''The diameter (of a circle) does not stand to the circumference as an integer to an integer''
Johann Heinrich Lambert (17281777).
Irrational numbers are numbers which cannot be expressed as a fraction of two integers (see Classification of numbers). Many famous numbers are known to be irrational.
Let P(x) be a polynomial of degree n

with (a_{0},a_{1},...,a_{n}) being integers and a_{n} ¹ 0, we have the important theorem
Theorem 1 If an irreducible fraction p/q is a root of P, then p divides a_{0} and q divides a_{n}.
Proof : Suppose x = p/q is root of P, then q^{n}P(p/q) = 0, hence

and from this relation q divides a_{n}p^{n}, but q and p are relatively prime so q divides a_{n}. The same way we have p divides a_{0}q^{n},... ·
corollary 1 The roots of the unitary polynomial

are either integers or irrational numbers.
Proof : In the previous theorem q divides a_{n} = 1, so q = ±1.·
corollary 2 The roots of

where m is prime, are irrational numbers.
corollary 3 Ö2,Ö3,Ö5,... are irrational numbers.
It is possible to give another very elementary proof that Ö2 is irrational. Suppose Ö2 = p/q, with (p,q) being relatively prime, it follows that p^{2} = 2q^{2} and then p must be of the form 2p^{¢}, so q^{2} = (4p^{¢2})/2 = 2p^{¢2} and finally q = 2q^{¢}, this contradicts the hypothesis that (p,q) are relatively prime.
It's probably with a similar proof that around 500 B.C.E, the Pythagoreans established the irrationality of Ö2. According to the Pythagorean theorem, this is equivalent to say that the side and the diagonal of a unit square are incommensurable. Such a discovery was so awful for them, that they tried to keep the secret of the existence of such numbers, but the truth spread very rapidly.
We focus on polynomials with the following form

where c_{k} are integers. We have the obvious properties

hence P_{n} and it's derivatives take integral values at x = 0. The same conclusion is true at x = 1 from the symetry relation P_{n}(x) = P_{n}(1x).
The following results on the exponential and logarithm functions were first established by Lambert in 1761 by different means from the one we choose.
Theorem 2 If a > 0 is an integer then e^{a} is irrational.
Proof : Suppose e^{a} = p/q, with (p,q) positive integers, consider the function

from the properties of P_{n}, I(0) and I(1) are integers. But

giving

hence the last integral is a non zero integer. And now the key point, from the bounds of P_{n}(x), we have the bounds for the integral

when n becomes large. This contradicts the fact that the integral was an integer.·
corollary 4 If p/q is a non nul rational number then e^{p/q}is irrational.
Proof : If e^{p/q}is rational so is (e^{p/q})^{q} = e^{p} which is impossible from the previous theorem when p > 0 (for p < 0, apply the theorem on 1/e^{p}).·
corollary 5 e,e^{2},Öe are irrational numbers.
Euler gave in 1737 a very elementary proof of the irrationality of e based on the sequence

The proof goes like that : suppose e = p/q with q > 1 then the number

is a non nul integer (from the left hand side of the identity) but is also equal to

so it's strictly less than 1.
Theorem 3 If p/q > 0 and p/q ¹ 1 is a positive rational number then log(p/q) is irrational
Proof : Suppose log(p/q) = a/b then taking the exponential of both sides gives

corollary 6 log(2),log(3) are irrational numbers.
The irrationality of p was established for the first time by Johann Heinrich Lambert in 1761 [1]. The proof was rather complex and based on a continued fraction for the tanx function. In 1794, Legendre proved the stronger result that p^{2} is irrational [2]. We prefer the more elementary proof given by Niven in 1947 [4].
This proof uses again Niven's polynomials to establish the irrationality of p^{2}.
Theorem 4 p^{2} is an irrational number.
Proof : Again, suppose p^{2} = p/q, with (p,q) positive integers, consider the function

from the properties of P_{n}, J(0) and J(1) are integers. But

giving

hence the last integral is a non zero integer. From the bounds of P_{n}(x), we have the bounds for the integral

when n becomes large. This contradicts the fact that the integral was an integer. ·
corollary 7 p is an irrational number.
Other results are available, we give a small selection without proof.
Theorem 5 If a and b (b > 1) are positive integers log_{a}(b) is irrational whenever a or b has a prime factor which the other lacks [6].
Theorem 6 p^{a} is irrational for any non zero integer a
Theorem 7 e^{p} is irrational (Gelfond 1929).
During the ''Journées arithmétiques de MarseilleLuminy'' (France), in 1978, R. Apery announced a proof of the irrationality of z(3) [5]. The proof was based on the fast converging sequence

Proving the irrationality of a given number is often a difficult problem. Some famous number are not known to be irrational, like the Euler's constant g, p^{e}, 2^{e}, z(5), ¼.
The idea is to use Niven's polynomials to find rational approximations for our constants. We have to compute

with different functions for g(x).
Here, let's take g(x) = 1/(1+x^{2}) and n = 4, an easy computation shows

and because the polynomial P_{n}(x) takes it's maximum at x = 1/2, we have the majoration

so

now with g(x) = 1/(1+x) and n = 4

and after the same kind of majoration for the integral we find

Again, we apply the previous method for g(x) = e^{x} and n = 4,

giving

Better approximations can be obtained by taking larger values of n.
mathematical constants and computation