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 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 2,3,5,... are irrational numbers.

It is possible to give another very elementary proof that 2 is irrational. Suppose 2 = p/q, with (p,q) being relatively prime, it follows that p2 = 2q2 and then p must be of the form 2p, so q2 = (4p2)/2 = 2p2 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 2. 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

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
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
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,e 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 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
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
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) =

k = 1 
1
k3
= 5
2


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), .

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) =
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) =
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

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) =
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) =
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)


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