Classify the following numbers as rational or irrational with justification:
$(i)$ $10.124124 \ldots$
$(ii)$ $1.010010001 \ldots$

(N/A) $(i)$ $10.124124 \ldots$ is a decimal expansion which is non-terminating and recurring. Since a number with a non-terminating recurring decimal expansion can be expressed in the form $p/q$ where $p, q$ are integers and $q \neq 0$,it is a rational number.
$(ii)$ $1.010010001 \ldots$ is a decimal expansion which is non-terminating and non-recurring. Since it cannot be expressed in the form $p/q$ where $p, q$ are integers and $q \neq 0$,it is an irrational number.