Irrationality proofs Irrationality proofs

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

Johann Heinrich Lambert (1728-1777).

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

P(x) = a0+a1x+...+anxn

with (a0,a1,...,an) being integers and an 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 an.

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

a0qn+a1qn-1p+...+an-1qpn-1 = -anpn

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

corollary 1 The roots of the unitary polynomial

a0+a1x+...+an-1xn-1+xn

are either integers or irrational numbers.

Proof : In the previous theorem q divides an = 1, so q = ±1.·

corollary 2 The roots of

x2-m = 0

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

Pn(x) = xn(1-x)n
n!
= 1
n!
2n
å
k = n 
ckxk

where ck are integers. We have the obvious properties

0 < Pn(x)
<
1
n!
    (0 < x < 1),
Pn(0)
=
0,
Pn(m)(0)
=
0     (m < n  and m > 2n),
Pn(m)(0)
=
m!
n!
cm     (n £ m £ 2n),

hence Pn and it's derivatives take integral values at x = 0. The same conclusion is true at x = 1 from the symetry relation Pn(x) = Pn(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 ea is irrational.

Proof : Suppose ea = p/q, with (p,q) positive integers, consider the function

I(x) = a2nPn(x)-a2n-1Pn¢(x)+...-aPn(2n-1)(x)+Pn(2n)(x),

from the properties of Pn, I(0) and I(1) are integers. But

q(eaxI(x))¢ = qeax(aI(x)+I¢(x)) = qeaxa2n+1Pn(x)

giving

q[ eaxI(x)]01 = q(eaI(1)-I(0)) = pI(1)-qI(0) = qa2n+1 ó
õ
1

0 
eaxPn(x)dx

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

0 < qa2n+1 ó
õ
1

0 
eaxPn(x)dx < qa2n(ea-1)
n!
< 1

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 ep/qis irrational.

Proof : If ep/qis rational so is (ep/q)q = ep which is impossible from the previous theorem when p > 0 (for p < 0, apply the theorem on 1/ep).·

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

e = 1+ 1
1!
+ 1
2!
+...+ 1
n!
+...

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

q!(e-1- 1
1!
- 1
2!
-...- 1
q!
) = q!( 1
(q+1)!
+ 1
(q+2)!
+...+ 1
(q+n)!
+...)

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

1
q+1
+ 1
(q+1)(q+2)
+...+ 1
(q+1)...(q+n)
+... < 1
2
+ 1
22
+...+ 1
2n
+... = 1

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

p/q = ea/b,
which is impossible from a previous corollary. ·

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

J(x) = qn(p2nPn(x)-p2n-2Pn(2)(x)+p2n-4Pn(4)(x)-...(-1)nPn(2n)(x)),

from the properties of Pn, J(0) and J(1) are integers. But

(J¢(x)sinpx-J(x)pcospx)¢
=
(J(2)(x)+p2J(x))sinpx = qnp2n+2Pn(x)sinpx
=
p2pnPn(x)sinpx

giving

1
p
[ J¢(x)sinpx-J(x)pcospx]01 = J(0)+J(1) = ppn ó
õ
1

0 
Pn(x)sinpxdx

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

0 < ppn ó
õ
1

0 
Pn(x)sinpxdx < ppn
n!
< 1

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 loga(b) is irrational whenever a or b has a prime factor which the other lacks [6].

Theorem 6 pa is irrational for any non zero integer a

Theorem 7 ep 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

z(3) = ¥
å
k = 1 
1
k3
= 5
2
¥
å
k = 1 
(-1)k-1 (k!)2
k3(2k)!

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, pe, 2e, 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

In(g) = ó
õ
1

0 
Pn(x)g(x)dx

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

I4(g) = ó
õ
1

0 
x4(1-x)4
1+x2
dx = 22
7
-p

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

I4(g) < 1
28
ó
õ
1

0 
dx
1+x2
< 1
28

so

0 < 22
7
-p < 1
256

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

I4(g) = ó
õ
1

0 
x4(1-x)4
1+x
dx = 16log(2)- 621
56

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

0 < log(2)- 621
896
< 1
4096

8.2  Approximation for e

Again, we apply the previous method for g(x) = ex and n = 4,

I4(g) = ó
õ
1

0 
x4(1-x)4exdx = 24024e-65304

giving

0 < e- 2721
1001
< e-1
256.24024
< 1
3075072

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)


Back to

mathematical constants and computation


File translated from TEX by TTH, version 2.32.
On 5 Jul 2000, 22:17.