Liouville Number - Uncountability

Uncountability

Consider, for example, the number

3.1400010000000000000000050000....

3.14(3 zeros)1(17 zeros)5(95 zeros)9(599 zeros)2...

where the digits are zero except in positions n! where the digit equals the nth digit following the decimal point in the decimal expansion of π.

This number, as well as any other non-terminating decimal with its non-zero digits similarly situated, satisfies the definition of Liouville number. Since the set of all sequences of non-null digits has the cardinality of the continuum, the same thing occurs with the set of all Liouville numbers. Moreover, the Liouville numbers form a dense subset of the set of real numbers.