a while on the geocities websites... but since sami started selling his solutions, he prolly asked drbuchanon 2 take them off... so they are gone.

 

[Grey Council]

hrm, how do you know it was Sami's solutions? I think Sami would type them up, but the ones Buchanan posted up are hand written.

 

[Grey Council]

whats a transcendental number? an irrational number?