criteria if given irrational number is or not transcedental It is known that constants $\pi$ and $e$ are irrational numbers but also transcedental. Where consist difference between irrationality and transcedentality. How we know that given irrational number is not tanscedental.     
 A: A number $x$ is irrational if there are no integers $a_0, a_1$ such that $a_1x + a_0 = 0$. That is, if there is no integer polynomial $P$ of degree 1 with $P(x)=0$.
A number $x$ is transcendental if there is no positive integer $n$ and no integers $a_0, \ldots a_n$ such that $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0 = 0$. That is, if there is no integer polynomial $P$ of any degree $n$ with $P(x)=0$.
All transcendental  numbers are irrational, because we can take $n=1$. Not all irrational numbers are transcendental. Non-transcendental numbers are called algebraic.  $\sqrt2$ is irrational, but not transcendental, because $(\sqrt2)^2 - 2 = 0$. (That is, $n=2, a_2 = 1, a_1 = 0, a_0=-2$.)
Nobody knows methods that work in general to show that a particular number is rational, irrational, or transcendental. (Many methods are known that work in particular cases.) $\pi$ and $e$ are known to be transcendental, but nobody knows the answer even for simple combinations of $\pi$ and $e$ such as $\pi+e$ or $\pi e$. The important constant $\gamma$ has been studied for hundreds of years, but nobody has yet proved that it is not rational.
Historically the first  example of a specific number known to be transcendental was Liouville's number, which is:
$$
\sum_{i=1}^\infty {1\over 10^{i!}} = \frac1{10^{\vphantom1}} + \frac1{10^2} + \frac1{10^6} + \frac1{10^{24}} +\cdots = 0.1100010000000000000000010\ldots
$$
The proof that Liouville's number is transcendental is particularly simple. If you want to see a proof that a number is transcendental, that is a good place to start.
