Just for some Minor Amusement  Printable Version + HP Forums (https://archived.hpcalc.org/museumforum) + Forum: HP Museum Forums (https://archived.hpcalc.org/museumforum/forum1.html) + Forum: Old HP Forum Archives (https://archived.hpcalc.org/museumforum/forum2.html) + Thread: Just for some Minor Amusement (/thread111191.html) 
Just for some Minor Amusement  Chuck  03302007 Teased a few colleagues today with a goofy little math problem... Which is bigger (i.e., magnitude), i^pi or pi^i ? One is quite obvious using DeMoivre's formulas. The other takes a little bit of paper and pencil (or you can wimp out and first try it on a calculator) ;) Also, can you geometrically explain pi^i? Hmmmmm.
Have fun. Edited: 30 Mar 2007, 12:20 a.m.
Re: Just for some Minor Amusement (i and pi)  Karl Schneider  03302007 Hello, Chuck  I've never seen that particular problem, but have worked similar ones.
pi^i = cos(ln(pi)) + i*sin(ln(pi)) The magnitude is unity in each case because cos^{2} x + sin^{2} x = 1 The HP15C handles these calculations with aplomb, if not blazing speed:
pi^i: i^pi: Here's an archived post of mine that some may find helpful: http://www.hpmuseum.org/cgisys/cgiwrap/hpmuseum/archv014.cgi?read=66246#66246
 KS Edited: 30 Mar 2007, 11:41 p.m. after one or more responses were posted
Re: Just for some Minor Amusement (i and pi)  Namir  03302007 Looking at your ln(pi) term made me curious at it's numerical value. With a few calculator keystrokes, I discovered that: ln(pi) = pi  2 (with a 0.1 % error) and e^pi = 20 + pi (with a 0.03 % error) Pi continues to be spookie!!!
Namir
Re: Just for some Minor Amusement (i and pi)  Valentin Albillo  03302007 Hi, Namir:
3.15098043851 and 2.71057757158to 12 decimal places. Rounding to a mere two places, they would be 3.15 and 2.71, agreeing with Pi and e to a single ulp. Best regards from V. Re: Just for some Minor Amusement (i and pi)  Paul Guertin  03302007 Quote:
Also see http://xkcd.com/c217.html .
Re: Just for some Minor Amusement (i and pi)  Chuck  03302007 Good work Karl. Seeing that we know i^pi and pi^i, I got to thinking about i^i. Seems that that turns out to be a REAL number. Too cool.
CHUCK
Re: Just for some Minor Amusement (i^i)  Karl Schneider  03302007 Hi, Chuck 
Quote: Yes indeed. The fact that i^i = e^(pi/2) was mentioned in the post from 2004 that I linked in my first response (as "j^j"); some discussion ensued as well.  KS
Edited: 30 Mar 2007, 11:34 p.m.
Re: Just for some Minor Amusement (i^i)  Chuck  03312007 Man. As soon I saw that, Karl, it rang a bell. I remember playing with i^i years ago, but forgot the actual value. Wish these brain cells would stop disappearing. :( Thanks for sparking my memory.
CHUCK
