''The diameter (of a circle) does not stand to the circumference as an integer to an integer''

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.

1  Roots of a polynomial

Let P(x) be a polynomial of degree n

with (a₀,a₁,...,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 a₀ and q divides a_n.

Proof : Suppose x = p/q is root of P, then qⁿP(p/q) = 0, hence

and from this relation q divides a_npⁿ, but q and p are relatively prime so q divides a_n. The same way we have p divides a₀qⁿ,... ·

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 p² = 2q² and then p must be of the form 2p^¢, so q² = (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.

2  Niven's polynomials

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(1-x).

3  Irrationality of e

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/qis irrational.

Proof : If e^p/qis 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,e²,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.

4  Irrationality of log(2)

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.

5  Irrationality of p

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 p² 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 p².

Theorem 4 p² is an irrational number.

Proof : Again, suppose p² = 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.

6  Other irrational numbers

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 Marseille-Luminy'' (France), in 1978, R. Apery announced a proof of the irrationality of z(3) [5]. The proof was based on the fast converging sequence

7  Open problems

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.

8  Some rational approximations

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).

8.1  Approximation for p and log(2)

Here, let's take g(x) = 1/(1+x²) 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

8.2  Approximation for e

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.

References

[1]

J. H. Lambert, Mémoire sur quelques propriétés remarquables des quantités transcendentes circulaires et logarithmiques, Histoire de l'Académie Royale des Sciences et des Belles-Lettres der Berlin, (1761), p. 265-276

[2]

A. M. Legendre, Eléments de géométrie, Didot, Paris, (1794)

[3]

H. Padé, Sur l'irrationalité des nombres e et p, Darboux Bull., (1888), vol. 12, p. 144-148

[4]

I. Niven, A Simple Proof that p is Irrational, Bulletin of the American Mathematical Society, (1947), vol. 53, p. 509

[5]

A. van der Poorten, A Proof that Euler Missed ..., Apéry's Proof of the Irrationality of z(3), The Mathematical Intelligencer, (1979), vol. 1, p. 195-203

[6]

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford Science Publications, (1979)

mathematical constants and computation