SOLUTION: is the number 2.123112311123....a rational or irrational? explain your answer.

 
Every rational number can either be represented by a terminating
decimal or by a decimal that eventually repeats the same finite 
sequence of digits forever.  Conversely every decimal that eventually
repeats the same finite sequence of digits forever represents a 
rational number.  {If you want to know why that is true, post again
asking why; otherwise you'll just have to accept it as true.}

The number 2.123112311123... is irrational. That's because this
decimal value, assuming that pattern of increasing the number of 
1's each time continues forever, can never repeat the same finite 
sequence of digits forever.  That's because you could always find
a row of 1's longer than any finite sequence of digits that could
repeat forever.
  
Edwin