Find a rational number and also an irrational number lying between the numbers
Question:

Find a rational number and also an irrational number lying between the numbers 0.3030030003 … and 0.3010010001 …

Solution:

Let 

$a=0.3030030003 \ldots$

$b=0.3010010001 \ldots$

Here decimal representation of a and b are non-terminating and non-repeating. So a and are irrational numbers. We observe that in first two decimal place of a and b have the same digit but digit in the third place of their decimal representation is distinct.

Therefore, a > b.

Hence one rational number is $0.3011$ lying between $0.3030030003 \ldots$ and $0.3010010001 \ldots$

And irrational number is $0.3020200200020000 \ldots$ lying between $0.3030030003 \ldots$ and $0.3010010001 \ldots$

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