"Adam P. Goucher" <apgoucher@gmx.com> wrote:
I had a simpler example in mind, namely one that follows from *only* the following facts about algebraic numbers:
-- The algebraic numbers form a field of characteristic zero; -- x and exp(x) cannot both be algebraic.
As did I. The solution I was thinking of is the unique real solution to x = exp(-x). Approximately 0.5671432. Decades ago, by the same reasoning, I realized that the unique real solution to x = cos(x) must be transcendental. Of course that means cos(cos(x)), cos(cos(cos(x))), cos(cos(cos(cos(x)))), etc., are all also transcendental, since they're all the same number. Approximately 0.7390851. (Puzzle: Is the solution to x = cos(x) still transcendental if you do it in degrees instead of radians?) Dan Asimov <asimov@msri.org> wrote:
Why did God make it so hard to prove transcendence?
That's nothing compared to normality. Again "nearly all" real numbers are normal, but it's not easy to prove that any specific number is normal. Pi, for instance, was proven irrational in the 18th century and transcendental in the 19th, but the 20th didn't accomplish anything with it except increase the number of known digits from a few hundred to a few hundred billion. And the 21st has done nothing except increase the number of known digits by another factor of 100 or so. Maybe the 22nd century will prove (or disprove) that it's normal. Or maybe pi's normality is literally unknowable. Are any specific numbers known to be normal? That's not clear to me. Wikipedia says, without references: .... This theorem established the existence of normal numbers. In 1917, Wacl/aw Sierpinski showed that it is possible to specify a particular such number. Becher and Figueira proved in 2002 that there is a computable absolutely normal number, however no digits of their number are known. What is the specification of Sierpinski's number? And what is the algorithm for computing Becher/Figueira's number, and why have no digits been computed? Thanks.