Question

find an irrational number between 2.3 and 2.4

Answer 1

2.3^2 = 5.29 
2.4^2 = 5.76 

So if we take the square root of a number between those, that we know has an irrational square root, we'll have such a number. 

5.5 = 11/2 is between 5.29 and 5.76, and we can easily see that 
√(11/2) = (√22)/2 is irrational, so I'd go with that one.

Answer 2

2.30 , 2.31 , 2.32 , 2. 33 , 2.34 , 2.35 , 2.36 , 2.37 , 2.38 , 2.39. etc..........