The Museum of HP Calculators

HP Forum Archive 17

 Just for some Minor AmusementMessage #1 Posted by Chuck on 30 Mar 2007, 12:19 a.m. 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)Message #2 Posted by Karl Schneider on 30 Mar 2007, 2:29 a.m.,in response to message #1 by Chuck Hello, Chuck -- I've never seen that particular problem, but have worked similar ones. ```pi^i = cos(ln(pi)) + i*sin(ln(pi)) = 0.41329 + i*0.91060 i^pi = cos(0.5*pi^2) + i*sin(0.5*pi^2) = 0.22058 - i*0.97537 ``` The magnitude is unity in each case because cos2 x + sin2 x = 1 The HP-15C handles these calculations with aplomb, if not blazing speed: ```pi^i: i^pi: g pi 1 1 Re<->Im Re<->Im g pi y^x y^x g ABS g ABS ``` Here's an archived post of mine that some may find helpful: -- 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)Message #3 Posted by Namir on 30 Mar 2007, 6:49 a.m.,in response to message #2 by Karl Schneider 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)Message #4 Posted by Valentin Albillo on 30 Mar 2007, 7:04 a.m.,in response to message #3 by Namir Hi, Namir: Just for the record, the numbers that exactly comply with your two simultaneous conditions are: ``` 3.15098043851 and 2.71057757158 ``` to 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)Message #5 Posted by Paul Guertin on 30 Mar 2007, 8:09 p.m.,in response to message #3 by Namir Quote: e^pi = 20 + pi (with a -0.03 % error) Also see http://xkcd.com/c217.html .

 Re: Just for some Minor Amusement (i and pi)Message #6 Posted by Chuck on 30 Mar 2007, 9:28 p.m.,in response to message #2 by Karl Schneider 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)Message #7 Posted by Karl Schneider on 30 Mar 2007, 11:32 p.m.,in response to message #6 by Chuck Hi, Chuck -- Quote: 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. 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)Message #8 Posted by Chuck on 31 Mar 2007, 12:45 a.m.,in response to message #7 by Karl Schneider 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

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