The 15 Most Famous Transcendental Numbers

In 1844, math genius Joseph Liouville (1809-1882) was the first to prove the existence of transcendental numbers. 

Hermite proved that the number e was transcendental in 1873.Lindeman proved that pi was transcendental in 1882. In 1882,  German mathematician F.

Lindemann proved that pi is transcendental. But are there other famous transcendental numbers. I made a list of the fifteen most famous transcendental numbers.

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: iⁱis 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.)