Re: Irrational numbers

There is a measure of regularity or normality of irrational numbers
called "approximate entropy" or ApEn developed by Steven Pincus and
various colleagues in the past few years. See the Proceedings of the
National Academy of Science, US in 1997 and 1998 for details, also
Science magazine in 1997. It turns out that the most irregular number is
pi, then the square root of 2, then 3. However, the square root of 2 has
one of the simplest CF expansions. Translating irrational numbers into
other number bases (binary, etc) can change the ApEn. It's a very
counter-intuitive area.

--John

John Chalmers wrote,

>There is a measure of regularity or normality of irrational numbers
>called "approximate entropy" or ApEn developed by Steven Pincus and
>various colleagues in the past few years. See the Proceedings of the
>National Academy of Science, US in 1997 and 1998 for details, also
>Science magazine in 1997. It turns out that the most irregular number is
>pi, then the square root of 2, then 3. However, the square root of 2 has
>one of the simplest CF expansions. Translating irrational numbers into
>other number bases (binary, etc) can change the ApEn. It's a very
>counter-intuitive area.

Bizarre! Do you really mean pi and not phi? And do you mean 3 and not
sqrt(3)?