While reading on Wikipedia about transcendental numbers, i asked myself:
Why is it so hard and difficult to prove that $e +\pi, \pi – e, \pi e,
\frac{\pi}{e}$ etc. are transcendental numbers?
Answer by @hardmath:
It is generally more difficult to prove a number is transcendental than to prove it is not transcendental, i.e. that it is algebraic. Showing a number x is algebraic amounts to proving it is the root of a polynomial with rational coefficients, and so one often can just exhibit the polynomial and show by computation that a particular x is its root. Proving number x is transcendental amounts to proving no rational polynomial exists with that number as a root, and this requires more work (because we will be "proving the negative", i.e. exhausting all possible polynomials).
We know that $\pi$ and $e$ are transcendental numbers, why we can't deduce that
$e + \pi \approx 5.859874482048838473822930854632165381954416493075065395941912…$
or $\pi – e \approx 0.423310825130748003102355911926840386439922305675146246007976…$
are also transcendental numbers?
Answer by @Matt Samuel:
For example, $\pi$ and $1−\pi$ are transcendental, but $\pi+(1−\pi)=1$ is not.
Since both of you gave imho a great answer, i don't know whom of you should become the credits…
Best Answer
For example, $\pi$ and $1-\pi$ are transcendental, but $\pi+(1-\pi)=1$ is not.