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 cos^{2} x + sin^{2} 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