Question

Find two rational numbers of the form $\dfrac{p}{q}$ between the numbers $0.2121121112.....$ and $0.2020020002.....$

Answer 1

Hint: We first find two rational numbers in the decimal form. We have the conditions of those rational numbers being either terminating or recurring non-terminating. We find the in between numbers and convert to fraction form.

Complete step-by-step solution:
We first find two rational numbers in the decimal form between the numbers $0.2121121112.....$ and $0.2020020002.....$ and then convert them to the fraction form of $\dfrac{p}{q}$.
We know that the characteristics of a rational number is that its decimal form will have to be either terminating or recurring non-terminating.
So, following the condition we get $0.21$ and $0.2\overline{1}$ as two rational numbers in the decimal form between the numbers $0.2121121112.....$ and $0.2020020002.....$
We now convert them in fraction form.
For $0.21$, we get $0.21=\dfrac{21}{100}$ and for $0.2\overline{1}$, we get $0.2\overline{1}=\dfrac{21-2}{90}=\dfrac{19}{90}$.
Therefore, two rational numbers of the form $\dfrac{p}{q}$ between the numbers $0.2121121112.....$ and $0.2020020002.....$ are $\dfrac{21}{100},\dfrac{19}{90}$.

Note: Any non-recurring decimal number will always be an irrational number. The number of both rational and irrational numbers between two given numbers is always infinite. We cannot convert the given number to fraction form to find the solution directly.