Can e & pi be easily related?
Bill Taylor
The recent thread reminded me of this old query:-
Is there a reasonably simple real-numbers equation connecting e & pi,
that can be used to calculate either one from the other?
-------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
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"I don't think so..." - Descartes' last words.
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math...@my-dejanews.com
In article <6mcte1$ggs$2...@cantuc.canterbury.ac.nz>,
mat...@math.canterbury.ac.nz (Bill Taylor) wrote:
>
> The recent thread reminded me of this old query:-
>
> Is there a reasonably simple real-numbers equation connecting e & pi,
> that can be used to calculate either one from the other?
>
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David A. Scott
In a previous article, math...@my-dejanews.com () says:
>In article <6mcte1$ggs$2...@cantuc.canterbury.ac.nz>,
> mat...@math.canterbury.ac.nz (Bill Taylor) wrote:
>>
>> The recent thread reminded me of this old query:-
>>
>> Is there a reasonably simple real-numbers equation connecting e & pi,
>> that can be used to calculate either one from the other?
>>
> NO!
>
e^(i(pi)) = -1
where ^ is the power operator.
and i is sqware root of -1
--
http://cryptography.org under /Misc location of scott16u.zip and now
the location of scott4u.zip no one soleved the 40bit key contest next
time i will write a easier version.
Adrian Cable
> >> Is there a reasonably simple real-numbers equation connecting e & pi,
> >> that can be used to calculate either one from the other?
> >>
> > NO!
> >
>
> e^(i(pi)) = -1
> where ^ is the power operator.
> and i is sqware root of -1
This isn't a "real-numbers equation". The i in the exponent should give
the game away. :)
As for an equation with only real coefficients relating e and pi, how
about:
x = inf.
sqrt(pi)/2 = integral (e^(-x^2)) dx
x = 0
But simpler still, get rid of e altogether:
x = 2
pi = integral (sqrt(4 - x^2)) dx
x = 0
This integral can be trivially derived from the properties of the
circle.
Thanks, cheers,
Adrian Cable.
Dario Alejandro Alpern
Adrian Cable wrote:
Or simpler still, without square roots:
x=1 dx
pi = 4 integral -------
x=0 1 + x^2
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Kaziu
Bill Taylor napisał(a) w wiadomości:
<6mcte1$ggs$2...@cantuc.canterbury.ac.nz>...
>Is there a reasonably simple real-numbers equation connecting e & pi,
>that can be used to calculate either one from the other?
Integral _-infinity ^+infinity ( exp (-x^2)) dx = sqrt (pi)
I know, i know : it isn't real-number equation, it's integral.
Kaziu
L Euler
>Is there a reasonably simple real-numbers equation connecting e & pi,
>that can be used to calculate either one from the other?
If we restrict ourselves to polynomial relationships, then you are asking
whether the field e is transcendental over Q(pi). I would assume so. But this
is a very difficult question: we had a hard enough time proving e itself was
transcendental! I don't even think we know whether e*pi is irrational,
although all our intuitions say no.
Jared
Adrian Cable
> If we restrict ourselves to polynomial relationships, then you are asking
> whether the field e is transcendental over Q(pi). I would assume so. But this
> is a very difficult question: we had a hard enough time proving e itself was
> transcendental! I don't even think we know whether e*pi is irrational,
> although all our intuitions say no.
Don't you mean that your intuition says yes, regarding whether e*pi is
irrational?
Thanks, cheers,
Adrian Cable.
itsall...@mailcity.com
In article <6mdqdm$d87$1...@news.ysu.edu>,
an...@yfn.ysu.edu (David A. Scott) wrote:
>
>
> In a previous article, math...@my-dejanews.com () says:
>
> >In article <6mcte1$ggs$2...@cantuc.canterbury.ac.nz>,
> > mat...@math.canterbury.ac.nz (Bill Taylor) wrote:
> >>
> >> The recent thread reminded me of this old query:-
> >>
> >> that can be used to calculate either one from the other?
> >>
> >
>
> e^(i(pi)) = -1
> where ^ is the power operator.
> and i is sqware root of -1
>
J. Mayer
itsall...@mailcity.com schrieb:
> In article <6mdqdm$d87$1...@news.ysu.edu>,
> an...@yfn.ysu.edu (David A. Scott) wrote:
> >
> >
> > In a previous article, math...@my-dejanews.com () says:
> >
> > >In article <6mcte1$ggs$2...@cantuc.canterbury.ac.nz>,
> > > mat...@math.canterbury.ac.nz (Bill Taylor) wrote:
> > >>
> > >> The recent thread reminded me of this old query:-
> > >>
> > >> Is there a reasonably simple real-numbers equation connecting e & pi,
> > >> that can be used to calculate either one from the other?
> > >>
> > > NO!
> > >
> >
> > e^(i(pi)) = -1
> > where ^ is the power operator.
> > and i is sqware root of -1
> >
> But he asked for a real number equation.
>
What about e=exp(-cos pi)?
JarWeinst
Indeed, showing that e*pi is rational would revolutionize computations for both
numbers, but owing to a reluctance of e*pi to show up in any meaningful (and
"nonestimational") formulas precludes any proofs or disproofs thus far. Same
for things like e/pi, pi^e, etc.
But wait, this gives me an idea. If we allow asymptotic considerations, we
have the following formula, which easily follows from Euler's formula for
zeta(m) and Sterling's approximation of m!:
pi*e ~ (m/2) * (2*sqrt(2*pi*m)/|B_m|)^(1/m)
where B_m is the mth Bernoulli number:
t/(exp(t)-1) = SUM(m=1..inf) B_m*t^m/m!.
Actually, since part of the above asymptotic relationship goes off to 1 by
itself, we can reduce it to
pi*e ~ (m/2) * |B_m|^(-1/m).
By "~" I mean that the ratio of the two sides tends to 1 as m goes to infinity
along even integers. For m = 700, the left side is
8.539734222, while the right side is 8.480313272. Which is kind of second-rate.
But if we keep in the (2*sqrt(2*pi*m))^1/m garbage as in the original formula,
the right side is 8.539732775. I have no idea if this is what you were looking
for as far as "easily related" -- Bernoulli numbers are by no means quick to
calculate -- but it's something.
Jared
Stephen
>>that can be used to calculate either one from the other?
pi = [exp (ln 4)] (arctan 1)
- Steve Pipkin
Kurt Foster
In <199806210105...@ladder03.news.aol.com>, L Euler said:
[snip]
. I don't even think we know whether e*pi is irrational, although all our
. intuitions say no.
.
We do know, however, that at least one of the numbers e + pi and e*pi is
transcendental. For Q(e, pi) is an extension of degree at most 2 of the
field Q(e + pi, e*pi), being obtained from it by adjoining the roots of
the quadratic x^2 - (e + pi)*x + e*pi = 0. It follows that the two fields
have the same transcendence degree over Q. Since e and pi are both
transcendental, Q(e, pi) has transcendence degree at least 1 over Q; and
therefore so does Q(e + pi, e*pi).
Steve Gray
>
>Is there a reasonably simple real-numbers equation connecting e & pi,
>that can be used to calculate either one from the other?
>
and pi (such as (e^e)/pi = e+2 or something, it would comprise a
previously unknown identity and is therefore extremely unlikely..
Kevin Brown
On 21 Jun 1998 jarw...@aol.com (JarWeinst) wrote:
> I don't even think we know whether e*pi is irrational,
It's known that e * pi is not rational, but we don't whether
e + pi is rational, and we certainly don't know if both
are transcendental, although we do know that at least one
of them must be.
By the way, here's an interesting little formula involving
the product of pi and e (which will be utterly unintelligible
if you're reading this with something other than a fixed
pitch font, and maybe even regardless):
_______
/ pi e 1 1 1
/ ----- = 1 + --- + ----- + ------- + ....
/ 2 1*3 1*3*5 1*3*5*7
+ 1
------------------
1
1 + ----------------
2
1 + --------------
3
1 + -----------
4
1 + --------
5
1 + ------
1 + ...
I probably don't even need to say who came up with this - which
raises an interesting question: Is any other mathematician's work
so instantly recongizable? For the record, Ramanujan posed the
above formula as question 541 in the Journal of Indian Math.
On an unrelated point, does anyone know where the first "3"
appears in the base-pi representation of e? I think the leading
digits are something like
e = 2.2021201002111...(3)?... base pi
where the kth digit is the largest integer d such that d/pi^k is less
than or equal to the current remainder. Similarly the representation
of pi in the base e is
pi = 10.101002020002111... base e
but of course here there can never be any digits other than 0,1,
or 2.
______________________________________________________________
| MathPages /*\ http://www.seanet.com/~ksbrown/ |
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feld...@bsi.fr
In article <358dcdf2...@news.seanet.com>,
ksb...@seanet.com wrote:
>
> On 21 Jun 1998 jarw...@aol.com (JarWeinst) wrote:
> > I don't even think we know whether e*pi is irrational,
> > although all our intuitions say [it is]... Indeed, showing that
> > e*pi is rational would revolutionize computations for both
> > numbers..
>
> It's known that e * pi is not rational,
I didn't know it (it it not a consequence of Ramanujan formula, as this give
only the irrationality of sqrt(pi*e/2)); do you have a reference?
Well, I dont know if this is of any use, but.:-) Anyway, one digit in about
20 should be a 3, and sure enough, in the first 150 digits, we (I mean
MapleV and me) found them in position 68, 76,90,110 and 130 (counting 2.20...
as positions 1, 2 and 3)
__
Jean-Pascal Laedermann
I think one of the most beautyful relation between these numbers is
e ^ (i*pi) + 1 = 0
The three operators : +, * , ^
The five constants : 0, 1, i, e, pi
Cheers.
Wilbert Dijkhof
Kevin Brown wrote:
That's a nice one, I haven't seen him before.
Do you know a proof for this?
Wilbert
itsall...@mailcity.com
In article <358D55B9...@hadiko.de>,
"J. Mayer" <j...@hadiko.de> wrote:
>
>
> itsall...@mailcity.com schrieb:
>
> > In article <6mdqdm$d87$1...@news.ysu.edu>,
> > an...@yfn.ysu.edu (David A. Scott) wrote:
> > >
> > >
> > > In a previous article, math...@my-dejanews.com () says:
> > >
> > > >In article <6mcte1$ggs$2...@cantuc.canterbury.ac.nz>,
> > > > mat...@math.canterbury.ac.nz (Bill Taylor) wrote:
> > > >>
> > > >> The recent thread reminded me of this old query:-
> > > >>
> > > >> that can be used to calculate either one from the other?
> > > >>
> > > >
> > >
> > > e^(i(pi)) = -1
> > > where ^ is the power operator.
> > >
> > But he asked for a real number equation.
> >
>
> What about e=exp(-cos pi)?
I suppose that's a link, but you can't isolate pi in terms of e, because the
logarithm of both sides will turn the left side to one.
Adrian Cable
> On an unrelated point, does anyone know where the first "3"
> appears in the base-pi representation of e? I think the leading
> digits are something like
>
> e = 2.2021201002111...(3)?... base pi
An interesting question. I once wrote a QBASIC program that would take a
real number in any base of any length and convert it to any other base,
including non-integral bases. I haven't looked at what e base pi is
like, but if you'd like, I'll post the program here.
Thanks, cheers,
Adrian Cable.
Radioac...@my-dejanews.com
In article <6mkbsl$d...@topdog.cs.umbc.edu>,
pip...@topdog.cs.umbc.edu (Stephen) wrote:
>
> >Bill Taylor napisał(a) w wiadomości:
> ><6mcte1$ggs$2...@cantuc.canterbury.ac.nz>...
> >>that can be used to calculate either one from the other?
>
>
> - Steve Pipkin
>
>
i don't think that helps calculating one from the other (or was it meant 2 b a
joke?)
Radioactive Man
Don Redmond
In article <358E62...@student.utwente.nl>, Wilbert Dijkhof
<w.j.d...@student.utwente.nl> wrote:
This originally solved in J. Indian Math. Soc. vol. 7, pp 17-20 (at least
according to the collected works.) A generalization of this, due to Ramanujan,
can be found in Ramanujan's Notebooks, part II, p. 166.
Don
Bill Taylor
ksb...@seanet.com (Kevin Brown) writes:
|> _______
|> / pi e 1 1 1
|> / ----- = 1 + --- + ----- + ------- + ....
|> / 2 1*3 1*3*5 1*3*5*7
|>
|>
|> + 1
|> ------------------
|> 1
|> 1 + ----------------
|> 2
|> 1 + --------------
|> 3
Fascinating!
Does either the series of the c-fraction have a nice value on its own?
-------------------------------------------------------------------------------
The answer may be right but it's not the answer I want
-------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
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Identity of indiscernibles: x = y <==> (P) Px <=> Py
Identity of indiscriminables: P = Q <==> (x) Px <=> Qx
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Cyrand
Bill Taylor wrote:
> ksb...@seanet.com (Kevin Brown) writes:
>
> |> _______
> |> / pi e 1 1 1
> |> / ----- = 1 + --- + ----- + ------- + ....
> |> / 2 1*3 1*3*5 1*3*5*7
> |>
> |>
> |> + 1
> |> ------------------
> |> 1
> |> 1 + ----------------
> |> 2
> |> 1 + --------------
> |> 3
I don't know if this has been mentioned in this thread(being new here, and this
being theonly piece of it I can see), but another relationship between e & pi is
ln(n*i) = ln(n) + i*( pi/2)
or for a more easy relation, if n = 1
ln(i) = i * (pi /2)
which is equivilant to,
e^i*(pi/2) = i
--
Email: cyr...@juno.com
"Strategy and appropriate use of the *available* resources does NOT equate to
godhood. --Barry Kearns"
Steven T. Smith
mat...@math.canterbury.ac.nz (Bill Taylor) writes:
> ksb...@seanet.com (Kevin Brown) writes:
> |> _______
> |> / pi e 1 1 1
> |> / ----- = 1 + --- + ----- + ------- + ....
> |> / 2 1*3 1*3*5 1*3*5*7
> |>
> |> + 1
> |> ------------------
> |> 1
> |> 1 + ----------------
> |> 2
> |> 1 + --------------
> |> 3
>
> Does either the series of the c-fraction have a nice value on its own?
The rather circular (haha) answer is
______
1 1 1 / pi e / \
1 + --- + ----- + ------- + ... = / ----- | 1 - erfc(1/sqrt(2)) |
1*3 1*3*5 1*3*5*7 / 2 \ /
Therefore,
1 2 3
--- --- --- ... = sqrt(pi*e/2)*erfc(1/sqrt(2))
1 + 1 + 1 +
Steven Smith
Bill Dubuque
Bill Taylor <mat...@math.canterbury.ac.nz> wrote:
|
| Is there a reasonably simple real-numbers equation connecting
| e & pi, that can be used to calculate either one from the other?
One of the most beautiful and mysterious approximations is
pi 3 1/sqrt(163) -30
e = ( 640320 + 744 ) - 5.177... 10
Check it on your calculator, then see my prior post [1]
for references to the deep and fascinating explanation
involving class fields, complex multiplication, modular
functions, Kronecker's Jugendtraum (youthful dream), etc.
Or behold Shanks' [2] amazing approximation
6 24 -r pi -163
pi = - log(2abcd) + -- e + 6.69... 10
r r
a = A+sqrt(A^2-1), b = B+sqrt(B^2-1), c = C+sqrt(C^2-1), d = D+sqrt(D^2-1),
A = 1/2 (1071 + 184 sqrt(34)), B = 1/2 (1553 + 266 sqrt(34))
C = 429 + 304 sqrt(2), D = 1/2 (627 + 442 sqrt(2)), r = sqrt(3502)
To improve an approximation P to pi use P + sin(P) or P + 2 cos(P/2),
which triples the accuracy, or use P + (2 sin P - tan P)/3, which
quintuples the accuracy, cf. [3]. For example, applying these to the
pi approximation from the first equation above triples the accuracy
from 10^-31 to 10^-93, and quintuples the accuracy to 10^-155.
Quintupling Shanks' approximation would give you over 800 digits.
-Bill Dubuque
[1] sci.math post of 1996/09/27 titled: Re: (pi+20)^i ~= -1, explain why
http://x11.dejanews.com/getdoc.xp?AN=185756777%26CONTEXT=898769821.149749811%26hitnum=0
[2] Shanks, Daniel. Dihedral quartic approximations and series for pi.
J. Number Theory 14 (1982), no. 3, 397--423.
MR 83k:12010 (Reviewer: Harvey Cohn) 12A70
[3] Shanks, Daniel. Improving an approximation for pi.
Amer. Math. Monthly 99 (1992), no. 3, 263.
MR 94a:11197 (Reviewer: W. W. Adams) 11Y60
Charles H. Giffen
How about:
\Gamma(1/2) = \int_0^\infty x^{-1/2} e^{-x} dx = \sqrt{\pi} ?
Or, which is essentially the same:
\int_{-\infty}^\infty e^{-t^2 / 2} dt = \sqrt{2\pi} ?
--Chuck Giffen