"Ladies and Gentlemen, in the one corner we have pi. First written about by Euclid, pi is the ratio of the circumference of any circle to its diameter. Pi's been baffling people for millennia with its ability to just keep going without end. And here's the thing folks, it's the same for ALL circles! you can have a circle so small you can't even see it, and it'll be pi times as big around as it is across. And then you can draw a circle around your house, and same thing. If you had a circle that went all the way around the universe, people, its circumference would be pi times its diameter."

"And in the other corner, the scrappy young e! e was studied by the mathemetician Euler in the 1700s, e certainly doesn't have as much experience as pi, and at a measly 2.71828, certainly gives up a weight advantage to pi's 3.14159, but don't let that fool you folks...e can go on just as bafflingly long as pi, and represents the limit as n approaches infinity of (1 + 1/n)ⁿ. It is the base for the natural logarithm! The graph of f(x)=e^x has a slope of exactly e^x at any value of x! That's right folks, it derives its OWN tangent!"

"AND WITH THAT, e takes an early jab at pi, which reels back but quickly multiplies itself by i and raises e to...-1! Oh, e is hurting people but quickly re-derives itself from itself showing pi the versatility of truly interdimensional mathematics..., pi confined merely to the cartesian plane is quickly lost in its meager foray into Complex mathematics, and tries to re-aquire geometric footing..."

"But Pi's not done here folks, the referee can't really comprehend this higher-calculus either and quickly breaks up the hold. Pi immediately starts spinning out sine waves, proving its utility to electrical and recording engineers worldwide, but e is unimpressed. Despite lying "undiscovered" for centuries longer than pi, e demonstrates its presence throughout nature in the shell of the nautilus, the expansion of the universe, and even the architectual designs of pi's Greek contemporaries."

"Pi stumbles back, dumbfounded and weakly shields itself with a large circle, but its bag of tricks appears to be running empty, and e is just getting started, as it begins to feed itself into continuously-compounding interest equations, showing off its economic maximizing potential. But it's still not done, it's now rubbing pi's face in the base-e numerical system, with the undisputed most efficient economy of numerical width times radix, leaving no doubt as to where the "natural logarithm" got its name."

"Now I must say here, folks that stylisticaly speaking, pi looks good, but e has the undeniable presence advantage, being found on every keyboard, while pi is inherently difficult to type, not even being included in the ASCII character set. Nobody's confusing e with dessert, either, and the game may be stacked here, but ln(e¹)=1, while ln(pi¹) is incredibly ugly."

"Pi is definitely outclassed here. It's a textbook case of a long-time champion growing overconfident in its size and inertia and getting out maneuvered by a leaner, meaner number who just wanted it more. The winner by undisputed decision and a KNOCKOUT is...."

"eeeeeeeeeeeeeeeeeeeeeeeeee"

This deathmatch brought to you by by the balanced ternary number 111, and the letter theta

An interesting and very much open question in number theory. The real numbers are a vector space (of infinite dimension, even of uncountable dimension!) over the field of rational numbers. As such, we can ask whether any set of real numbers is linearly independent. Is the set {1,e,π} linearly independent?

That is, do there exist integers a,b such that a*e+b*π is an integer?

We don't know.

Can we get it with a=b? That is, "is e+π a rational number? Is e-π?"

We don't know.

Of course, it would be absurd if the answer to any of these questions was "yes". I doubt any mathematician seriously countenances the possibility. But we have nothing remotely like a proof.

What about algebraic dependence? Is there a polynomial with integer coefficients P(x,y)∈Z[x,y] for which P(e,π)=0?

We don't know.

It is fairly easy to prove irrationality of e and of various powers of e. Niven's proof of the irrationality of pi is completely elementary. A theorem of Lindemayer tells us that various powers of e are even transcendental numbers. ζ(3)=∑_n≥11/n³ is known to be irrational. Since e and π are transcendental, so are e+sqrt(2), e-17sqrt(3), π*sqrt(5), and the like; in particular, we know they're irrational.

And that's about all we know. Proving irrationality is hard.