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 ▼ Chuck Senior Member Posts: 320 Threads: 59 Joined: Dec 2006 03-30-2007, 12:19 AM 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. ▼ Karl Schneider Posting Freak Posts: 1,792 Threads: 62 Joined: Jan 2005 03-30-2007, 02:29 AM 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 ▼ Namir Posting Freak Posts: 2,247 Threads: 200 Joined: Jun 2005 03-30-2007, 06:49 AM 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 ▼ Valentin Albillo Posting Freak Posts: 1,755 Threads: 112 Joined: Jan 2005 03-30-2007, 07:04 AM 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. Paul Guertin Member Posts: 79 Threads: 5 Joined: Jun 2007 03-30-2007, 08:09 PM Quote: e^pi = 20 + pi (with a -0.03 % error) Also see http://xkcd.com/c217.html . Chuck Senior Member Posts: 320 Threads: 59 Joined: Dec 2006 03-30-2007, 09:28 PM 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 ▼ Karl Schneider Posting Freak Posts: 1,792 Threads: 62 Joined: Jan 2005 03-30-2007, 11:32 PM 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. ▼ Chuck Senior Member Posts: 320 Threads: 59 Joined: Dec 2006 03-31-2007, 12:45 AM 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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