is

Question

is

nincompoop · Answer

$$\pi^e$$

nincompoop · Answer

irrational?

anonymous · Answer

shouldn't be both pi and e are rational numbers

anonymous · Answer

the answer i think is like 22.5 or something close

anonymous · Answer

the answer i think is like 22.5 or something close

zepdrix · Answer

@dedee Pi is rational...? :o

anonymous · Answer

HA sorry your right it is not

anonymous · Answer

i have been working so long i had a brain fart and for a slight second i was confusing rational with real numbers

campbell_st · Answer

both pi is irrational and so is e  so pi^e will also be irrational.... it can't be written as a simple fraction.

nincompoop · Answer

An irrational number to an irrational power can be rational

campbell_st · Answer

well I've learnt something...

anonymous · Answer

I know an irrational number to an imaginary power can be rational, but an irrational to an irrational? Yeah, that should be possible with the right numbers.

π^e, though? I doubt it.

anonymous · Answer

Proving that it is irrational is beyond my abilities. Since both numbers are transcendental,  I don't know if usual methods will apply.

lilai3 · Answer

i'm pretty sure pi is rational, because it does not terminate, or repeat. Repeating decimals are in the irrational category, while the terminating decimals are in the rational. So my guess would be that pi is rational, because it goes on in a non-repeating and non-terminating way. (By terminating, it means eding, such as 0.25, it ends.)

campbell_st · Answer

lol.. 0.33333... is a repeating decimal that is rational... its  1/3

campbell_st · Answer

and repeating decimal can be written as a rational number.... 

no repeating decimals like pi.. are irrational... and its in a group called transendental numbers.

anonymous · Answer

it seems rational to say that pi^e is irrational because they both are irrational numbers but i dont think anyone has proved this (sometimes irrational numbers multiplied by irrational numbers result in a rational number)

nincompoop · Answer

irrational
$$\sqrt{2}$$

rational
$$\left( \sqrt{2}^{\sqrt{2}} \right)^{\sqrt{2}}$$

pottersheep · Answer

Do people really think pi i rational?

anonymous · Answer

@lilai3 "i'm pretty sure pi is rational, because it does not terminate, or repeat. Repeating decimals are in the irrational category, ..."

I think you have the terms 'rational' and 'irrational' mixed up here.

nincompoop · Answer

to clear some misconception: pi = irrational. it cannot be written in normal fraction a/b where a is an integer and b is a nonzero integer. http://mathworld.wolfram.com/RationalNumber.html http://mathworld.wolfram.com/IrrationalNumber.html

anonymous · Answer

Yes. It seems the challenge is to somehow prove that $\pi^e$ is irrational. The two numbers do have a clearly defined relationship involving imaginary numbers (see Euler).
My guess is that since something like $e^{i\pi}$ is rational, it would take an excursion into the complex plane to get e and π to rationalize together, so dealing with only real multiples of e and π, the power would still be irrational.

Good luck on that. As far as I know, it is still an open question.

nincompoop · Answer

It is still an OPEN QUESTION indeed.  No one has been able to prove or disprove the rationality or irrationality of it.  I posted the question since no one has on Open Study at least, and I thought it might pique the interest of the quasi math-gods and goddesses.

anonymous · Answer

It piques my interest; I am very curious. But, I am no math-god. This is outside my areas of expertise.