Is ln(√e^π)/π a rational number?

[johann1301]

is ln(√e^π)/π a rational number?

where π =3.14...

 

[WWGD]

Wow, e^∏ alone not hard-enough? You could write a whole paper, if not a small book in answering this.

 

[johann1301]

Isn't it a half?

 

[WWGD]

You may be right; I guess I jumped the gun.

 

[johann1301]

but my textbook says its an irrational number, how can that be?

 

[PeroK]

but my textbook says its an irrational number, how can that be?

Your textbook could be wrong!

 

[ellipsis]

$$
\begin{align}

\frac{ \ln{ \sqrt{ e^\pi } } }{\pi} &= \frac{ \ln{ e^\frac{\pi}{2} } }{\pi}\\

&= \frac{1}{\pi} \frac{\pi}{2}\\

&= \frac{1}{2}

\end{align}
$$
The $\sqrt{e^\pi}$ is equivalent to $e^{\frac{\pi}{2}}$, so the natural log cancels with $e$ and you're left with $\frac{(\frac{\pi}{2})}{\pi}$ which is $\frac{1}{2}$.