Numbers like e = 2.718..., pi = 3.1415..., and 2**(sqrt 3) are transcendental. These numbers are not the solutions of polynomial expressions having rational coefficients. The digits of pi and e never end, nor has anyone detected an orderly pattern in their arrangement.
After conducting a brief survey of readers, I made a list of the fifteen most famous transcendental numbers.
pi = 3.1415 ...
e = 2.718 ...
Euler's constant, gamma = 0.577215 ... = lim midsub < n rarrow infinity > (1 + 1/2 + 1/3 + 1/4 + ... + 1/n - ln(n)) (Not proven to be transcendental, but generally believed to be by mathematicians.)
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.)
Liouville's number 0.110001000000000000000001000 ... which has a one in the 1st, 2nd, 6th, 24th, etc. places and zeros elsewhere.
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.)
Chapernowne's number, 0.12345678910111213141516171819202122232425... This is constructed by concatenating the digits of the positive integers. (Can you see the pattern?)
Special values of the zeta function, such as zeta (3). (Transcendental functions can usually be expected to give transcendental results at rational points.)
ln(2).
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 100 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.)
e ** pi
pi ** e (Not proven to be transcendental, but generally believed to be by mathematicians.)
Morse-Thue's number, 0.01101001 ...
i ** i (Here i is the imaginary number sqrt(-1). 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: i ** i is equal to e ** (- pi / 2 ) and several other values. Consider i ** i = e ** ( i log i ) = e ** ( i times i pi / 2 ) . Since log is multivalued, there are other possible values for i ** i .
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. 6692016091029906718532038204662016172581855774757686327456513430041343302 1131473713868974402394801381716598485518981513440862714202793252231244298889 089085994493546323671341153248171 4219947455644365823793202009561058330575458 617652222070385410646749494284981453391726200568755665952339875603825637225
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.
