----- Original Message ---- From: Guy Haworth <g.haworth@reading.ac.uk> To: math-fun@mailman.xmission.com Sent: Wednesday, October 15, 2008 6:38:55 AM Subject: [math-fun] 'Manifest' transcendental number? Apologies if this is trivial, but what is the easiest transcendental number to demonstrate _is_ a transcendental number? G ______________________________________________ A real number r is said to be approximable to order n if there exists a positive constant c, depending on r, such that are infinitely many rational numbers (h/k) satisfying | r - h/k | < c / k^n . A theorem of Liouville states that an algebraic number of degree n is not approximable to order n+1 or higher. A Liouville number L is a number that possesses rational approximations of arbitrarily high order. For every positive integer n, there exists a rational h[n] / k[n] such that | L - h[n] / k[n] | < 1 / k[n]^n . Liouville numbers are therefore transcendental. These were the first transcendental numbers to be explicitly constructed. Example: sum( 2^(-n!), n=1..infinity). The numbers e and pi can also be easily shown to be transcendental. Proofs of all this can be found in Niven's Carus Monograph, "Irrational Numbers". Niven also states without proof that according to Roth's theorem, an algebraic number is not approximable to order greater than 2. Gene