Transcendental number

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…

Solutions Collecting From Web of "Transcendental number"

For example, $\pi$ and $1-\pi$ are transcendental, but $\pi+(1-\pi)=1$ is not.

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:

$$(2^{\sqrt{2}})^{\sqrt{2}} = 2^{(\sqrt{2})^2}=4$$

A difficulty of showing those numbers transcendental lies in the fact that they are not numbers that seem to occur in a natural way.

Sure, they are the sum, product and so on of two of the most common constants, but in “actual mathematics” they rarely appear in this form, and as illustrated in other answers summing and multiplying might not preserve transcendence.

But on the good side $e^{\pi}$ is known transcendental!

(Also $e+\pi$ or $e\pi$ is transcendental as otherwise $\pi$ and $e$ being roots of the polynomial $x^2 – (e + \pi )x + e \pi= (x-e)(x – \pi)$ yields a contradiction to them being transcendent.)