Fraction digit tables
Attached images depict binary and ternary digits of irreducible fractions in the (0, 1) interval. Fractions are ordered first by denominators and then by numerators. Take numbers formed by digits lying on the main diagonal. By Cantor's diagonal argument, a complement of the binary diagonal number is irrational. I'm pretty sure that the diagonal numbers are actually transcendental, but can't prove it.

Binary diagonal number:

Binary: 0.111000111001010110000110111110101011100111111011011010000100011 ...

Decimal: 0.8890003549695230139346719690879129310538480210179543414543465370 ...

Continued fraction: 0, 1, 8, 110, 1, 1, 1, 4, 1, 3, 33, 3, 4, 1, 26, 1, 37, 1, 23, 2, 117, 1, 1, 5, 1, 2, 20, 12, 1, 2, 1, 4, 1, ...

Ternary diagonal number:

Ternary: 0.1002211021120000111220000002000011111210202200100001202102121112 ...

Decimal: 0.3682120994629780785967399570406755660508922444001004151748993819 ...

Continued fraction: 0, 2, 1, 2, 1, 1, 12, 1, 2, 9, 2, 6, 5, 1, 2, 2, 7, 5, 1, 1, 2, 2, 2, 6, 7, 1, 41, 1, 4, 2, 2, 3, 1, ...