Is Pi multiplied by Euler's Number Rational?

Open Question

It is not known whether the product of $\pi$ (pi) and Euler's number $e$:

$\pi \times e$

is rational or irrational.

Progress

By:

Transcendence of Sum or Product of Transcendentals

Euler's Number is Transcendental

Pi is Transcendental

at least one of $\pi + e$ and $\pi e$ is transcendental.

Also, see Schanuel's Conjecture Implies Transcendence of Pi by Euler's Number.

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Miscellaneous Problems: $47$
• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.29$: Liouville ($\text {1809}$ – $\text {1882}$)
• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.18$: Algebraic and Transcendental Numbers. $e$ is Transcendental