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.
criteria if given irrational number is or not transcedental
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1(Since it took me a moment to see M.B.'s 'easy to see', a quick fleshing-out: consider the polynomial $(x-e)(x-\pi) = x^2-(e+\pi)x+e\pi$; if both the coefficients $e+\pi$ and $e\pi$ were rational (or even algebraic) then the roots of this polynomial (namely $e$ and $\pi$) would both be algebraic.) – 2012-05-24
1 Answers
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.