Just for some Minor Amusement


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

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


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)


e^pi = 20 + pi (with a -0.03 % error)

Pi continues to be spookie!!!



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.

e^pi = 20 + pi (with a -0.03 % error)

Also see http://xkcd.com/c217.html .


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.



Hi, Chuck --

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.


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