# The Grey Labyrinth is a collection of puzzles, riddles, mind games, paradoxes and other intellectually challenging diversions. Related topics: puzzle games, logic puzzles, lateral thinking puzzles, philosophy, mind benders, brain teasers, word problems, conundrums, 3d puzzles, spatial reasoning, intelligence tests, mathematical diversions, paradoxes, physics problems, reasoning, math, science.

Author Message
DrJones
Daedalian Member

 Posted: Wed Feb 27, 2002 1:32 pm    Post subject: 1 Maybe this is not the best place to post it, but I didn't know where to do it. I was playing with my calculator, when I discovered that: PI * Cos 30 +-= e Do you know any better relation between "Pi" and "e"? Is this an important thing, or just a curiosity? GoodBye!
ralphmerridew
Daedalian Member

 Posted: Wed Feb 27, 2002 1:34 pm    Post subject: 2 The best relation between pi and e is almost certainly e^(i pi) == -1
groza528
No Place Like Home

 Posted: Wed Feb 27, 2002 1:41 pm    Post subject: 3 hehe, that's what I said in OT. It may get confusing to have to check two forums to read our responses though.
ZutAlors!
Daedalian Member

 Posted: Wed Feb 27, 2002 1:50 pm    Post subject: 4 And, incidentally, PI*cos(3) != e PI*cos(30) = 2.720699046 e = 2.718281828 Close (within 99.9%), but with enough judicious playing with available constants, you can approximate nearly anything.
dave10000
Tinhorn

 Posted: Wed Feb 27, 2002 6:13 pm    Post subject: 5 1000*[sin(Pi*1/49)*sin(Pi*2/47)]/Pi = e (to .000104% accuracy!) Find this and more at The Inverse Symbolic Calculator: http://www.cecm.sfu.ca/projects/ISC/ISCmain.html
Chuck
Daedalian Member

 Posted: Wed Feb 27, 2002 9:01 pm    Post subject: 6 12/5 + 1/pi = e approximately.
GH
Daedalian Member

 Posted: Wed Feb 27, 2002 10:16 pm    Post subject: 7 e/pi * pi/e = 1 *exactly*
ralphmerridew
Daedalian Member

 Posted: Wed Feb 27, 2002 10:20 pm    Post subject: 8 I think that e^(pi sqrt(163)) is very close to an integer (the difference is < 10^-12)
Griffin
Daedalian Member

 Posted: Wed Feb 27, 2002 10:23 pm    Post subject: 9 .0009 * pi^7 ~ e
HyToFry
Drama queen

 Posted: Wed Feb 27, 2002 10:27 pm    Post subject: 10 Griffen lives?
HappyMutant
Daedalian again

 Posted: Wed Feb 27, 2002 10:34 pm    Post subject: 11 Explain why e^(i*pi) = 1. (It's fun. I did it in calculus when no one was looking...) While you're at it, solve for i^(i). ------------------ Brunch - you'll love it. It's not quite breakfast, it's not quite lunch; but it comes with a slice of cantaloupe at the end.
ralphmerridew
Daedalian Member

 Posted: Thu Feb 28, 2002 2:12 am    Post subject: 12 That should be -1 in your post.
tigg
Daedalian Member

 Posted: Thu Feb 28, 2002 3:27 am    Post subject: 13 I think i^i is ambiguous. for any integer k, e^(-(4k+1)pi/2) = i^i.
mole
Subterranean Member

 Posted: Thu Feb 28, 2002 3:32 am    Post subject: 14 i ^ i = (e^(i.pi/2))^i = e^i(i.pi/2) = e^-pi/2
groza528
No Place Like Home

 Posted: Thu Feb 28, 2002 5:17 pm    Post subject: 15 I proved last year that e^(pi*i)=-1. The easiest way to show it is to work backwards. Upon learning about radians and polar notation I happened to noticed that cis(pi)=-1 (where cis(x)= cos(x)+i*sin(x)) and wondered if there was a connection. There was. I started with the proposition that perhaps e^x=cis(x/i) and my calculator confirmed this. So I found the algebra behind it. e^x=cis(x/i) original equation. x/i=-xi, combining that step with expansion yields e^x=cos(-xi)+i*sin(-xi) I used basic knowledge about sin and cos to get e^x=cos(xi)-i*sin(xi) After that I eliminated the i with hyperbolic trig. e^x=cosh(x)+sinh(x) Using the definitions of these hyperbolic trig functions e^x=(e^x+e^(-x))/2+(e^x-e^(-x))/2 Then simplify e^x=e^x so e^(pi*i)=cis(pi)=-1 [This message has been edited by groza528 (edited 03-01-2002 06:51 AM).]
ralphmerridew
Daedalian Member

 Posted: Thu Feb 28, 2002 5:59 pm    Post subject: 16 groza, your post uses circular reasoning, since the relations between cosh(x) and cos(ix) are proven using e^(ix) == cis(x). The usual proofs are generally either: 1) Use various methods to show code:``` e^x = sum (n=0 to oo, x^n/n!) sin(x) = sum(n=0 to oo, (-1)^n x^(2n+1) /(2n+1)!) cos(x) = sum(n=0 to oo, (-1)^n x^(2n)/(2n)!) ``` Then compare the formulas for e^ix and cis(x) 2) Using differential equations: y'' + y == 0, y(0)=1, y'(0)=i, y twice differentiable everywhere has a unique solution. But both y(x)=e^(ix) and y(x)=cis(x) satisfy it. Therefore both functions are equal everywhere. (Edit to fix mistake noted by tigg) [This message has been edited by ralphmerridew (edited 02-28-2002 03:24 PM).]
tigg
Daedalian Member

 Posted: Thu Feb 28, 2002 6:13 pm    Post subject: 17 for 2), I think you want y'(0) = i
groza528
No Place Like Home

 Posted: Thu Feb 28, 2002 7:01 pm    Post subject: 18 Thanks ralphmerridew, I didn't know that. I guess I did use circular reasoning. I just used my calculator as a reference that cos(i)=cosh(1) etc.
Jen Aside
Daedalian Member

 Posted: Thu Feb 28, 2002 7:21 pm    Post subject: 19 Pi and e... together, they make a tasty treat!
Ghost Post
Icarian Member

 Posted: Fri Mar 01, 2002 7:54 pm    Post subject: 20 didnt einstein discover something like this. it was apparantly his faviriote equation, because it relates two mysteries of maths, or something like that!
Sparticus
Spourk's Insignificant Other

 Posted: Mon Mar 04, 2002 7:39 pm    Post subject: 21 ralphmerridew: you're a C++ programmer, aren't you? ------------------ "'Tis better to be thought an idiot than to open one's mouth and erase all doubt...No, my mouth was not open."
ralphmerridew
Daedalian Member

 Posted: Mon Mar 04, 2002 8:15 pm    Post subject: 22 I am a programmer, but I prefer Java & C; I greatly dislike C++. Why do you ask?
groza528
No Place Like Home

 Posted: Mon Mar 04, 2002 8:26 pm    Post subject: 23 I think it was the ==. I don't know much about programming, but I think == is used in C++
CzarJ
Hot babe

 Posted: Mon Mar 04, 2002 11:12 pm    Post subject: 24 Well, yes, but surely in other languages as well. ------------------ Unslumping yourself is not easily done.
ctrlaltdel
Member of the Daedalians

 Posted: Tue Mar 05, 2002 8:36 am    Post subject: 25 how does pi relate to i??? they are cousins, everybody knows that...
Chuck
Daedalian Member

 Posted: Tue Mar 05, 2002 3:36 pm    Post subject: 26 193 × e + 45 × pi = 666. Well, it's close.
tigg
Daedalian Member

 Posted: Tue Mar 05, 2002 4:20 pm    Post subject: 27 Chuck- you're a strange man..
Lucky Wizard
Daedalian Member

 Posted: Thu Mar 07, 2002 12:20 am    Post subject: 28 I noticed today (while fooling with my calculator) that this is approximately true: code:``` pi 2 (e )-pi=(5 )-5 ``` johnny, for the record, the e^(i*pi)=-1 thing was discovered by Euler, not Einstein.
 Display posts from previous: All Posts1 Day7 Days2 Weeks1 Month3 Months6 Months1 Year by All usersChuckctrlaltdelCzarJdave10000DrJonesGHGhost PostGriffingroza528HappyMutantHyToFryJen AsideLucky WizardmoleralphmerridewSparticustiggZutAlors! Oldest FirstNewest First
 All times are GMT Page 1 of 1

 Jump to: Select a forum Puzzles and Games----------------Grey Labyrinth PuzzlesVisitor Submitted PuzzlesVisitor GamesMafia Games Miscellaneous----------------Off-TopicVisitor Submitted NewsScience, Art, and CulturePoll Tournaments Administration----------------Grey Labyrinth NewsFeature Requests / Site Problems
You cannot post new topics in this forum
You can reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum
You cannot vote in polls in this forum