(on pi^e ~ e^pi) Joe N says: > Simply [a mental picture pf] the log (ln) curve, with points where it intersects (e,1) and (pi,log(pi)) noted. >...assuming a little background of course. > If there's simpler picture that "says it all" I'd like to know.
R Hansen asks: Ok, so let me ask you: How does your picture show that e^pi > pi^e? >Don't you have to provide the justification?
Justification? Depends on context. For a class of calculus students, sure.
I'd invite you or others who don't "see it" to ponder a little more. I don't offer this as a "proof without words" that is utterly obvious. It takes, as you would agree, just a few more "ingredients" to tease the meaning out of it. And then you might have a mini-ah ha! Or not.
I'm partial to this solution, obviously for personal reasons, but it illustrates what I like about visual reasoning. The picture is an information carrier, but as with any info carrier it needs a (trained) decoder on the other side. Still, as an info carrier its elegant and compact and aesthetic. When I think of this problem, that's all I need to remember -- its a single "chunk" in my mind, not a bunch of lines of algebra.
As a hint, you might want to add at least one or two more straight lines mentally into the picture (as a mental concept, its not supposed to be static!) Obvious ones.
(You might object I didn't give you the whole picture, but I'm trying to make a point. Mental pictures are not static. As an "info carrier", though I don't need the extra baggage of those other lines. For chunking, they are pretty much implied. Sort of there but not there. There are three points of interest: 0, log at e, and log at pi. Only so many lines present themselves.)