|e^pi < > pi^e ? [Minor update]|
Message #28 Posted by Karl Schneider on 28 Oct 2009, 12:41 a.m.,
in response to message #27 by Gerson W. Barbosa
A year or two ago, a Forum participant offered a small "challenge" of sorts for young math students: Knowing that pi and e were both near 3 in magnitude, and pi > e, how would one go about deducing whether e^pi or pi^e had the greater value, without actually calculating them?
Here's a straightforward approach I think most of us would follow:
e^pi ? pi^e
ln (e^pi) ? ln (pi^e)
pi ? e * ln (pi)
e * (pi/e) ? e * ln (pi)
Because to raise e to a given higher value requires a greater
multiplier than exponent, pi/e > ln (pi).
Applying this from the bottom upwards, we conclude that e^pi > pi^e.
e^pi = 23.1406926328
pi^e = 22.4591577184
e^pi is only about 3% greater than pi^e.
Similar, simpler numbers:
2^3 = 8
3^2 = 9
3^4 = 81
4^3 = 64
Paul Dale posted:
Can anyone do likewise for pi^e = 22.459157718361...
My 15c gives 22.45915771 for this. Correctly rounded it should be 22.45915772.
If anyone can present a reasonably-straightforward way to obtain this actual value of pi^e to ten digits on a pre-Saturn HP, I'd like to see it, because I'm stumped...
On an HP-42S with both pi and e rounded to nine decimal digits (10 significant digits), pi^e = 22.4591577145, so it appears that the HP-15C's result is the best possible, given its limitations.
Edited: 1 Nov 2009, 3:48 p.m. after one or more responses were posted