source file: mills2.txt Date: Tue, 19 Dec 1995 06:20:47 -0800 Subject: repeating digits ( transcendental numbers ) From: Franz Kriftner Brian Mc Laren writes: > It has been speculated that the digits in the > decimal expansion of transcendental numbers > never repeat. One subscriber has even stated as > much on this forum. However, this proposition > has never been proven mathematically. > I'm quite sure this has been proven but as I do not know any proof i will prove it myself: any number with only a finite number of non-zeros is rational proof: if there are only zeros after the decimal point it's integer and we're done else just write down all the digits up to ( and including ) the last non-zero leave out the decimal point and divide by a 1 with as many zeros as there were digits after the decimal point the result is the same number written as a fraction of two integers we're done any number the digits of which repeat is rational proof: any number the digits of which repeat every n digits is an integer multiple of 0.0...010...010... ... ( a 1 at every nth position ) this 0.0...010...010...010... ... can also be written as 1/9...9 ( 1 divided by a number with n 9s ) an integer multiple of a rational number is rational so we're done any number the digits of which repeat starting from some position is just the sum of a number with only finitely many digits and another one the digits of which always repeat so we know for sure that that if the digits in the decimal expansion of a number repeat this number must be rational as a transcendental number can never be rational its digits never repeat --------------------------------------------------- by repeating I mean: starting from some position a fixed sequence of digits repeats forever forgive me if I took you ( Brian Mc Laren ) wrong if you only meant that some sequences of digits repeat sometimes it is easy to prove that this will always happen as there are only finitely many different sequences of a given length there must be a repeating sequence of this length for example: there are 100 sequences of two digits so after 202 digits ( 101 sequences ) there must have been a repetition --------------------------------------------------- BTW: I really appreciate Brian's postings ______ ______ _(0) /___/_/ _(0) /___/_/ / \_/ |\ / \_/ |\ | \ F R A N Z K R I F T N E R | \ / | Franz.Kriftner@source.co.at / | _/ _| _/ _| Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 19 Dec 1995 21:43 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id MAA02562; Tue, 19 Dec 1995 12:42:58 -0800 Date: Tue, 19 Dec 1995 12:42:58 -0800 Message-Id: <0099B1E344A24A92.4D16@ezh.nl> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu