Los 15 números Trascendentes más famosos ( De Gaussianos )

1. pi = 3.1415 ...
2. e = 2.718 ...
3. Euler's constant, gamma = 0.577215 ... = lim n -> infinity > (1 + 1/2 + 1/3 + 1/4 + ... + 1/n - ln(n)) (Not proven to be transcendental, but generally believed to be by mathematicians.)
4. Catalan's constant, G = sum (-1)^k / (2k + 1 )^2 = 1 - 1/9 + 1/25 - 1/49 + ... (Not proven to be transcendental, but generally believed to be by mathematicians.)
5. Liouville's number 0.110001000000000000000001000 ... which has a one in the 1st, 2nd, 6th, 24th, etc. places and zeros elsewhere.
6. Chaitin's "constant", the probability that a random algorithm halts. (Noam Elkies of Harvard notes that not only is this number transcendental but it is also incomputable.)
7. Chapernowne's number, 0.12345678910111213141516171819202122232425... This is constructed by concatenating the digits of the positive integers. (Can you see the pattern?)
8. Special values of the zeta function, such as zeta (3). (Transcendental functions can usually be expected to give transcendental results at rational points.)
9. ln(2).
10. Hilbert's number, 2^{(sqrt 2 )}. (This is called Hilbert's number because the proof of whether or not it is transcendental was one of Hilbert's famous problems. In fact, according to the Gelfond-Schneider theorem, any number of the form a^b is transcendental where a and b are algebraic (a ne 0, a ne 1 ) and b is not a rational number. Many trigonometric or hyperbolic functions of non-zero algebraic numbers are transcendental.)
11. e^pi
12. pi^e (Not proven to be transcendental, but generally believed to be by mathematicians.)
13. Morse-Thue's number, 0.01101001 ...
14. iⁱ = 0.207879576... (Here i is the imaginary number sqrt(-1). Isn't this a real beauty? How many people have actually considered rasing i to the i power? If a is algebraic and b is algebraic but irrational then a^b is transcendental. Since i is algebraic but irrational, the theorem applies. Note also: iⁱ is equal to e^{(- pi / 2 )} and several other values. Consider iⁱ = e^{(i log i )} = e^{( i times i pi / 2 )} . Since log is multivalued, there are other possible values for iⁱ. 
    Here is how you can compute the value of iⁱ = 0.207879576...

    1. Since e^(ix) = Cos x + i Sin x, then let x = Pi/2. 
    2. Then e^(iPi/2) = i = Cos Pi/2 + i Sin Pi/2; since Cos Pi/2 = Cos 90 deg. = 0. But Sin 90 = 1 and i Sin 90 deg. = (i)*(1) = i. 
    3. Therefore e^(iPi/2) = i. 
    4. Take the ith power of both sides, the right side being i^i and the left side = [e^(iPi/2)]^i = e^(-Pi/2). 
    5. Therefore i^i = e^(-Pi/2) = .207879576...

15. Feigenbaum numbers, e.g. 4.669 ... . (These are related to properties of dynamical systems with period-doubling. The ratio of successive differences between period-doubling bifurcation parameters approaches the number 4.669 ... , and it has been discovered in many physical systems before they enter the chaotic regime. It has not been proven to be transcendental, but is generally believed to be.)

Keith Briggs from the Mathematics Department of the University of Melbourne in Australia computed what he believes to be the world-record for the number of digits for the Feigenbaum number:

4. 669201609102990671853203820466201617258185577475768632745651 343004134330211314737138689744023948013817165984855189815134 408627142027932522312442988890890859944935463236713411532481 714219947455644365823793202009561058330575458617652222070385 410646749494284981453391726200568755665952339875603825637225

Briggs carried out the computation using special-purpose software designed by David Bailey of NASA Ames running on an IBM RISC System/6000. The computation required a few hours of computation time.