Can someone help me with this Discrete Math Problem?

Question

For the two numbers given below, which are rational, and which are irrational? Express those which are rational in the form p q with p, q ∈ Z.

a. 0.a1a2a3..., where for n = 1, 2, 3, ..., the value of an is the number 0, 1, 2, or 3 which is the remainder on dividing n by 4.

b. 0.101001000100001000001 . . .

Answer 1

a.

The number would be 0.1230123012301230...

Note how 1230 repeats.

The fraction, since it includes 4 repeating numbers, is easily shown to be 1230/9999. This can be reduced to 410/3333

b. Irrational. A rational number's fractional representation either terminates or has a repeating pattern. This number will not, as there are more and more 0s between 1s.