is $(1+\pi)/(1-\pi)$ a transcendental number? I know it's a open problem to show if $\frac{\pi}{e}$ is a trans. number or not. But what about quotient between numbers in function only of $\pi$, which is trans, such as $\frac{1+\pi}{1-\pi}$ or $\frac{1+i\pi}{1-i\pi}$. Wolfram says all theses numbers are trans. $\frac{1+\pi}{1-\pi} = -1- \frac{2}{\pi-1}$ I'm summing a not trans number with a trans. That implies the result is trans? Also, how to be sure that $\frac{2}{\pi-1}$ is trans? Is a division between "equal" trans numbers also trans? by "equal" I'm mean when both numbers in the division are written in function of $\pi$. (except cases like $\pi/\pi=1$) $\frac{1+i\pi}{1-i\pi} = -1+ \frac{2i}{\pi-i}$. Same problem. The same question can be asked about $e$ instead of $\pi$.

$1-{2\over{\pi-1}}$ is algebraic implies that ${1\over{\pi-1}}=a$ where $a$ is algebraic, this implies that $\pi-1=1/a$ and $\pi=1+1/a$ contradiction