All Ten Digit Numbers are Plentiful

The number pi contains all ten digits in its decimal form as does e, sqrt(2) and most familiar irrationals.  It is more difficult to find relatively simple fractions that contain all ten digits.  It is somewhat surprising that nearly all real numbers contain all ten digits.  Attached is a “proof” of that result.  This discussion wanders near the boundary of “measure theory” and work done by the mathematician E. Borel. Click on TenDigOnly to see the discussion.

TenDigOnly