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.
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:
http://www.hpmuseum.org/cgi-sys/cgiwrap/hpmuseum/archv014.cgi?read=66246#66246
-- KS
Edited: 30 Mar 2007, 11:41 p.m. after one or more responses were posted
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
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.
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
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.
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