Talk:E

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An interesting note that the nth derivative of xn is n! (quite obvious when you think about it too), so that may be a better conceptual meaning for the denominators in the talor-series expansion of ex than thinking in combinatorial terms - ie what does it mean conceptually to divide x^n by it's own nth derivative? --Nad

Yes, I think your right, it is actually the derivative property that is important and that is why the terms on the denominator are factorials.
f'(x) = f(x) = \int_-\inf^\inf f(x) dx
So each term in the polynomial taylor (Maclaurin series) is scaled by its powers factorial to maintain the derivative property. That is at any point y along y=exp(x) its derivative is exp(x), and so is the area under the curve from -inf to that point x. --Sven
I still don't understand the meaning of that scale-factor, what does n! actually mean when used to scale xn down? --16:38, 12 July 2007 (NZST)

Gamma function properties. This section could be expanded, the gamma function, and distribution has many properties which link it to other statistical distributions etc.

  • Γ(1)=1
  • Γ(1/2) = sqrt(π)
  • Γ(k+1) = kΓ(k)
  • If k is a positive integer, Γ(k) = (k-1)!
  • Gamma(k=1, λ) gives expenential(λ) distribution.
  • iid exponential(λ) = Gamma(λ , k)
  • χv (chisq)= Gamma(k=v/2, λ=1/2)

Sorry didn't latexify the formulas --Sven 10:17, 26 Oct 2006 (NZDT)


From R.Knott's Golden Mean page

tan x 	= cos x and, since tan x = sin x / cos x, we have:
sin x 	=(cos x)2
	=1-(sin x)2 because (sin x)2+(cos x)2=1.
or	(sin x)2 + sin x = 1
and solving this as a quadratic in sin x, we find
sin x = (-1+sqrt5)/2 or
sin x = (-1-sqrt5)/2
It's the geometric way of creating the quadratic structure yielding phi...
The real component of 2eiπ/5 is φ because φ = 2cos(π/5) = 1.618...
And also, the imaginary component of 2eiπ/10 is Φ since Φ = 2sin(π/10) = 0.618... = φ-1 = 1/φ

Are you sure the notation is correct? The definition on wikipedia:Golden ratio and Wolfram is that lower case φ = 1.618 034. , and upper case Φ is the Golden ratio conjugate where Φ = 1/φ = φ -1 = 0.618 034. --Sven 11:58, 26 Aug 2006 (NZST)

Yeah, you're right - I was just going from that powers-of-phi page which has it wrong... --Nad 11:04, 26 Aug 2006 (NZST)
Bear in mind that I am biased, as I have been questioning notation on Wikipedia:Talk golden ratio :p --Sven 12:25, 26 Aug 2006 (NZST)
These guys say φ = 1.618... and Φ = 0.618...
These guys say φ = 0.618... and Φ = 1.618...

Interesting, the closure of the math tag is missing the / in the docie e^x = \sum_{n = 0}^{\infty} {x^n \over n!} = 1 + x + {x^2 \over 2!} + {x^3 \over 3!} + {x^4 \over 4!} + \cdots

Sven 09:48, 16 Nov 2005 (NZDT)

Is the relationship (7/5) * (pi/e) exact, or is it an approximation? --Sven 10:19, 22 Aug 2006 (NZST)

I had always thought it was exact, but....

+sandbox.pl

gives...

perl.php

Yes, thats what I got in R. Someone on 'pedia mentioned that the relationship was coincidence --Sven 14:41, 22 Aug 2006 (NZST)

ok, now this is interesting. pi on 32-bit computers is an approximation (22 decimal places in). E is a tailor series expansion, so I tried seeing which iteration of E was closest to Phi, and I found it was the 7th order tailor series expansion (8.84 X 107);

+GoldenRatio.R

print(phi-phi2)
[1] 1.569904260434463e-05
> print(pi)
[1] 3.141592653589793
> print(exp(1))
[1] 2.718281828459045
> print(phiCalc - phi)
[1]  5.8108086876296028e-01  1.4125789726038929e-01  3.1302154384746705e-02
[4]  5.9277521826752722e-03  9.4627444975325936e-04  1.1899787172575671e-04
[7]  8.8451503788000707e-07 -1.3878442173043126e-05 -1.5518754122645362e-05
[10] -1.5682785134529809e-05 -1.5697697043126624e-05 -1.5698939702435410e-05
[13] -1.5699035291527608e-05 -1.5699042119621254e-05 -1.5699042574590649e-05
[16] -1.5699042603234403e-05 -1.5699042604566671e-05 -1.5699042604566671e-05
[19] -1.5699042604566671e-05 -1.5699042604566671e-05

This is a bit off the wall, but does this relate in any way to the maximum how many electron orbits that are allowed in molecules?

Not sure what all that above is about - you'll need to explain it a bit more fully for the non-math's fellas like me. The outer orbital shell of an atom is a dynamic equilibrium around the positively charged nucleus. The shells are from the quantisation of the electron energy levels into discrete harmonics. --Nad 12:00, 23 Aug 2006 (NZST)
Also, if you could get the server to execute R on command-line to stdout, I can add in-article R execution to the wiki too like with C, PERL, Java... (the PERL above is executed in real time, it doesn't use embedded result) --Nad 12:02, 23 Aug 2006 (NZST)
Yes can do that sometime for the Redhat Post Install script, just needs a source install of the R programming language. All I did with that equation was looked at making pi as accurate as possible for trying to calculate phi using that formula, trying different approximations of e using different tailor series expansions from 1 to 20 order, and found that the 7th order approximation in the calculation is most similar to the true value of phi. I then started wondering why the 7th order approximation was closest. It was left field I know but I was interested if seven somehow had anything to do with quantisation of the electron energy levels shells in the chemistry of molecules, looking on 'Pedia five shells states are possible (so its bs)... By the way, where/who did you find out that equation from again? --Sven 14:03, 23 Aug 2006 (NZST)
It was from Dan Winter - yes I know, the poor fello's in a lot of trouble :-( --Nad 11:11, 24 Aug 2006 (NZST)
Yes, I was reading some of the alligations against him, all seems to stem from [Stan Tenen's allegations of plagiarism --Sven 08:25, 25 Aug 2006 (NZST)

Ok check this shit out, http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi2DGeomTrig.html#phipipowers.

phi = 2*cos(pi/5)
phi = 1/(2*sin(pi/10))

Very close approximations... --Sven 14:53, 23 Aug 2006 (NZST)

Now that stuff looks pretty cool! --Nad 15:40, 23 Aug 2006 (NZST)
That's excellent - that allows a similar loop to building e from the talor series, but adding π each time - and shows that there's a simple relation involving the conversion of 1/5 from angular to rectangular. --Nad 16:09, 23 Aug 2006 (NZST)

type this into Google: ((7 * pi) / (5 * e)) - ((1 + sqrt(5)) / 2) --Sven 16:30, 23 Aug 2006 (NZST)

Doesn't search anything remotely related to the formula at all, purely syntactic no semantic understanding... --Nad 15:38, 23 Aug 2006 (NZST)
No I am meaning from my summary of the edit that google can act as a rudimendary calculator, so it calculates;

((7 * pi) / (5 * e)) - ((1 + sqrt(5)) / 2) = -1.56990426 × 10-5

e.g. 2*cos(pi/5) - (1+sqrt(5))/2 = 0 in Google --Sven 20:31, 23 Aug 2006 (NZST)


More interesting stuff...

φ 0 + φ 0 = 2
φ2 + φ-2 = 3 
φ4 + φ-4 = 7
φ6 + φ-6 = 18
φ8 + φ-8 = 47
...
and
φ1 - φ-1=1
φ3 - φ-3=4
φ5 - φ-5=11
φ7 - φ-7=29
φ9 - φ-9=76
and
φn = φn-1n-2
φ-n = φ-n+2-n+1
These can be substituted into the above equations
+phi-power.pl
perl.php

--Sven 08:27, 25 Aug 2006 (NZST)

Some R code

<R>

  1. sequences with max larger than 8 seem to suffer from numerical
  2. approximation errors

n <- 10^seq(0,8, length=101)

x <- 5

y <- (1+x/n)^n

  1. Lower bounded

plot(n,y, type="l", log="x", lty=3) n

  1. Taylor series

n <- c() x <- 1 for(i in 1:9) {

 n[i] <- (x^(i-1))/factorial(i-1)

} n sum(n) exp(1)

sum(outer(n,n)) exp(2) </R>