Answer without paper or computers: which is bigger, pi^e or e^pi?
This question is interesting.
It’s so, … unreal.
I remember seeing a proof for this. e^pi is bigger, … no?
Been a long time since college. I might have been able to work it out once upon a time. I figure if you compare functions using e you might have a go.
Okay. Let’s try to construct a proof. I remember this from college and I remember the logic of it.
Let’s see: If F(X) = X – e(ln(X))
Let’s see if I can still derive.
f'(x) = 1 – e/x.
For x>e, this is positive. Of course.
For all situations where x>e, F(X)>F(e) is = 0.
Okay, this too me a while, but:
pi > E
e^x is a function that increases. Implication: e^pi is > than pi^e.
Missed some logic, but this is what I remember. This was actually an exam question for 3rd year. Damn. Thanks for the recall.
There’s a way to figure this out with just logic, once you know the primary fact about e, which is that it is the limit of (1 + 1/n)^n.
I’ll give it some thought tomorrow. PCG and I are watching a horror movie.
Saw your new comment after sending mine. You’re basically right. The point is that exponentiation is obviously more powerful once the base is big enough, and the place it gets big enough is e. For example, 10^11 = 10 * (10^10); 11^10 = (1.1^10) * (10^10) which is about e * 10^10 because of the limit definition for e, so since 10>e 10^11 > 11^10. When the base is actually e you can’t quite conclude this rigorously without looking at the derivative to make sure the second-order effects have the right sign, but I’ll just wave my hands and say I’ve shown enough.
Judging by how I imagine the slopes of the two curves, it’s e^Pi. And yes, exponentiation is increasing the slope of the curve faster. This could be done with analysis, but I don’t have time for it right now. You do a function like f(x)=e^x-x^e and find out the intersection points and then depending of x in relation to e, you can see where you are. I don’t know if I will have time to finish, but whatever.
f'(x)=e^x-e*x^(e-1) => the points of intersection are x=e and x=1. Since the exponential grows faster than the power, this is an increasing function, for x>e(which Pi is). Since f is increasing and Pi>e, then f(Pi)>0 => e^Pi>Pi^e. QED
That puzzle got a good response even though it was a case of “it’s 11:58pm, what can I do in 1 minute that’s postworthy so I won’t skip a day?” The previous day’s post on GRE scores involved a lot more work and no one responded.
By the way, the numerical answers to 2 places are: e^pi = 23.14, pi^e=22.46, close enough that this would be quite difficult to solve numerically by hand.
I have been running a faboulous mathematics puzzle site. Mathalon . Come and help me to build it better.
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