something new on the "pi and e" debate... (1 Viewer)

[who_loves_maths]

sorry for making this thread on "pi and e" again, but i had no choice on account of the last one being closed.

Anyhow, just found out something interesting.

Originally Posted by buchanan
Perhaps all that is now required is to in fact prove (ln(π +1) -2)/(1 - ln(π)) is irrational.

Originally Posted by no_arg
all you've done is rewrite the question!

Clearly [ln(pi +1) -2]/(1 - ln(pi))=4

It is perfectly possible for the ln of an irrational to be rational!...

Originally Posted by no_arg
Does it really matter whether or not I actually believe that [ln(pi +1) -2]/(1 - ln(pi))=4?

Just went to look for some info on 'pi' on Wolfram, and was surprised at a line on the website:

Wolfram - Pi

specifically, it was this line:

It is not known if (pi+e), pi/e, or ln(pi) are irrational.

this might mean that the problem of showing pi^4 + pi^5 = e^6 is an approximation isn't as simple to prove (ie. without calculator) as showing that [ln(pi +1) -2]/(1 - ln(pi)) is irrational...

the fact that 'ln(pi)' is not known in the maths community to be irrational or rational yet can mean that proving either 1) [ln(pi +1) -2]/(1 - ln(pi)), or 2) [6 - ln(pi +1)]/ln(pi) to be irrational is not yet even possible (certainly beyond the reach of HSCers on this forum i think).

so buchanan, i guess you can stop trying that 'dead-end' now


the only way to use [ln(pi +1) -2]/(1 - ln(pi)), or [6 - ln(pi +1)]/ln(pi) to prove the case against no_arg is then to show that they are, as i said before, non-integral - which will suffice as a proof for this particular problem of approximation.

alternatively, if the "equality" of [ln(pi +1) -2]/(1 - ln(pi)) = [6 - ln(pi +1)]/ln(pi) can be proven false without using a calculator, then it also follows that the original "equality" of pi^4 + pi^5 = e^6 is a mere approximation...

 

[brett86]

Pi^4+Pi^5=e^6 is an exact formula

lol, i find it amusing that right after a number of people proved ur argument was wrong u decided to attacking me

ill quote what i put in the "Oops" thread:

lol, im not discouraging people from doing maths, what i am trying to do is get u to admit that pi^4 + pi^5 is not equal to e^6

im angry cause u claimed pi^4 + pi^5 = e^6 which is obviously wrong and hsc-er might believe it

also, u keep saying that buchanans argument is invalid when its perfectly correct

and btw, i encourage everyone whos interested to post whatever creative ways they come up with to disprove no_args ridiculous claims

 

[brett86]

also, to stop everyone from repeating themselves over and over again i have make this point

everyone has accepted that pi^4 + pi^5 is not equal to e^6 as this has been shown many times in the previous 2 threads

further discussion should only be on creating new ways of making no_arg look like an idiot by disproving his stupid claims

 

[brett86]

Pi^4+Pi^5=e^6 is an exact formula

π < 3.14159266 & 2.718281828 < e.

So

π^4+π^5

< 3.14159266^4+3.14159266^5

= 403.4287797363725532846657585016733756588576 (exactly)

< 403.42879308396500147612667658990386685186886530139770 4704

= 2.718281828^6 (exactly)

< e^6

this was on:

http://mathworld.wolfram.com/eApproximations.html

a site from Wolfram Research

do u really expect people to believe u over them

hence, no_arg is wrong

comment about no_arg:

he is arguing for the sake of it, (like some prick at my school), the answer is obvious, of course he expects us to take his word over ANY number of experts

 

[KFunk]

Aren't you a naughty boy!
Still trying to think after two attempts at censoring this issue!
Whatever you have to say you better say it quick before stormtrooper brett86
arrives to make sure we are all of the same mind!
Quick KcLake you better close this one down as well!
We can't have people exercising independent thought.
Oh no we can't let that happen!

Fight the power dude.

 

[who_loves_maths]

^ lol, actually, it's very coincidental that you should mention "continued fractions" there no_arg, i was just about to post something about it myself, but you beat me to it .

anyways, i was reading through some info on 'pi' and 'e', and a very interesting fact that came up was on the representation of transcendental numbers by continued fractions:

Continued Fractions

apparently, 'e' is a number that is an infinite periodic continued fraction, which means that its continued fraction representation has a [quote] "recognisable pattern", and this in turn means that:

Knowing this regularity permits us to compute e to any degree of precision...

i'm not sure myself, but being able to compute 'e' to "any degree of precision" using fractions (ie. rational numbers) reminds me of something to do with 'Liouville Numbers' ?
i forgot exactly what Liouville Numbers are, but i think they are transcendental numbers that can be approximated extremely accurately with rational numbers? in which case, that would most likely make 'e' a Louiville Number...

now also on that site is the fact that:

Unlike phi and e, the transcendental number pi cannot be expressed in a general formula for bn...

- meaning that one cannot approximate 'pi' with infinite continued fractions to any large degree of precision.
and i have an inkling that this has something to do with the fact that [quote from Wolfram]:

It is also known that pi is not a Liouville number.

so IF it is true that 'e' is a Liouville number, combined with the fact that 'pi' is not a Liouville number, then maybe that's the discrepancy that can be used to prove that (pi^4 + pi^5) = e^6 is a mere approximation.
since on the LHS, you have a non-Liouville number raised to integral powers, and on the RHS, you have a Liouville number also raised to integral power...
maybe the fact that 'pi cannot be represented by infinite regular, and rational, continued fractions, whilst 'e' can, means that the equality of LHS = RHS in this case is simply not possible, hence it must be false ...

 

[who_loves_maths]

^ really? oh well, there goes my attempted 'method', lol

but buchanan, can you plz show me why 'e' is not a Liouville number? or point me to where it says that... thnx


ok, now if what you say is true, then the Liouville approach is useless, but the continued fraction approach, as i outlined in my last post, is still very much valid...

the fact that 'pi' can't be represented by regular infinite continued fractions, but that 'e' can {which is a know fact, unlike the L no.s}, then that can still be very well used to show that LHS = RHS is false...

 

[brett86]

im glad no_arg is gone, even though the calculator proofs werent exactly creative, they were still valid

no_arg didnt accept them because he claimed rounding off caused the answers to be inaccurate, so then people calculated values exactly and he still didnt accept them

he continued to claim that pi^4 + pi^5 = e^6 after it was disproved over and over again and when it was clear to everyone his argument was wrong he started to attack me

if a hsc-er believed that pi^4 + pi^5 = e^6 was right and used it in an exam they would be severely penalised, no_arg didnt seem to care about this and continued to say pi^4 + pi^5 = e^6 without any justification

im glad and im sure many others are too, that no_arg is gone. hopefully his disappearance will be permanent

 

[haboozin]

if a hsc-er believed that pi^4 + pi^5 = e^6 was right and used it in an exam they would be severely penalised, no_arg didnt seem to care about this and continued to say pi^4 + pi^5 = e^6 without any justification

well no HSC question is gonna ask PROVE e^6 = pi^4 + pi^5 (only one question could related to this and that was 2001 Q8 biii i think and it was a lead up question so you would be stupid to state something like this anyway)

if a hsc question has a question involving e and u used the above equality the numbers are so close (since they say to 2dp or wateva) that they wont even notice what you did..


haha severly penalized.

 

[lucifel]

i got quoted! GO me!! Yay!. I have nothing to add to this discussion, it's pointless, and the thready should be closed and buried, this topic never given rise again.

 

[brett86]

if a hsc question has a question involving e and u used the above equality the numbers are so close (since they say to 2dp or wateva) that they wont even notice what you did..

in maths they usually require students to put things into "exact" form

and anyway, its very wrong to say ur sure somethings correct when it isnt on a site thats meant to assist those being tested on the topics discussed

lucifel, ill quote u again:

it's pointless, and the thready should be closed and buried, this topic never given rise again

i completely agree, i dont see how anyone can add anything new to this discussion, i too await the closure of this thread

 

[brett86]

just realised, ive left myself open to more attack by no_arg, i bet he'll put something like this up again lol:

Whatever you have to say you better say it quick before stormtrooper brett86 arrives to make sure we are all of the same mind!
Quick McLake you better close this one down as well!

hahahahahahaha

 

[lucifel]

cos you know throwing pointless insults like that over the net is cool.

 

[Trefoil]

Of all the posts in this series of threads, no_arg's are the most useful.

Lies My Calculator and Computer Told Me

 

[acmilan]

Of all the posts in this series of threads, no_arg's are the most useful.

Lies My Calculator and Computer Told Me

I agree. Tomorrow or Friday I will try and go to the usyd School of Mathematics library and see if I can find any references on this approximation. My lecturer recommended a reference:

thanks for your message and interest

perhaps try

Pi, a source book / [edited by] Lennart Berggren, Jonathan Borwein,
Peter Borwein. maths library, level 8 carslaw, call number: 513.1 13

there are people who have devoted their life to trying to
understand pi and its relationships to other numbers!!

at some stage i will show you some ways to approximate pi
which use series

best wishes, david

 

[yrtherenonames]

Sounds fair enough to me

 

[turtle_2468]

From the last 2 years I've spent moderating, if I were a betting man, I'd probably say that he'll be back. In approximately 3 months... then the cycle will begin anew.

 

[Templar]

The point is not to reply against him and just let it die. No amount of logic would ever convince him.

 

[brett86]

From what I've read he did still have a valid point though about how trusting people are of machines. His continued comments were intended to encourage people to consider other methods of proof, rather than simply proving one can push buttons

lol, no_arg was never trying to encourage people to consider other methods of proof, the only reason he didnt accept buchanans proof was because it proved he was wrong

i mean, everyone makes mistakes and theres nothing wrong with making mistakes but when ur shown to be incorrect u should admit to it and not present misleading information to try and cover it up

when calculators and computers perform big calculations there can be errors due to rounding off during the calculation, but in maths software such as mathematica, it rounds off at the end and if dont want it to round off u can tell it this and it gives an exact answer

e.g.

entering 3.14159266^4 in mathematica gives 97.4091

however getting rid of the decimal in the input and asking it to give the answer to 34 sig figs (32dp) we can get an exact answer

N[(314159266*10^(-8))^4, 34] = 97.40909182902901737436094415467536

if u check this on paper ull find that its exactly correct (ive checked the values presented of 3.14159266^4, 3.14159266^5, 2.71828182^6 on paper)

so i dont see a problem with buchanans answer, all his calculations were exactly correct

You really are keen to stop people discussing mathematical concepts aren't you!

Whatever you have to say you better say it quick before stormtrooper brett86arrives to make sure we are all of the same mind!
Quick McLake you better close this one down as well!
We can't have people exercising independent thought.
Oh no we can't let that happen!

no_args claims that i was trying to stop people from learning were ridiculous and were only said because i backed up buchanans argument that showed no_arg was wrong

i hope people who read no_args post were able to see the real reason he made those comments because in many of my past posts ive encouraged people to think creatively when solving problems in maths

Of all the posts in this series of threads, no_arg's are the most useful

Idyll and Trefoil, if u want to thank people for contributing to this forum, u should thank Slide Rule, FinalFantasy, who_loves_maths, KFunk, McLake, acmilan, turtle_2468 and others like them, not thank idiots like no_arg

 

[no_arg]

I notice that you have not closed up your quotes around the numbers in the Gardner ref above!
What does the reference actually say?
It is of course true that the numbers are equal to 4 decimal places!