From: Gerry Myerson Subject: Re: Pi Date: Fri, 07 Apr 2000 14:33:33 +1000 Newsgroups: sci.math Summary: [missing] In article <38EC2777.6B21186D@math.ucla.edu>, Mike Oliver wrote: > Bill Taylor wrote: > > I've been trying to find a way to construct a number which IS normal > > to all bases, using a similar sort of method, but I've never been able > > to quite make it work. The trouble is, while it's easy to keep adding > > digits to UNstabilize the balance, it's hard to keep adding them to > > be sure to MAINTAIN a balance, for more than one base at a time. > > > > Can anyone help me with such a construction please? > > Would it be enough to give an explicit definition of a > single real, the definition being, say, at worst Delta^1_1, > together with a proof that the number is normal to all bases? [snip construction] I know this has come up here before so I tried to search DejaNews. I found this: ************************************* Forum: sci.math Subject: Re: A Normal Number, Please! Date: 09/10/1997 Author: Ray Mines If you look at TITLE: Constructive analysis AUTHOR: Errett Bishop, Douglas Bridges. CALL NO: QA299.8 B57 1985 you will find that they show if N is set of positive Lebesgue measure, then it is possible to construct a real number that is in N. If you let N be the set of all normal numbers (normal in all bases) and apply their construction you will get a Cauchy sequence that will converge to a number that is normal in all bases. One word of warning: This number will not be as easy to write down as the Champernowne number. -- Ray Mines ray@nmsu.edu Mathematical Sciences New Mexico State University Las Cruces, New Mexico 88003 ************************************** I also found a reference to a post by Benjamin Tilly on 5 May 1996 in the thread, Another Normal Number question, but I wasn't able to convince DejaNews that there ever was such a post. Gerry Myerson (gerry@mpce.mq.edu.au)