Does every irrational number have a 2 in its decimal expansion?

Answer 1

No; the only condition for qualyfing as an irrational number is that the same pattern of digits doesn't repeat over and over again, as it does with a rational number. For example, 8/7 is a rational number; the decimal expansion is 1.142857 142857 142857 ... As you see, the same pattern of digits repeats over and over. The number may start with different digits, but if after a while the same pattern repeats again and again, the number is rational.

The following number is irrational: 0.101001000100001000001 ... The pattern doesn't repeat, because a zero is added every time. And, in this example, the decimal expansion doesn't contain any digits other than 0 and 1.