Why is it so hard to prove a number is transcendental? 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...

It is mainly because transcedental numbers behave so weird. For example, would you think that a transcedental number raised to an irrational is an integer? Well, it is possible:
$$(2^{\sqrt{2}})^{\sqrt{2}} = 2^{(\sqrt{2})^2}=4$$
