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Definition 1 Rational numbers can be expressed as fraction of integers r=p/q, where (p,q) are integers (p is the numerator and q the denominator).
Definition 2 The set of rational numbers is denoted Q.
Numbers like 7/5,1/1999,31416/10000 are rational numbers. During ancient time the mathematicians believed that all numbers were rational, Pythagoreans (a mathematical and philosophical school in Greece during the VIth century B.C.) based their system on such a belief. It was a scandal when they proved that Ö2 could not be written as a fraction and it's why such numbers were called irrational.
Rational numbers can be defined by a finite amount of information (the numerator and denominator), their complexity is finite.
A basic property of rational numbers is that their decimal
expansion is finite (regular numbers) or periodic (non regular
numbers). Let us consider:

We have the property, if l denotes the length of the period and if p and q are relatively prime, that l(p/q)=l(1/q) for any non regular number and that l(1/q) is at
most q1. And the previous case becomes:

After Ö2 it was a new challenge for mathematicians to find other irrational numbers among the famous constants. Let us quote some major results:
Some irrationality proofs of classical constants can be found in [5] or [6].
Irrational numbers can be classified in two categories, the Algebraic numbers and the Transcendental numbers.
Definition 3
An algebraic number x is the root of a polynomial equation with integers
coefficients :
a_{n}x^{n}+a_{n1}x^{n1}+...+a_{1}x+a_{0}=0.
If n is the smallest possible degree of the polynomial, x is called an algebraic number of degree n. For example, the algebraic numbers of degree n=1 define the set of rational numbers. Square roots of integers which are not perfect squares are algebraic numbers of degree 2. All the solutions of the following equations are algebraic numbers of degree 2 or 3 for the last example:

In other words, Ö2,j = (1+Ö5)/2,Ö{51}4,... are algebraic numbers.
In a certain sense, algebraic numbers are represented by a finite collection of integers (the coefficients (a_{0},a_{1},...,a_{n}) of the polynomial), so their complexity may be considered as finite.
It's possible to show that the sum, the product, the quotient of algebraic numbers are also algebraic numbers (in mathematical terms, the set of algebraic numbers which is sometimes denoted A, is a field).
Definition 4 A numbers which is not root of any polynomial equation with integer coefficients is called a transcendental number.
Definition 5 Transcendental numbers are non algebraic numbers.
In 1844, Joseph Liouville (18091882) [12] was the first to prove the existence of such numbers and he gave an important and elementary characterization of a category of transcendental numbers, the Liouville numbers:
Theorem 6
The non rational number x is a Liouville number if, for every positive
integer n there exist two integers (p,q) such as
and such a number is transcendental.
ê
ê
x
p
ê
ê
<
1
q > 1,
He used this result to exhibit the first concrete transcendental numbers. One of those numbers is famous

Theorem 7
For any algebraic and non rational number x of degree n, there exist a
positive constant C, such as for any couple of integers (p,q) with q > 0
ê
ê
x
p
ê
ê
>
1
.
In 1873, Georg Cantor (18451918) proved that the transcendental numbers are non denumerable, whereas the algebraic numbers are denumerable. In other words, there are much more transcendental numbers than algebraic numbers. Because the proof is elementary we give it.
Theorem 8 The set of algebraic numbers is denumerable.
Proof. For any polynomial P(x)=a_{n}x^{n}+a_{n1}x^{n1}+...+a_{1}x+a_{0} with
integer coefficients of degree n, define it's height h(P) as the integer


Proving that a given number is transcendental is usually a very hard problem. The two most famous historical results are :
An important property of those numbers is the impossibility to construct any transcendental number with geometric construction (with only the help of a compass and a ruler). The celebrated problem of squaring the circle which goes back to ancient times and occupied generations of mathematicians was showed to have no solution with the proof of the transcendence of p.
In 1934, Aleksandr Gelfond (19061968) and Theodor Schneider (1911) independently solved one of the famous Hilbert's problems (the Seventh Problem: ''Irrationality and transcendence of certain numbers'') which were cited at the International Congress of Mathematicians at Paris in 1900.
Theorem 9
(GelfondSchneider). Let a an algebraic number (a ¹ 0,a ¹ 1) and b an algebraic and irrational number, then
a^{b} is transcendental.
Corollary 10 The numbers 2^{Ö2}(the GelfondSchneider constant) and e^{p} are transcendental numbers.
In 1966, Alan Baker (1939) succeeded in giving a vast
generalization of GelfondSchneider theorem : any finite sum (if
not zero)

The sum of a transcendental number and an algebraic number is also transcendental [2]. Some isolated results are available:
It is not known whether p+e, g (Euler's constant), B (Brun's constant), z(3), ... are transcendental. We don't even know if g,z(5) nor G (Catalan's constant) are irrational.
Only numerical estimation are available, Brent & McMillan showed in 1980 [3] that if g = p/q then q > 10^{15000}. This result has been improved to q > 10^{242080} by Papanikolaou in 1997.
In 1988, Bailey also established that no polynomial with small integer coefficients (of Euclidian norm smaller than 10^{9}) of degree 8 or less exists which vanishes on the constants p+e, g,e^{g},... [1].